ring_theory.adjoin.basic | Mathlib porting status

Changes since the initial port

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

Changes in mathlib3

a35ddf20

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

69c6a5a1

bump to nightly-2024-03-17-08

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

22f01ce4

Diff

@@ -202,7 +202,7 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
   apply le_antisymm
   · intro r hr; rcases Subsemiring.mem_closure_iff_exists_list.1 hr with ⟨L, HL, rfl⟩; clear hr
     induction' L with hd tl ih; · exact zero_mem _
-    rw [List.forall_mem_cons] at HL 
+    rw [List.forall_mem_cons] at HL
     rw [List.map_cons, List.sum_cons]
     refine' Submodule.add_mem _ _ (ih HL.2)
     replace HL := HL.1; clear ih tl
@@ -211,7 +211,7 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
       rcases this with ⟨z, r, hr, hzr⟩; rw [← hzr]
       exact smul_mem _ _ (subset_span hr)
     induction' hd with hd tl ih; · exact ⟨1, 1, (Submonoid.closure s).one_mem', one_smul _ _⟩
-    rw [List.forall_mem_cons] at HL 
+    rw [List.forall_mem_cons] at HL
     rcases ih HL.2 with ⟨z, r, hr, hzr⟩; rw [List.prod_cons, ← hzr]
     rcases HL.1 with (⟨hd, rfl⟩ | hs)
     · refine' ⟨hd * z, r, hr, _⟩
@@ -404,8 +404,8 @@ theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B]
     [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
     (hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B' :=
   by
-  change x ∈ (adjoin R s).toSubmodule at hx 
-  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx 
+  change x ∈ (adjoin R s).toSubmodule at hx
+  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
   rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • ↑i) = x⟩
   choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.Prop
   use l.support.sup n₁

69c6a5a1

bump to nightly-2023-09-24-06

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

5655435f

Diff

@@ -3,10 +3,10 @@ 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.Algebra.Algebra.Operations
-import Mathbin.Algebra.Algebra.Subalgebra.Tower
-import Mathbin.LinearAlgebra.Prod
-import Mathbin.LinearAlgebra.Finsupp
+import Algebra.Algebra.Operations
+import Algebra.Algebra.Subalgebra.Tower
+import LinearAlgebra.Prod
+import LinearAlgebra.Finsupp
 
 #align_import ring_theory.adjoin.basic from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"

69c6a5a1

bump to nightly-2023-08-17-09

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

60285dcc

Diff

@@ -104,7 +104,7 @@ theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : 
     (Hmul : ∀ x y, p x → p y → p (x * y)) : p x :=
   let S : Subalgebra R A :=
     { carrier := p
-      mul_mem' := Hmul
+      hMul_mem' := Hmul
       add_mem' := Hadd
       algebraMap_mem' := Halg }
   adjoin_le (show s ≤ S from Hs) h
@@ -215,7 +215,7 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
     rcases ih HL.2 with ⟨z, r, hr, hzr⟩; rw [List.prod_cons, ← hzr]
     rcases HL.1 with (⟨hd, rfl⟩ | hs)
     · refine' ⟨hd * z, r, hr, _⟩
-      rw [Algebra.smul_def, Algebra.smul_def, (algebraMap _ _).map_mul, _root_.mul_assoc]
+      rw [Algebra.smul_def, Algebra.smul_def, (algebraMap _ _).map_hMul, _root_.mul_assoc]
     ·
       exact
         ⟨z, hd * r, Submonoid.mul_mem _ (Submonoid.subset_closure hs) hr,

69c6a5a1

bump to nightly-2023-07-20-09

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

9c56d0dd

Diff

@@ -2,17 +2,14 @@
 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.adjoin.basic
-! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.Algebra.Operations
 import Mathbin.Algebra.Algebra.Subalgebra.Tower
 import Mathbin.LinearAlgebra.Prod
 import Mathbin.LinearAlgebra.Finsupp
 
+#align_import ring_theory.adjoin.basic from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
+
 /-!
 # Adjoining elements to form subalgebras

69c6a5a1

bump to nightly-2023-06-28-03

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

79a0bd3b

Diff

@@ -268,7 +268,7 @@ theorem adjoin_image (f : A →ₐ[R] B) (s : Set A) : adjoin R (f '' s) = (adjo
 theorem adjoin_insert_adjoin (x : A) : adjoin R (insert x ↑(adjoin R s)) = adjoin R (insert x s) :=
   le_antisymm
     (adjoin_le
-      (Set.insert_subset.mpr
+      (Set.insert_subset_iff.mpr
         ⟨subset_adjoin (Set.mem_insert _ _), adjoin_mono (Set.subset_insert _ _)⟩))
     (Algebra.adjoin_mono (Set.insert_subset_insert Algebra.subset_adjoin))
 #align algebra.adjoin_insert_adjoin Algebra.adjoin_insert_adjoin
@@ -307,7 +307,7 @@ theorem adjoin_inl_union_inr_eq_prod (s) (t) :
   by
   apply le_antisymm
   ·
-    simp only [adjoin_le_iff, Set.insert_subset, Subalgebra.zero_mem, Subalgebra.one_mem,
+    simp only [adjoin_le_iff, Set.insert_subset_iff, Subalgebra.zero_mem, Subalgebra.one_mem,
       subset_adjoin,-- the rest comes from `squeeze_simp`
       Set.union_subset_iff,
       LinearMap.coe_inl, Set.mk_preimage_prod_right, Set.image_subset_iff, SetLike.mem_coe,

69c6a5a1

bump to nightly-2023-06-13-21

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

829520ac

Diff

@@ -45,29 +45,41 @@ variable [CommSemiring R] [Semiring A] [Semiring B]
 
 variable [Algebra R A] [Algebra R B] {s t : Set A}
 
+#print Algebra.subset_adjoin /-
 theorem subset_adjoin : s ⊆ adjoin R s :=
   Algebra.gc.le_u_l s
 #align algebra.subset_adjoin Algebra.subset_adjoin
+-/
 
+#print Algebra.adjoin_le /-
 theorem adjoin_le {S : Subalgebra R A} (H : s ⊆ S) : adjoin R s ≤ S :=
   Algebra.gc.l_le H
 #align algebra.adjoin_le Algebra.adjoin_le
+-/
 
+#print Algebra.adjoin_eq_sInf /-
 theorem adjoin_eq_sInf : adjoin R s = sInf {p | s ⊆ p} :=
   le_antisymm (le_sInf fun _ h => adjoin_le h) (sInf_le subset_adjoin)
 #align algebra.adjoin_eq_Inf Algebra.adjoin_eq_sInf
+-/
 
+#print Algebra.adjoin_le_iff /-
 theorem adjoin_le_iff {S : Subalgebra R A} : adjoin R s ≤ S ↔ s ⊆ S :=
   Algebra.gc _ _
 #align algebra.adjoin_le_iff Algebra.adjoin_le_iff
+-/
 
+#print Algebra.adjoin_mono /-
 theorem adjoin_mono (H : s ⊆ t) : adjoin R s ≤ adjoin R t :=
   Algebra.gc.monotone_l H
 #align algebra.adjoin_mono Algebra.adjoin_mono
+-/
 
+#print Algebra.adjoin_eq_of_le /-
 theorem adjoin_eq_of_le (S : Subalgebra R A) (h₁ : s ⊆ S) (h₂ : S ≤ adjoin R s) : adjoin R s = S :=
   le_antisymm (adjoin_le h₁) h₂
 #align algebra.adjoin_eq_of_le Algebra.adjoin_eq_of_le
+-/
 
 #print Algebra.adjoin_eq /-
 theorem adjoin_eq (S : Subalgebra R A) : adjoin R ↑S = S :=
@@ -75,15 +87,20 @@ theorem adjoin_eq (S : Subalgebra R A) : adjoin R ↑S = S :=
 #align algebra.adjoin_eq Algebra.adjoin_eq
 -/
 
+#print Algebra.adjoin_iUnion /-
 theorem adjoin_iUnion {α : Type _} (s : α → Set A) :
     adjoin R (Set.iUnion s) = ⨆ i : α, adjoin R (s i) :=
   (@Algebra.gc R A _ _ _).l_iSup
 #align algebra.adjoin_Union Algebra.adjoin_iUnion
+-/
 
+#print Algebra.adjoin_attach_biUnion /-
 theorem adjoin_attach_biUnion [DecidableEq A] {α : Type _} {s : Finset α} (f : s → Finset A) :
     adjoin R (s.attach.biUnion f : Set A) = ⨆ x, adjoin R (f x) := by simpa [adjoin_Union]
 #align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_biUnion
+-/
 
+#print Algebra.adjoin_induction /-
 @[elab_as_elim]
 theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : ∀ x ∈ s, p x)
     (Halg : ∀ r, p (algebraMap R A r)) (Hadd : ∀ x y, p x → p y → p (x + y))
@@ -95,7 +112,9 @@ theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : 
       algebraMap_mem' := Halg }
   adjoin_le (show s ≤ S from Hs) h
 #align algebra.adjoin_induction Algebra.adjoin_induction
+-/
 
+#print Algebra.adjoin_induction₂ /-
 /-- Induction principle for the algebra generated by a set `s`: show that `p x y` holds for any
 `x y ∈ adjoin R s` given that that it holds for `x y ∈ s` and that it satisfies a number of
 natural properties. -/
@@ -120,7 +139,9 @@ theorem adjoin_induction₂ {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin
         (fun x y => Hadd_left x y ((algebraMap R A) r)) fun x y =>
         Hmul_left x y ((algebraMap R A) r)
 #align algebra.adjoin_induction₂ Algebra.adjoin_induction₂
+-/
 
+#print Algebra.adjoin_induction' /-
 /-- The difference with `algebra.adjoin_induction` is that this acts on the subtype. -/
 theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin h⟩)
     (Halg : ∀ r, p (algebraMap R _ r)) (Hadd : ∀ x y, p x → p y → p (x + y))
@@ -138,7 +159,9 @@ theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s),
         Exists.elim hx fun hx' hx =>
           Exists.elim hy fun hy' hy => ⟨Subalgebra.mul_mem _ hx' hy', Hmul _ _ hx hy⟩
 #align algebra.adjoin_induction' Algebra.adjoin_induction'
+-/
 
+#print Algebra.adjoin_adjoin_coe_preimage /-
 @[simp]
 theorem adjoin_adjoin_coe_preimage {s : Set A} : adjoin R ((coe : adjoin R s → A) ⁻¹' s) = ⊤ :=
   by
@@ -150,6 +173,7 @@ theorem adjoin_adjoin_coe_preimage {s : Set A} : adjoin R ((coe : adjoin R s →
   · exact Subalgebra.add_mem _
   · exact Subalgebra.mul_mem _
 #align algebra.adjoin_adjoin_coe_preimage Algebra.adjoin_adjoin_coe_preimage
+-/
 
 #print Algebra.adjoin_union /-
 theorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=
@@ -159,18 +183,23 @@ theorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin
 
 variable (R A)
 
+#print Algebra.adjoin_empty /-
 @[simp]
 theorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=
   show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact Algebra.gc
 #align algebra.adjoin_empty Algebra.adjoin_empty
+-/
 
+#print Algebra.adjoin_univ /-
 @[simp]
 theorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=
   eq_top_iff.2 fun x => subset_adjoin <| Set.mem_univ _
 #align algebra.adjoin_univ Algebra.adjoin_univ
+-/
 
 variable (R) {A} (s)
 
+#print Algebra.adjoin_eq_span /-
 theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s) :=
   by
   apply le_antisymm
@@ -198,20 +227,27 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
   change Submonoid.closure s ≤ (adjoin R s).toSubsemiring.toSubmonoid
   exact Submonoid.closure_le.2 subset_adjoin
 #align algebra.adjoin_eq_span Algebra.adjoin_eq_span
+-/
 
+#print Algebra.span_le_adjoin /-
 theorem span_le_adjoin (s : Set A) : span R s ≤ (adjoin R s).toSubmodule :=
   span_le.mpr subset_adjoin
 #align algebra.span_le_adjoin Algebra.span_le_adjoin
+-/
 
+#print Algebra.adjoin_toSubmodule_le /-
 theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
     (adjoin R s).toSubmodule ≤ t ↔ ↑(Submonoid.closure s) ⊆ (t : Set A) := by
   rw [adjoin_eq_span, span_le]
 #align algebra.adjoin_to_submodule_le Algebra.adjoin_toSubmodule_le
+-/
 
+#print Algebra.adjoin_eq_span_of_subset /-
 theorem adjoin_eq_span_of_subset {s : Set A} (hs : ↑(Submonoid.closure s) ⊆ (span R s : Set A)) :
     (adjoin R s).toSubmodule = span R s :=
   le_antisymm ((adjoin_toSubmodule_le R).mpr hs) (span_le_adjoin R s)
 #align algebra.adjoin_eq_span_of_subset Algebra.adjoin_eq_span_of_subset
+-/
 
 #print Algebra.adjoin_span /-
 @[simp]
@@ -239,11 +275,14 @@ theorem adjoin_insert_adjoin (x : A) : adjoin R (insert x ↑(adjoin R s)) = adj
 -/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print Algebra.adjoin_prod_le /-
 theorem adjoin_prod_le (s : Set A) (t : Set B) :
     adjoin R (s ×ˢ t) ≤ (adjoin R s).Prod (adjoin R t) :=
   adjoin_le <| Set.prod_mono subset_adjoin subset_adjoin
 #align algebra.adjoin_prod_le Algebra.adjoin_prod_le
+-/
 
+#print Algebra.mem_adjoin_of_map_mul /-
 theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a₂, f (a₁ * a₂) = f a₁ * f a₂)
     (h : x ∈ adjoin R s) : f x ∈ adjoin R (f '' (s ∪ {1})) :=
   by
@@ -259,7 +298,9 @@ theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a
   rw [algebra_map_eq_smul_one]
   exact f.map_smul _ _
 #align algebra.mem_adjoin_of_map_mul Algebra.mem_adjoin_of_map_mul
+-/
 
+#print Algebra.adjoin_inl_union_inr_eq_prod /-
 theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     adjoin R (LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1})) =
       (adjoin R s).Prod (adjoin R t) :=
@@ -282,7 +323,9 @@ theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     replace Hb : ((0 : A), b) ∈ P := adjoin_mono (Set.subset_union_right _ _) Hb
     simpa using Subalgebra.add_mem _ Ha Hb
 #align algebra.adjoin_inl_union_inr_eq_prod Algebra.adjoin_inl_union_inr_eq_prod
+-/
 
+#print Algebra.adjoinCommSemiringOfComm /-
 /-- If all elements of `s : set A` commute pairwise, then `adjoin R s` is a commutative
 semiring.  -/
 def adjoinCommSemiringOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) :
@@ -299,14 +342,19 @@ def adjoinCommSemiringOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a *
           (fun x₁ x₂ y₁ h₁ h₂ => by rw [mul_assoc, h₂, ← mul_assoc y₁, ← h₁, mul_assoc x₁])
           fun x₁ x₂ y₁ h₁ h₂ => by rw [mul_assoc x₂, ← h₂, ← mul_assoc x₂, ← h₁, ← mul_assoc] }
 #align algebra.adjoin_comm_semiring_of_comm Algebra.adjoinCommSemiringOfComm
+-/
 
+#print Algebra.adjoin_singleton_one /-
 theorem adjoin_singleton_one : adjoin R ({1} : Set A) = ⊥ :=
   eq_bot_iff.2 <| adjoin_le <| Set.singleton_subset_iff.2 <| SetLike.mem_coe.2 <| one_mem _
 #align algebra.adjoin_singleton_one Algebra.adjoin_singleton_one
+-/
 
+#print Algebra.self_mem_adjoin_singleton /-
 theorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=
   Algebra.subset_adjoin (Set.mem_singleton_iff.mpr rfl)
 #align algebra.self_mem_adjoin_singleton Algebra.self_mem_adjoin_singleton
+-/
 
 end Semiring
 
@@ -331,12 +379,14 @@ theorem adjoin_union_eq_adjoin_adjoin :
 #align algebra.adjoin_union_eq_adjoin_adjoin Algebra.adjoin_union_eq_adjoin_adjoin
 -/
 
+#print Algebra.adjoin_union_coe_submodule /-
 theorem adjoin_union_coe_submodule :
     (adjoin R (s ∪ t)).toSubmodule = (adjoin R s).toSubmodule * (adjoin R t).toSubmodule :=
   by
   rw [adjoin_eq_span, adjoin_eq_span, adjoin_eq_span, span_mul_span]
   congr 1 with z; simp [Submonoid.closure_union, Submonoid.mem_sup, Set.mem_mul]
 #align algebra.adjoin_union_coe_submodule Algebra.adjoin_union_coe_submodule
+-/
 
 #print Algebra.adjoin_adjoin_of_tower /-
 theorem adjoin_adjoin_of_tower [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
@@ -352,6 +402,7 @@ theorem adjoin_adjoin_of_tower [Semiring B] [Algebra R B] [Algebra A B] [IsScala
 
 variable {R}
 
+#print Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin /-
 theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]
     [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
     (hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B' :=
@@ -376,11 +427,14 @@ theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B]
   rw [← Submonoid.closure_eq B'.to_submonoid]
   apply Submonoid.closure_mono hs (n₂ a)
 #align algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin
+-/
 
+#print Algebra.pow_smul_mem_adjoin_smul /-
 theorem pow_smul_mem_adjoin_smul (r : R) (s : Set A) {x : A} (hx : x ∈ adjoin R s) :
     ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ adjoin R (r • s) :=
   pow_smul_mem_of_smul_subset_of_mem_adjoin r s _ subset_adjoin hx (Subalgebra.algebraMap_mem _ _)
 #align algebra.pow_smul_mem_adjoin_smul Algebra.pow_smul_mem_adjoin_smul
+-/
 
 end CommSemiring
 
@@ -392,11 +446,14 @@ variable [Algebra R A] {s t : Set A}
 
 variable {R s t}
 
+#print Algebra.adjoin_int /-
 theorem adjoin_int (s : Set R) : adjoin ℤ s = subalgebraOfSubring (Subring.closure s) :=
   le_antisymm (adjoin_le Subring.subset_closure)
     (Subring.closure_le.2 subset_adjoin : Subring.closure s ≤ (adjoin ℤ s).toSubring)
 #align algebra.adjoin_int Algebra.adjoin_int
+-/
 
+#print Algebra.mem_adjoin_iff /-
 theorem mem_adjoin_iff {s : Set A} {x : A} :
     x ∈ adjoin R s ↔ x ∈ Subring.closure (Set.range (algebraMap R A) ∪ s) :=
   ⟨fun hx =>
@@ -405,6 +462,7 @@ theorem mem_adjoin_iff {s : Set A} {x : A} :
     suffices Subring.closure (Set.range ⇑(algebraMap R A) ∪ s) ≤ (adjoin R s).toSubring from @this x
     Subring.closure_le.2 Subsemiring.subset_closure⟩
 #align algebra.mem_adjoin_iff Algebra.mem_adjoin_iff
+-/
 
 #print Algebra.adjoin_eq_ring_closure /-
 theorem adjoin_eq_ring_closure (s : Set A) :
@@ -415,12 +473,14 @@ theorem adjoin_eq_ring_closure (s : Set A) :
 
 variable (R)
 
+#print Algebra.adjoinCommRingOfComm /-
 /-- If all elements of `s : set A` commute pairwise, then `adjoin R s` is a commutative
 ring.  -/
 def adjoinCommRingOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) :
     CommRing (adjoin R s) :=
   { (adjoin R s).toRing, adjoinCommSemiringOfComm R hcomm with }
 #align algebra.adjoin_comm_ring_of_comm Algebra.adjoinCommRingOfComm
+-/
 
 end Ring
 
@@ -438,15 +498,19 @@ theorem map_adjoin (φ : A →ₐ[R] B) (s : Set A) : (adjoin R s).map φ = adjo
 #align alg_hom.map_adjoin AlgHom.map_adjoin
 -/
 
+#print AlgHom.adjoin_le_equalizer /-
 theorem adjoin_le_equalizer (φ₁ φ₂ : A →ₐ[R] B) {s : Set A} (h : s.EqOn φ₁ φ₂) :
     adjoin R s ≤ φ₁.equalizer φ₂ :=
   adjoin_le h
 #align alg_hom.adjoin_le_equalizer AlgHom.adjoin_le_equalizer
+-/
 
+#print AlgHom.ext_of_adjoin_eq_top /-
 theorem ext_of_adjoin_eq_top {s : Set A} (h : adjoin R s = ⊤) ⦃φ₁ φ₂ : A →ₐ[R] B⦄
     (hs : s.EqOn φ₁ φ₂) : φ₁ = φ₂ :=
   ext fun x => adjoin_le_equalizer φ₁ φ₂ hs <| h.symm ▸ trivial
 #align alg_hom.ext_of_adjoin_eq_top AlgHom.ext_of_adjoin_eq_top
+-/
 
 end AlgHom

69c6a5a1

bump to nightly-2023-06-04-18

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

3fb4268b

Diff

@@ -53,7 +53,7 @@ theorem adjoin_le {S : Subalgebra R A} (H : s ⊆ S) : adjoin R s ≤ S :=
   Algebra.gc.l_le H
 #align algebra.adjoin_le Algebra.adjoin_le
 
-theorem adjoin_eq_sInf : adjoin R s = sInf { p | s ⊆ p } :=
+theorem adjoin_eq_sInf : adjoin R s = sInf {p | s ⊆ p} :=
   le_antisymm (le_sInf fun _ h => adjoin_le h) (sInf_le subset_adjoin)
 #align algebra.adjoin_eq_Inf Algebra.adjoin_eq_sInf

69c6a5a1

bump to nightly-2023-06-01-16

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

b9c87c70

Diff

@@ -176,16 +176,16 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
   apply le_antisymm
   · intro r hr; rcases Subsemiring.mem_closure_iff_exists_list.1 hr with ⟨L, HL, rfl⟩; clear hr
     induction' L with hd tl ih; · exact zero_mem _
-    rw [List.forall_mem_cons] at HL
+    rw [List.forall_mem_cons] at HL 
     rw [List.map_cons, List.sum_cons]
     refine' Submodule.add_mem _ _ (ih HL.2)
     replace HL := HL.1; clear ih tl
-    suffices ∃ (z r : _)(hr : r ∈ Submonoid.closure s), SMul.smul z r = List.prod hd
+    suffices ∃ (z r : _) (hr : r ∈ Submonoid.closure s), SMul.smul z r = List.prod hd
       by
       rcases this with ⟨z, r, hr, hzr⟩; rw [← hzr]
       exact smul_mem _ _ (subset_span hr)
     induction' hd with hd tl ih; · exact ⟨1, 1, (Submonoid.closure s).one_mem', one_smul _ _⟩
-    rw [List.forall_mem_cons] at HL
+    rw [List.forall_mem_cons] at HL 
     rcases ih HL.2 with ⟨z, r, hr, hzr⟩; rw [List.prod_cons, ← hzr]
     rcases HL.1 with (⟨hd, rfl⟩ | hs)
     · refine' ⟨hd * z, r, hr, _⟩
@@ -356,8 +356,8 @@ theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B]
     [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
     (hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B' :=
   by
-  change x ∈ (adjoin R s).toSubmodule at hx
-  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
+  change x ∈ (adjoin R s).toSubmodule at hx 
+  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx 
   rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • ↑i) = x⟩
   choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.Prop
   use l.support.sup n₁

69c6a5a1

bump to nightly-2023-06-01-04

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

4d9eaa85

Diff

@@ -438,12 +438,10 @@ theorem map_adjoin (φ : A →ₐ[R] B) (s : Set A) : (adjoin R s).map φ = adjo
 #align alg_hom.map_adjoin AlgHom.map_adjoin
 -/
 
-#print AlgHom.adjoin_le_equalizer /-
 theorem adjoin_le_equalizer (φ₁ φ₂ : A →ₐ[R] B) {s : Set A} (h : s.EqOn φ₁ φ₂) :
     adjoin R s ≤ φ₁.equalizer φ₂ :=
   adjoin_le h
 #align alg_hom.adjoin_le_equalizer AlgHom.adjoin_le_equalizer
--/
 
 theorem ext_of_adjoin_eq_top {s : Set A} (h : adjoin R s = ⊤) ⦃φ₁ φ₂ : A →ₐ[R] B⦄
     (hs : s.EqOn φ₁ φ₂) : φ₁ = φ₂ :=

69c6a5a1

bump to nightly-2023-05-28-18

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

8d353152

Diff

@@ -31,7 +31,7 @@ adjoin, algebra
 
 universe u v w
 
-open Pointwise
+open scoped Pointwise
 
 open Submodule Subsemiring

69c6a5a1

bump to nightly-2023-05-28-06

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

ba5d8b97

Diff

@@ -45,62 +45,26 @@ variable [CommSemiring R] [Semiring A] [Semiring B]
 
 variable [Algebra R A] [Algebra R B] {s t : Set A}
 
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 theorem subset_adjoin : s ⊆ adjoin R s :=
   Algebra.gc.le_u_l s
 #align algebra.subset_adjoin Algebra.subset_adjoin
 
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 theorem adjoin_le {S : Subalgebra R A} (H : s ⊆ S) : adjoin R s ≤ S :=
   Algebra.gc.l_le H
 #align algebra.adjoin_le Algebra.adjoin_le
 
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 theorem adjoin_eq_sInf : adjoin R s = sInf { p | s ⊆ p } :=
   le_antisymm (le_sInf fun _ h => adjoin_le h) (sInf_le subset_adjoin)
 #align algebra.adjoin_eq_Inf Algebra.adjoin_eq_sInf
 
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 theorem adjoin_le_iff {S : Subalgebra R A} : adjoin R s ≤ S ↔ s ⊆ S :=
   Algebra.gc _ _
 #align algebra.adjoin_le_iff Algebra.adjoin_le_iff
 
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 theorem adjoin_mono (H : s ⊆ t) : adjoin R s ≤ adjoin R t :=
   Algebra.gc.monotone_l H
 #align algebra.adjoin_mono Algebra.adjoin_mono
 
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 theorem adjoin_eq_of_le (S : Subalgebra R A) (h₁ : s ⊆ S) (h₂ : S ≤ adjoin R s) : adjoin R s = S :=
   le_antisymm (adjoin_le h₁) h₂
 #align algebra.adjoin_eq_of_le Algebra.adjoin_eq_of_le
@@ -111,33 +75,15 @@ theorem adjoin_eq (S : Subalgebra R A) : adjoin R ↑S = S :=
 #align algebra.adjoin_eq Algebra.adjoin_eq
 -/
 
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 theorem adjoin_iUnion {α : Type _} (s : α → Set A) :
     adjoin R (Set.iUnion s) = ⨆ i : α, adjoin R (s i) :=
   (@Algebra.gc R A _ _ _).l_iSup
 #align algebra.adjoin_Union Algebra.adjoin_iUnion
 
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 theorem adjoin_attach_biUnion [DecidableEq A] {α : Type _} {s : Finset α} (f : s → Finset A) :
     adjoin R (s.attach.biUnion f : Set A) = ⨆ x, adjoin R (f x) := by simpa [adjoin_Union]
 #align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_biUnion
 
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 @[elab_as_elim]
 theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : ∀ x ∈ s, p x)
     (Halg : ∀ r, p (algebraMap R A r)) (Hadd : ∀ x y, p x → p y → p (x + y))
@@ -150,9 +96,6 @@ theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : 
   adjoin_le (show s ≤ S from Hs) h
 #align algebra.adjoin_induction Algebra.adjoin_induction
 
-/- warning: algebra.adjoin_induction₂ -> Algebra.adjoin_induction₂ is a dubious translation:
-<too large>
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 /-- Induction principle for the algebra generated by a set `s`: show that `p x y` holds for any
 `x y ∈ adjoin R s` given that that it holds for `x y ∈ s` and that it satisfies a number of
 natural properties. -/
@@ -178,9 +121,6 @@ theorem adjoin_induction₂ {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin
         Hmul_left x y ((algebraMap R A) r)
 #align algebra.adjoin_induction₂ Algebra.adjoin_induction₂
 
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 /-- The difference with `algebra.adjoin_induction` is that this acts on the subtype. -/
 theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin h⟩)
     (Halg : ∀ r, p (algebraMap R _ r)) (Hadd : ∀ x y, p x → p y → p (x + y))
@@ -199,12 +139,6 @@ theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s),
           Exists.elim hy fun hy' hy => ⟨Subalgebra.mul_mem _ hx' hy', Hmul _ _ hx hy⟩
 #align algebra.adjoin_induction' Algebra.adjoin_induction'
 
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 @[simp]
 theorem adjoin_adjoin_coe_preimage {s : Set A} : adjoin R ((coe : adjoin R s → A) ⁻¹' s) = ⊤ :=
   by
@@ -225,23 +159,11 @@ theorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin
 
 variable (R A)
 
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 @[simp]
 theorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=
   show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact Algebra.gc
 #align algebra.adjoin_empty Algebra.adjoin_empty
 
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 @[simp]
 theorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=
   eq_top_iff.2 fun x => subset_adjoin <| Set.mem_univ _
@@ -249,9 +171,6 @@ theorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=
 
 variable (R) {A} (s)
 
-/- warning: algebra.adjoin_eq_span -> Algebra.adjoin_eq_span is a dubious translation:
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 theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s) :=
   by
   apply le_antisymm
@@ -280,24 +199,15 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
   exact Submonoid.closure_le.2 subset_adjoin
 #align algebra.adjoin_eq_span Algebra.adjoin_eq_span
 
-/- warning: algebra.span_le_adjoin -> Algebra.span_le_adjoin is a dubious translation:
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 theorem span_le_adjoin (s : Set A) : span R s ≤ (adjoin R s).toSubmodule :=
   span_le.mpr subset_adjoin
 #align algebra.span_le_adjoin Algebra.span_le_adjoin
 
-/- warning: algebra.adjoin_to_submodule_le -> Algebra.adjoin_toSubmodule_le is a dubious translation:
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 theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
     (adjoin R s).toSubmodule ≤ t ↔ ↑(Submonoid.closure s) ⊆ (t : Set A) := by
   rw [adjoin_eq_span, span_le]
 #align algebra.adjoin_to_submodule_le Algebra.adjoin_toSubmodule_le
 
-/- warning: algebra.adjoin_eq_span_of_subset -> Algebra.adjoin_eq_span_of_subset is a dubious translation:
-<too large>
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 theorem adjoin_eq_span_of_subset {s : Set A} (hs : ↑(Submonoid.closure s) ⊆ (span R s : Set A)) :
     (adjoin R s).toSubmodule = span R s :=
   le_antisymm ((adjoin_toSubmodule_le R).mpr hs) (span_le_adjoin R s)
@@ -328,21 +238,12 @@ theorem adjoin_insert_adjoin (x : A) : adjoin R (insert x ↑(adjoin R s)) = adj
 #align algebra.adjoin_insert_adjoin Algebra.adjoin_insert_adjoin
 -/
 
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-Case conversion may be inaccurate. Consider using '#align algebra.adjoin_prod_le Algebra.adjoin_prod_leₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem adjoin_prod_le (s : Set A) (t : Set B) :
     adjoin R (s ×ˢ t) ≤ (adjoin R s).Prod (adjoin R t) :=
   adjoin_le <| Set.prod_mono subset_adjoin subset_adjoin
 #align algebra.adjoin_prod_le Algebra.adjoin_prod_le
 
-/- warning: algebra.mem_adjoin_of_map_mul -> Algebra.mem_adjoin_of_map_mul is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align algebra.mem_adjoin_of_map_mul Algebra.mem_adjoin_of_map_mulₓ'. -/
 theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a₂, f (a₁ * a₂) = f a₁ * f a₂)
     (h : x ∈ adjoin R s) : f x ∈ adjoin R (f '' (s ∪ {1})) :=
   by
@@ -359,9 +260,6 @@ theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a
   exact f.map_smul _ _
 #align algebra.mem_adjoin_of_map_mul Algebra.mem_adjoin_of_map_mul
 
-/- warning: algebra.adjoin_inl_union_inr_eq_prod -> Algebra.adjoin_inl_union_inr_eq_prod is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align algebra.adjoin_inl_union_inr_eq_prod Algebra.adjoin_inl_union_inr_eq_prodₓ'. -/
 theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     adjoin R (LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1})) =
       (adjoin R s).Prod (adjoin R t) :=
@@ -385,12 +283,6 @@ theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     simpa using Subalgebra.add_mem _ Ha Hb
 #align algebra.adjoin_inl_union_inr_eq_prod Algebra.adjoin_inl_union_inr_eq_prod
 
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 /-- If all elements of `s : set A` commute pairwise, then `adjoin R s` is a commutative
 semiring.  -/
 def adjoinCommSemiringOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) :
@@ -408,22 +300,10 @@ def adjoinCommSemiringOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a *
           fun x₁ x₂ y₁ h₁ h₂ => by rw [mul_assoc x₂, ← h₂, ← mul_assoc x₂, ← h₁, ← mul_assoc] }
 #align algebra.adjoin_comm_semiring_of_comm Algebra.adjoinCommSemiringOfComm
 
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 theorem adjoin_singleton_one : adjoin R ({1} : Set A) = ⊥ :=
   eq_bot_iff.2 <| adjoin_le <| Set.singleton_subset_iff.2 <| SetLike.mem_coe.2 <| one_mem _
 #align algebra.adjoin_singleton_one Algebra.adjoin_singleton_one
 
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 theorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=
   Algebra.subset_adjoin (Set.mem_singleton_iff.mpr rfl)
 #align algebra.self_mem_adjoin_singleton Algebra.self_mem_adjoin_singleton
@@ -451,9 +331,6 @@ theorem adjoin_union_eq_adjoin_adjoin :
 #align algebra.adjoin_union_eq_adjoin_adjoin Algebra.adjoin_union_eq_adjoin_adjoin
 -/
 
-/- warning: algebra.adjoin_union_coe_submodule -> Algebra.adjoin_union_coe_submodule is a dubious translation:
-<too large>
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 theorem adjoin_union_coe_submodule :
     (adjoin R (s ∪ t)).toSubmodule = (adjoin R s).toSubmodule * (adjoin R t).toSubmodule :=
   by
@@ -475,9 +352,6 @@ theorem adjoin_adjoin_of_tower [Semiring B] [Algebra R B] [Algebra A B] [IsScala
 
 variable {R}
 
-/- warning: algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin -> Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoinₓ'. -/
 theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]
     [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
     (hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B' :=
@@ -503,9 +377,6 @@ theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B]
   apply Submonoid.closure_mono hs (n₂ a)
 #align algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin
 
-/- warning: algebra.pow_smul_mem_adjoin_smul -> Algebra.pow_smul_mem_adjoin_smul is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align algebra.pow_smul_mem_adjoin_smul Algebra.pow_smul_mem_adjoin_smulₓ'. -/
 theorem pow_smul_mem_adjoin_smul (r : R) (s : Set A) {x : A} (hx : x ∈ adjoin R s) :
     ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ adjoin R (r • s) :=
   pow_smul_mem_of_smul_subset_of_mem_adjoin r s _ subset_adjoin hx (Subalgebra.algebraMap_mem _ _)
@@ -521,23 +392,11 @@ variable [Algebra R A] {s t : Set A}
 
 variable {R s t}
 
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 theorem adjoin_int (s : Set R) : adjoin ℤ s = subalgebraOfSubring (Subring.closure s) :=
   le_antisymm (adjoin_le Subring.subset_closure)
     (Subring.closure_le.2 subset_adjoin : Subring.closure s ≤ (adjoin ℤ s).toSubring)
 #align algebra.adjoin_int Algebra.adjoin_int
 
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-Case conversion may be inaccurate. Consider using '#align algebra.mem_adjoin_iff Algebra.mem_adjoin_iffₓ'. -/
 theorem mem_adjoin_iff {s : Set A} {x : A} :
     x ∈ adjoin R s ↔ x ∈ Subring.closure (Set.range (algebraMap R A) ∪ s) :=
   ⟨fun hx =>
@@ -556,12 +415,6 @@ theorem adjoin_eq_ring_closure (s : Set A) :
 
 variable (R)
 
-/- warning: algebra.adjoin_comm_ring_of_comm -> Algebra.adjoinCommRingOfComm is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align algebra.adjoin_comm_ring_of_comm Algebra.adjoinCommRingOfCommₓ'. -/
 /-- If all elements of `s : set A` commute pairwise, then `adjoin R s` is a commutative
 ring.  -/
 def adjoinCommRingOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) :
@@ -592,9 +445,6 @@ theorem adjoin_le_equalizer (φ₁ φ₂ : A →ₐ[R] B) {s : Set A} (h : s.EqO
 #align alg_hom.adjoin_le_equalizer AlgHom.adjoin_le_equalizer
 -/
 
-/- warning: alg_hom.ext_of_adjoin_eq_top -> AlgHom.ext_of_adjoin_eq_top is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align alg_hom.ext_of_adjoin_eq_top AlgHom.ext_of_adjoin_eq_topₓ'. -/
 theorem ext_of_adjoin_eq_top {s : Set A} (h : adjoin R s = ⊤) ⦃φ₁ φ₂ : A →ₐ[R] B⦄
     (hs : s.EqOn φ₁ φ₂) : φ₁ = φ₂ :=
   ext fun x => adjoin_le_equalizer φ₁ φ₂ hs <| h.symm ▸ trivial

69c6a5a1

bump to nightly-2023-05-28-04

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

6797a637

Diff

@@ -233,9 +233,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_empty Algebra.adjoin_emptyₓ'. -/
 @[simp]
 theorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=
-  show adjoin R ⊥ = ⊥ by
-    apply GaloisConnection.l_bot
-    exact Algebra.gc
+  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact Algebra.gc
 #align algebra.adjoin_empty Algebra.adjoin_empty
 
 /- warning: algebra.adjoin_univ -> Algebra.adjoin_univ is a dubious translation:
@@ -257,26 +255,19 @@ Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span
 theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s) :=
   by
   apply le_antisymm
-  · intro r hr
-    rcases Subsemiring.mem_closure_iff_exists_list.1 hr with ⟨L, HL, rfl⟩
-    clear hr
-    induction' L with hd tl ih
-    · exact zero_mem _
+  · intro r hr; rcases Subsemiring.mem_closure_iff_exists_list.1 hr with ⟨L, HL, rfl⟩; clear hr
+    induction' L with hd tl ih; · exact zero_mem _
     rw [List.forall_mem_cons] at HL
     rw [List.map_cons, List.sum_cons]
     refine' Submodule.add_mem _ _ (ih HL.2)
-    replace HL := HL.1
-    clear ih tl
+    replace HL := HL.1; clear ih tl
     suffices ∃ (z r : _)(hr : r ∈ Submonoid.closure s), SMul.smul z r = List.prod hd
       by
-      rcases this with ⟨z, r, hr, hzr⟩
-      rw [← hzr]
+      rcases this with ⟨z, r, hr, hzr⟩; rw [← hzr]
       exact smul_mem _ _ (subset_span hr)
-    induction' hd with hd tl ih
-    · exact ⟨1, 1, (Submonoid.closure s).one_mem', one_smul _ _⟩
+    induction' hd with hd tl ih; · exact ⟨1, 1, (Submonoid.closure s).one_mem', one_smul _ _⟩
     rw [List.forall_mem_cons] at HL
-    rcases ih HL.2 with ⟨z, r, hr, hzr⟩
-    rw [List.prod_cons, ← hzr]
+    rcases ih HL.2 with ⟨z, r, hr, hzr⟩; rw [List.prod_cons, ← hzr]
     rcases HL.1 with (⟨hd, rfl⟩ | hs)
     · refine' ⟨hd * z, r, hr, _⟩
       rw [Algebra.smul_def, Algebra.smul_def, (algebraMap _ _).map_mul, _root_.mul_assoc]

69c6a5a1

bump to nightly-2023-05-28-03

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

6c6011cf

Diff

@@ -151,10 +151,7 @@ theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : 
 #align algebra.adjoin_induction Algebra.adjoin_induction
 
 /- warning: algebra.adjoin_induction₂ -> Algebra.adjoin_induction₂ is a dubious translation:
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 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_induction₂ Algebra.adjoin_induction₂ₓ'. -/
 /-- Induction principle for the algebra generated by a set `s`: show that `p x y` holds for any
 `x y ∈ adjoin R s` given that that it holds for `x y ∈ s` and that it satisfies a number of
@@ -182,10 +179,7 @@ theorem adjoin_induction₂ {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin
 #align algebra.adjoin_induction₂ Algebra.adjoin_induction₂
 
 /- warning: algebra.adjoin_induction' -> Algebra.adjoin_induction' is a dubious translation:
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(Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))), p x)
+<too large>
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_induction' Algebra.adjoin_induction'ₓ'. -/
 /-- The difference with `algebra.adjoin_induction` is that this acts on the subtype. -/
 theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin h⟩)
@@ -258,10 +252,7 @@ theorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=
 variable (R) {A} (s)
 
 /- warning: algebra.adjoin_eq_span -> Algebra.adjoin_eq_span is a dubious translation:
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(Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _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.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A 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+<too large>
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span Algebra.adjoin_eq_spanₓ'. -/
 theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s) :=
   by
@@ -299,20 +290,14 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
 #align algebra.adjoin_eq_span Algebra.adjoin_eq_span
 
 /- warning: algebra.span_le_adjoin -> Algebra.span_le_adjoin is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align algebra.span_le_adjoin Algebra.span_le_adjoinₓ'. -/
 theorem span_le_adjoin (s : Set A) : span R s ≤ (adjoin R s).toSubmodule :=
   span_le.mpr subset_adjoin
 #align algebra.span_le_adjoin Algebra.span_le_adjoin
 
 /- warning: algebra.adjoin_to_submodule_le -> Algebra.adjoin_toSubmodule_le is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_to_submodule_le Algebra.adjoin_toSubmodule_leₓ'. -/
 theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
     (adjoin R s).toSubmodule ≤ t ↔ ↑(Submonoid.closure s) ⊆ (t : Set A) := by
@@ -320,10 +305,7 @@ theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
 #align algebra.adjoin_to_submodule_le Algebra.adjoin_toSubmodule_le
 
 /- warning: algebra.adjoin_eq_span_of_subset -> Algebra.adjoin_eq_span_of_subset is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span_of_subset Algebra.adjoin_eq_span_of_subsetₓ'. -/
 theorem adjoin_eq_span_of_subset {s : Set A} (hs : ↑(Submonoid.closure s) ⊆ (span R s : Set A)) :
     (adjoin R s).toSubmodule = span R s :=
@@ -368,10 +350,7 @@ theorem adjoin_prod_le (s : Set A) (t : Set B) :
 #align algebra.adjoin_prod_le Algebra.adjoin_prod_le
 
 /- warning: algebra.mem_adjoin_of_map_mul -> Algebra.mem_adjoin_of_map_mul is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align algebra.mem_adjoin_of_map_mul Algebra.mem_adjoin_of_map_mulₓ'. -/
 theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a₂, f (a₁ * a₂) = f a₁ * f a₂)
     (h : x ∈ adjoin R s) : f x ∈ adjoin R (f '' (s ∪ {1})) :=
@@ -390,10 +369,7 @@ theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a
 #align algebra.mem_adjoin_of_map_mul Algebra.mem_adjoin_of_map_mul
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_inl_union_inr_eq_prod Algebra.adjoin_inl_union_inr_eq_prodₓ'. -/
 theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     adjoin R (LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1})) =
@@ -485,10 +461,7 @@ theorem adjoin_union_eq_adjoin_adjoin :
 -/
 
 /- warning: algebra.adjoin_union_coe_submodule -> Algebra.adjoin_union_coe_submodule is a dubious translation:
-lean 3 declaration is
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(CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A 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(CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) 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 _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_union_coe_submodule Algebra.adjoin_union_coe_submoduleₓ'. -/
 theorem adjoin_union_coe_submodule :
     (adjoin R (s ∪ t)).toSubmodule = (adjoin R s).toSubmodule * (adjoin R t).toSubmodule :=
@@ -512,10 +485,7 @@ theorem adjoin_adjoin_of_tower [Semiring B] [Algebra R B] [Algebra A B] [IsScala
 variable {R}
 
 /- warning: algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin -> Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] [_inst_4 : CommSemiring.{u3} B] [_inst_5 : Algebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_6 : Algebra.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_7 : 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 (CommSemiring.toSemiring.{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 _inst_1))))) 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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))) (Algebra.toModule.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)))))] (r : A) (s : Set.{u3} B) (B' : Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5), (HasSubset.Subset.{u3} (Set.{u3} B) (Set.hasSubset.{u3} B) (SMul.smul.{u2, u3} A (Set.{u3} B) (Set.smulSet.{u2, u3} A B (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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A 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_inst_2))))) r n) x) B'))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoinₓ'. -/
 theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]
     [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
@@ -543,10 +513,7 @@ theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B]
 #align algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin
 
 /- warning: algebra.pow_smul_mem_adjoin_smul -> Algebra.pow_smul_mem_adjoin_smul is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align algebra.pow_smul_mem_adjoin_smul Algebra.pow_smul_mem_adjoin_smulₓ'. -/
 theorem pow_smul_mem_adjoin_smul (r : R) (s : Set A) {x : A} (hx : x ∈ adjoin R s) :
     ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ adjoin R (r • s) :=
@@ -635,10 +602,7 @@ theorem adjoin_le_equalizer (φ₁ φ₂ : A →ₐ[R] B) {s : Set A} (h : s.EqO
 -/
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align alg_hom.ext_of_adjoin_eq_top AlgHom.ext_of_adjoin_eq_topₓ'. -/
 theorem ext_of_adjoin_eq_top {s : Set A} (h : adjoin R s = ⊤) ⦃φ₁ φ₂ : A →ₐ[R] B⦄
     (hs : s.EqOn φ₁ φ₂) : φ₁ = φ₂ :=

69c6a5a1

bump to nightly-2023-05-24-06

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

fbc7b476

Diff

@@ -371,7 +371,7 @@ theorem adjoin_prod_le (s : Set A) (t : Set B) :
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] {s : Set.{u2} A} {x : A} {f : LinearMap.{u1, u1, u2, u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)}, (forall (a₁ : A) (a₂ : A), Eq.{succ u3} B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ 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_inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) => A -> B) (LinearMap.hasCoeToFun.{u1, u1, u2, u3} R R A B (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f) (Union.union.{u2} (Set.{u2} A) (Set.hasUnion.{u2} A) s (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) (OfNat.ofNat.{u2} A 1 (OfNat.mk.{u2} A 1 (One.one.{u2} A (AddMonoidWithOne.toOne.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))))))))
 but is expected to have type
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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f a₂))) -> (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (Membership.mem.{u3, u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : A) => B) x) (Subalgebra.{u1, u3} R B _inst_1 _inst_3 _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 _inst_3 _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (LinearMap.{u1, u1, u2, u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : A) => B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u3} R R A B (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f x) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 (Set.image.{u2, u3} A B (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (LinearMap.{u1, u1, u2, u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : A) => B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u3} R R A B (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f) (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) (OfNat.ofNat.{u2} A 1 (One.toOfNat1.{u2} A (Semiring.toOne.{u2} A _inst_2))))))))
+  forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] {s : Set.{u2} A} {x : A} {f : LinearMap.{u1, u1, u2, u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)}, (forall (a₁ : A) (a₂ : A), Eq.{succ u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : A) => B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u3} R R A B (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f) (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) (OfNat.ofNat.{u2} A 1 (One.toOfNat1.{u2} A (Semiring.toOne.{u2} A _inst_2))))))))
 Case conversion may be inaccurate. Consider using '#align algebra.mem_adjoin_of_map_mul Algebra.mem_adjoin_of_map_mulₓ'. -/
 theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a₂, f (a₁ * a₂) = f a₁ * f a₂)
     (h : x ∈ adjoin R s) : f x ∈ adjoin R (f '' (s ∪ {1})) :=
@@ -393,7 +393,7 @@ theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] (s : Set.{u2} A) (t : Set.{u3} B), Eq.{succ (max u2 u3)} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Algebra.adjoin.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5) (Union.union.{max u2 u3} (Set.{max u2 u3} (Prod.{u2, u3} A B)) (Set.hasUnion.{max u2 u3} (Prod.{u2, u3} A B)) (Set.image.{u2, max u2 u3} A (Prod.{u2, u3} A B) (coeFn.{max (succ u2) (succ (max u2 u3)), max (succ u2) (succ (max u2 u3))} (LinearMap.{u1, u1, u2, max u2 u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) 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 but is expected to have type
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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) B (fun (_x : B) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : B) => Prod.{u2, u3} A B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u3, max u2 u3} R R B (Prod.{u2, u3} A B) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Prod.instAddCommMonoidSum.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (LinearMap.inr.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) (Union.union.{u3} (Set.{u3} B) (Set.instUnionSet.{u3} B) t (Singleton.singleton.{u3, u3} B (Set.{u3} B) (Set.instSingletonSet.{u3} B) (OfNat.ofNat.{u3} B 1 (One.toOfNat1.{u3} B (Semiring.toOne.{u3} B _inst_3)))))))) (Subalgebra.prod.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 t))
+  forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] (s : Set.{u2} A) (t : Set.{u3} B), Eq.{max (succ u2) (succ u3)} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Algebra.adjoin.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5) (Union.union.{max u2 u3} (Set.{max u2 u3} (Prod.{u2, u3} A B)) (Set.instUnionSet.{max u2 u3} (Prod.{u2, u3} A B)) (Set.image.{u2, max u2 u3} A (Prod.{u2, u3} A B) (FunLike.coe.{max (succ u2) (succ u3), succ u2, max (succ u2) (succ u3)} (LinearMap.{u1, u1, u2, max u3 u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A (Prod.{u2, u3} A B) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Prod.instAddCommMonoidSum.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : A) => Prod.{u2, u3} A B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, max u2 u3} R R A (Prod.{u2, u3} A B) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Prod.instAddCommMonoidSum.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (LinearMap.inl.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) (OfNat.ofNat.{u2} A 1 (One.toOfNat1.{u2} A (Semiring.toOne.{u2} A _inst_2)))))) (Set.image.{u3, max u2 u3} B (Prod.{u2, u3} A B) (FunLike.coe.{max (succ u2) (succ u3), succ u3, max (succ u2) (succ u3)} (LinearMap.{u1, u1, u3, max u3 u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) B (Prod.{u2, u3} A B) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Prod.instAddCommMonoidSum.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) B (fun (_x : B) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : B) => Prod.{u2, u3} A B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u3, max u2 u3} R R B (Prod.{u2, u3} A B) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Prod.instAddCommMonoidSum.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (LinearMap.inr.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) (Union.union.{u3} (Set.{u3} B) (Set.instUnionSet.{u3} B) t (Singleton.singleton.{u3, u3} B (Set.{u3} B) (Set.instSingletonSet.{u3} B) (OfNat.ofNat.{u3} B 1 (One.toOfNat1.{u3} B (Semiring.toOne.{u3} B _inst_3)))))))) (Subalgebra.prod.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 t))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_inl_union_inr_eq_prod Algebra.adjoin_inl_union_inr_eq_prodₓ'. -/
 theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     adjoin R (LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1})) =
@@ -638,7 +638,7 @@ theorem adjoin_le_equalizer (φ₁ φ₂ : A →ₐ[R] B) {s : Set A} (h : s.EqO
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] {s : Set.{u2} A}, (Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Top.top.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toHasTop.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) -> (forall {{φ₁ : AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5}} {{φ₂ : AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5}}, (Set.EqOn.{u2, u3} A B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) (fun (_x : AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) => A -> B) ([anonymous].{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) φ₁) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) (fun (_x : AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) => A -> B) ([anonymous].{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) φ₂) s) -> (Eq.{max (succ u2) (succ u3)} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) φ₁ φ₂))
 but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] {s : Set.{u2} A}, (Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Top.top.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toTop.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) -> (forall {{φ₁ : AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5}} {{φ₂ : AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5}}, (Set.EqOn.{u2, u3} A B (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) _x) (SMulHomClass.toFunLike.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (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 _inst_2))))) (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 _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (SMulZeroClass.toSMul.{u1, u3} R B (AddMonoid.toZero.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))))) (DistribSMul.toSMulZeroClass.{u1, u3} R B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))))) (DistribMulAction.toDistribSMul.{u1, u3} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u2, u3, max u2 u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) (AlgHom.algHomClass.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5))))) φ₁) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) _x) (SMulHomClass.toFunLike.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (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 _inst_2))))) (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 _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (SMulZeroClass.toSMul.{u1, u3} R B (AddMonoid.toZero.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))))) (DistribSMul.toSMulZeroClass.{u1, u3} R B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))))) (DistribMulAction.toDistribSMul.{u1, u3} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u2, u3, max u2 u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) (AlgHom.algHomClass.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5))))) φ₂) s) -> (Eq.{max (succ u2) (succ u3)} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) φ₁ φ₂))
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] {s : Set.{u2} A}, (Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Top.top.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toTop.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) -> (forall {{φ₁ : AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5}} {{φ₂ : AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5}}, (Set.EqOn.{u2, u3} A B (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => B) _x) (SMulHomClass.toFunLike.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (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 _inst_2))))) (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 _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (SMulZeroClass.toSMul.{u1, u3} R B (AddMonoid.toZero.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))))) (DistribSMul.toSMulZeroClass.{u1, u3} R B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))))) (DistribMulAction.toDistribSMul.{u1, u3} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u2, u3, max u2 u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) (AlgHom.algHomClass.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5))))) φ₁) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => B) _x) (SMulHomClass.toFunLike.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (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 _inst_2))))) (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 _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (SMulZeroClass.toSMul.{u1, u3} R B (AddMonoid.toZero.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))))) (DistribSMul.toSMulZeroClass.{u1, u3} R B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))))) (DistribMulAction.toDistribSMul.{u1, u3} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u2, u3, max u2 u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) (AlgHom.algHomClass.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5))))) φ₂) s) -> (Eq.{max (succ u2) (succ u3)} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) φ₁ φ₂))
 Case conversion may be inaccurate. Consider using '#align alg_hom.ext_of_adjoin_eq_top AlgHom.ext_of_adjoin_eq_topₓ'. -/
 theorem ext_of_adjoin_eq_top {s : Set A} (h : adjoin R s = ⊤) ⦃φ₁ φ₂ : A →ₐ[R] B⦄
     (hs : s.EqOn φ₁ φ₂) : φ₁ = φ₂ :=

69c6a5a1

bump to nightly-2023-05-23-03

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

ed98987c

Diff

@@ -185,7 +185,7 @@ theorem adjoin_induction₂ {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {p : (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> Prop}, (forall (x : A) (h : Membership.Mem.{u2, u2} A (Set.{u2} A) (Set.hasMem.{u2} A) x s), p (Subtype.mk.{succ u2} A (fun (x : A) => Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) x (Algebra.subset_adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s x h))) -> (forall (r : R), p (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) (fun (_x : RingHom.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) => R -> (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (RingHom.hasCoeToFun.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) (algebraMap.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) r)) -> (forall (x : coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (y : coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)), (p x) -> (p y) -> (p (HAdd.hAdd.{u2, u2, u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (instHAdd.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (AddMemClass.add.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (AddZeroClass.toHasAdd.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (AddSubmonoidClass.to_addMemClass.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SubsemiringClass.to_addSubmonoidClass.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Semiring.toNonAssocSemiring.{u2} A _inst_2) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Subalgebra.subsemiringClass.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) x y))) -> (forall (x : coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (y : coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)), (p x) -> (p y) -> (p (HMul.hMul.{u2, u2, u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (instHMul.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (MulMemClass.mul.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (MulOneClass.toHasMul.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SubmonoidClass.to_mulMemClass.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SubsemiringClass.to_submonoidClass.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Semiring.toNonAssocSemiring.{u2} A _inst_2) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Subalgebra.subsemiringClass.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) x y))) -> (forall (x : coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)), p x)
 but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {p : (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) -> Prop}, (forall (x : A) (h : Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x s), p (Subtype.mk.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) x (Algebra.subset_adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s x h))) -> (forall (r : R), p (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A 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_inst_1 _inst_2 _inst_4 s))), (p x) -> (p y) -> (p (HMul.hMul.{u2, u2, u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 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(Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))), p x)
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {p : (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) -> Prop}, (forall (x : A) (h : Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x s), p (Subtype.mk.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) x (Algebra.subset_adjoin.{u1, u2} R A _inst_1 _inst_2 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(Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Subsemiring.toNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2) (Subalgebra.toSubsemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))))))) x y))) -> (forall (x : Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (y : Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))), (p x) -> (p y) -> (p (HMul.hMul.{u2, u2, u2} 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(instHMul.{u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Submonoid.mul.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Subsemiring.toSubmonoid.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2) (Subalgebra.toSubsemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))))) x y))) -> (forall (x : Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))), p x)
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_induction' Algebra.adjoin_induction'ₓ'. -/
 /-- The difference with `algebra.adjoin_induction` is that this acts on the subtype. -/
 theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin h⟩)
@@ -209,7 +209,7 @@ theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s),
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, Eq.{succ u2} (Subalgebra.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Algebra.adjoin.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Set.preimage.{u2, u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) A ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) A (HasLiftT.mk.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) A (CoeTCₓ.coe.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) A (coeBase.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) A (coeSubtype.{succ u2} A (fun (x : A) => Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))))))) s)) (Top.top.{u2} (Subalgebra.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (CompleteLattice.toHasTop.{u2} (Subalgebra.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Algebra.Subalgebra.completeLattice.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))))
 but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, Eq.{succ u2} (Subalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Algebra.adjoin.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Set.preimage.{u2, u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) A (Subtype.val.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) s)) (Top.top.{u2} (Subalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (CompleteLattice.toTop.{u2} (Subalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))))
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, Eq.{succ u2} (Subalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Algebra.adjoin.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Set.preimage.{u2, u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) A (Subtype.val.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) s)) (Top.top.{u2} (Subalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (CompleteLattice.toTop.{u2} (Subalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_adjoin_coe_preimage Algebra.adjoin_adjoin_coe_preimageₓ'. -/
 @[simp]
 theorem adjoin_adjoin_coe_preimage {s : Set A} : adjoin R ((coe : adjoin R s → A) ⁻¹' s) = ⊤ :=

69c6a5a1

bump to nightly-2023-05-19-16

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

a31df521

Diff

@@ -136,7 +136,7 @@ theorem adjoin_attach_biUnion [DecidableEq A] {α : Type _} {s : Finset α} (f :
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {p : A -> Prop} {x : A}, (Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (forall (x : A), (Membership.Mem.{u2, u2} A (Set.{u2} A) (Set.hasMem.{u2} A) x s) -> (p x)) -> (forall (r : R), p (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (fun (_x : RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) => R -> A) (RingHom.hasCoeToFun.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r)) -> (forall (x : A) (y : A), (p x) -> (p y) -> (p (HAdd.hAdd.{u2, u2, u2} A A A (instHAdd.{u2} A (Distrib.toHasAdd.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) x y))) -> (forall (x : A) (y : A), (p x) -> (p y) -> (p (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (Distrib.toHasMul.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) x y))) -> (p x)
 but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {p : A -> Prop} {x : A}, (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (forall (x : A), (Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x s) -> (p x)) -> (forall (r : R), p (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => A) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2) (RingHom.instRingHomClassRingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r)) -> (forall (x : A) (y : A), (p x) -> (p y) -> (p (HAdd.hAdd.{u2, u2, u2} A A A (instHAdd.{u2} A (Distrib.toAdd.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) x y))) -> (forall (x : A) (y : A), (p x) -> (p y) -> (p (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) x y))) -> (p x)
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {p : A -> Prop} {x : A}, (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (forall (x : A), (Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x s) -> (p x)) -> (forall (r : R), p (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => A) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2) (RingHom.instRingHomClassRingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r)) -> (forall (x : A) (y : A), (p x) -> (p y) -> (p (HAdd.hAdd.{u2, u2, u2} A A A (instHAdd.{u2} A (Distrib.toAdd.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) x y))) -> (forall (x : A) (y : A), (p x) -> (p y) -> (p (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) x y))) -> (p x)
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_induction Algebra.adjoin_inductionₓ'. -/
 @[elab_as_elim]
 theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : ∀ x ∈ s, p x)
@@ -154,7 +154,7 @@ theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : 
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {p : A -> A -> Prop} {a : A} {b : A}, (Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) a (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) b (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (forall (x : A), (Membership.Mem.{u2, u2} A (Set.{u2} A) (Set.hasMem.{u2} A) x s) -> (forall (y : A), (Membership.Mem.{u2, u2} A (Set.{u2} A) (Set.hasMem.{u2} A) y s) -> (p x y))) -> (forall (r₁ : R) (r₂ : R), p (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (fun (_x : RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) => R -> A) (RingHom.hasCoeToFun.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r₁) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (fun (_x : RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) => R -> A) (RingHom.hasCoeToFun.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r₂)) -> (forall (r : R) (x : A), (Membership.Mem.{u2, u2} A (Set.{u2} A) (Set.hasMem.{u2} A) x s) -> (p (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (fun (_x : RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) => R -> A) (RingHom.hasCoeToFun.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r) x)) -> (forall (r : R) (x : A), (Membership.Mem.{u2, u2} A (Set.{u2} A) (Set.hasMem.{u2} A) x s) -> (p x (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (fun (_x : RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) => R -> A) (RingHom.hasCoeToFun.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r))) -> (forall (x₁ : A) (x₂ : A) (y : A), (p x₁ y) -> (p x₂ y) -> (p (HAdd.hAdd.{u2, u2, u2} A A A (instHAdd.{u2} A (Distrib.toHasAdd.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) x₁ x₂) y)) -> (forall (x : A) (y₁ : A) (y₂ : A), (p x y₁) -> (p x y₂) -> (p x (HAdd.hAdd.{u2, u2, u2} A A A (instHAdd.{u2} A (Distrib.toHasAdd.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) y₁ y₂))) -> (forall (x₁ : A) (x₂ : A) (y : A), (p x₁ y) -> (p x₂ y) -> (p (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (Distrib.toHasMul.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) x₁ x₂) y)) -> (forall (x : A) (y₁ : A) (y₂ : A), (p x y₁) -> (p x y₂) -> (p x (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (Distrib.toHasMul.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) y₁ y₂))) -> (p a b)
 but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {p : A -> A -> Prop} {a : A} {b : A}, (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) a (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) b (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (forall (x : A), (Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x s) -> (forall (y : A), (Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) y s) -> (p x y))) -> (forall (r₁ : R) (r₂ : R), p (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => A) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2) (RingHom.instRingHomClassRingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r₁) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2) (RingHom.instRingHomClassRingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r₂)) -> (forall (r : R) (x : A), (Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x s) -> (p (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R (fun (_x : R) => (fun 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(RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2) (RingHom.instRingHomClassRingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r) x)) -> (forall (r : R) (x : A), (Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x s) -> (p x (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => A) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2) (RingHom.instRingHomClassRingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r))) -> (forall (x₁ : A) (x₂ : A) (y : A), (p x₁ y) -> (p x₂ y) -> (p (HAdd.hAdd.{u2, u2, u2} A A A (instHAdd.{u2} A (Distrib.toAdd.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) x₁ x₂) y)) -> (forall (x : A) (y₁ : A) (y₂ : A), (p x y₁) -> (p x y₂) -> (p x (HAdd.hAdd.{u2, u2, u2} A A A (instHAdd.{u2} A (Distrib.toAdd.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) y₁ y₂))) -> (forall (x₁ : A) (x₂ : A) (y : A), (p x₁ y) -> (p x₂ y) -> (p (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) x₁ x₂) y)) -> (forall (x : A) (y₁ : A) (y₂ : A), (p x y₁) -> (p x y₂) -> (p x (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) y₁ y₂))) -> (p a b)
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_induction₂ Algebra.adjoin_induction₂ₓ'. -/
 /-- Induction principle for the algebra generated by a set `s`: show that `p x y` holds for any
 `x y ∈ adjoin R s` given that that it holds for `x y ∈ s` and that it satisfies a number of
@@ -185,7 +185,7 @@ theorem adjoin_induction₂ {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {p : (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> Prop}, (forall (x : A) (h : Membership.Mem.{u2, u2} A (Set.{u2} A) (Set.hasMem.{u2} A) x s), p (Subtype.mk.{succ u2} A (fun (x : A) => Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) x (Algebra.subset_adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s x h))) -> (forall (r : R), p (coeFn.{max (succ u1) (succ 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_inst_1 _inst_2 _inst_4 s))) (RingHom.hasCoeToFun.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)))) (algebraMap.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) r)) -> (forall (x : coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (y : coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)), (p x) -> (p y) -> (p (HAdd.hAdd.{u2, u2, u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (instHAdd.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (AddMemClass.add.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (AddZeroClass.toHasAdd.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (AddSubmonoidClass.to_addMemClass.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SubsemiringClass.to_addSubmonoidClass.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Semiring.toNonAssocSemiring.{u2} A _inst_2) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Subalgebra.subsemiringClass.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) x y))) -> (forall (x : coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (y : coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)), (p x) -> (p y) -> (p (HMul.hMul.{u2, u2, u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (instHMul.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (MulMemClass.mul.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (MulOneClass.toHasMul.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SubmonoidClass.to_mulMemClass.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SubsemiringClass.to_submonoidClass.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Semiring.toNonAssocSemiring.{u2} A _inst_2) (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Subalgebra.subsemiringClass.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) x y))) -> (forall (x : coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)), p x)
 but is expected to have type
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Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (y : Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))), (p x) -> (p y) -> (p (HAdd.hAdd.{u2, u2, u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (instHAdd.{u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Distrib.toAdd.{u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (NonUnitalNonAssocSemiring.toDistrib.{u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Subsemiring.toNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2) (Subalgebra.toSubsemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))))))) x y))) -> (forall (x : Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (y : Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))), (p x) -> (p y) -> (p (HMul.hMul.{u2, u2, u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (instHMul.{u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))) (Submonoid.mul.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Subsemiring.toSubmonoid.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2) (Subalgebra.toSubsemiring.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))))) x y))) -> (forall (x : Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))), p x)
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_induction' Algebra.adjoin_induction'ₓ'. -/
 /-- The difference with `algebra.adjoin_induction` is that this acts on the subtype. -/
 theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin h⟩)
@@ -261,7 +261,7 @@ variable (R) {A} (s)
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A 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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _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.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _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.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span Algebra.adjoin_eq_spanₓ'. -/
 theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s) :=
   by
@@ -302,7 +302,7 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _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.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _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.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
 Case conversion may be inaccurate. Consider using '#align algebra.span_le_adjoin Algebra.span_le_adjoinₓ'. -/
 theorem span_le_adjoin (s : Set A) : span R s ≤ (adjoin R s).toSubmodule :=
   span_le.mpr subset_adjoin
@@ -312,7 +312,7 @@ theorem span_le_adjoin (s : Set A) : span R s ≤ (adjoin R s).toSubmodule :=
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R 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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} 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A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) 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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) t))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Preorder.toLE.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (PartialOrder.toPreorder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (OmegaCompletePartialOrder.toPartialOrder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A 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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) t) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) t))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Preorder.toLE.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (PartialOrder.toPreorder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (OmegaCompletePartialOrder.toPartialOrder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A 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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 _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A 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(x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _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.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) t) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) t))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_to_submodule_le Algebra.adjoin_toSubmodule_leₓ'. -/
 theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
     (adjoin R s).toSubmodule ≤ t ↔ ↑(Submonoid.closure s) ⊆ (t : Set A) := by
@@ -323,7 +323,7 @@ theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A 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(Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _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.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _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.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span_of_subset Algebra.adjoin_eq_span_of_subsetₓ'. -/
 theorem adjoin_eq_span_of_subset {s : Set A} (hs : ↑(Submonoid.closure s) ⊆ (span R s : Set A)) :
     (adjoin R s).toSubmodule = span R s :=
@@ -488,7 +488,7 @@ theorem adjoin_union_eq_adjoin_adjoin :
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] (s : Set.{u2} A) (t : Set.{u2} A), Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 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_inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A 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_inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 s)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))))) => (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] (s : Set.{u2} A) (t : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s t))) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A 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_inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))))) (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] (s : Set.{u2} A) (t : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s t))) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))))) (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))))) (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) 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 _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) 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 _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A 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_inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A 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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))))) (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) 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 _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) 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 _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_union_coe_submodule Algebra.adjoin_union_coe_submoduleₓ'. -/
 theorem adjoin_union_coe_submodule :
     (adjoin R (s ∪ t)).toSubmodule = (adjoin R s).toSubmodule * (adjoin R t).toSubmodule :=
@@ -515,7 +515,7 @@ variable {R}
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] [_inst_4 : CommSemiring.{u3} B] [_inst_5 : Algebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_6 : Algebra.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_7 : 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 (CommSemiring.toSemiring.{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 _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 (CommSemiring.toSemiring.{u2} A _inst_2))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{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 (CommSemiring.toSemiring.{u2} A _inst_2))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))) (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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (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 _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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))) (Algebra.toModule.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (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 _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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))) (Algebra.toModule.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)))))] (r : A) (s : Set.{u3} B) (B' : Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5), (HasSubset.Subset.{u3} (Set.{u3} B) (Set.hasSubset.{u3} B) (SMul.smul.{u2, u3} A (Set.{u3} B) (Set.smulSet.{u2, u3} A B (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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (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 _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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))) (Algebra.toModule.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6)))))) r s) ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (Set.{u3} B) (HasLiftT.mk.{succ u3, succ u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (Set.{u3} B) (CoeTCₓ.coe.{succ u3, succ u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (Set.{u3} B) (SetLike.Set.hasCoeT.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.setLike.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)))) B')) -> (forall {x : B}, (Membership.Mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.hasMem.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.setLike.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) x (Algebra.adjoin.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5 s)) -> (Membership.Mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.hasMem.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.setLike.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) (algebraMap.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6) r) B') -> (Exists.{1} Nat (fun (n₀ : Nat) => forall (n : Nat), (GE.ge.{0} Nat Nat.hasLe n n₀) -> (Membership.Mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.hasMem.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.setLike.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) (SMul.smul.{u2, u3} A B (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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (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 _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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))) (Algebra.toModule.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6))))) (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2))))) r n) x) B'))))
 but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] [_inst_4 : CommSemiring.{u3} B] [_inst_5 : Algebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_6 : Algebra.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_7 : IsScalarTower.{u1, u2, u3} R A B (Algebra.toSMul.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.toSMul.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6) (Algebra.toSMul.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)] (r : A) (s : Set.{u3} B) (B' : Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5), (HasSubset.Subset.{u3} (Set.{u3} B) (Set.instHasSubsetSet.{u3} B) (HSMul.hSMul.{u2, u3, u3} A (Set.{u3} B) (Set.{u3} B) (instHSMul.{u2, u3} A (Set.{u3} B) (Set.smulSet.{u2, u3} A B (Algebra.toSMul.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6))) r s) (SetLike.coe.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B')) -> (forall {x : B}, (Membership.mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) x (Algebra.adjoin.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5 s)) -> (Membership.mem.{u3, u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : A) => B) r) (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : A) => B) _x) (MulHomClass.toFunLike.{max u2 u3, u2, u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A B (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u3, u2, u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u3, u2, u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))))) (algebraMap.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6) r) B') -> (Exists.{1} Nat (fun (n₀ : Nat) => forall (n : Nat), (GE.ge.{0} Nat instLENat n n₀) -> (Membership.mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) (HSMul.hSMul.{u2, u3, u3} A B B (instHSMul.{u2, u3} A B (Algebra.toSMul.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6)) (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2))))) r n) x) B'))))
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] [_inst_4 : CommSemiring.{u3} B] [_inst_5 : Algebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_6 : Algebra.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_7 : IsScalarTower.{u1, u2, u3} R A B (Algebra.toSMul.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.toSMul.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6) (Algebra.toSMul.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)] (r : A) (s : Set.{u3} B) (B' : Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5), (HasSubset.Subset.{u3} (Set.{u3} B) (Set.instHasSubsetSet.{u3} B) (HSMul.hSMul.{u2, u3, u3} A (Set.{u3} B) (Set.{u3} B) (instHSMul.{u2, u3} A (Set.{u3} B) (Set.smulSet.{u2, u3} A B (Algebra.toSMul.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6))) r s) (SetLike.coe.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B')) -> (forall {x : B}, (Membership.mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) x (Algebra.adjoin.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5 s)) -> (Membership.mem.{u3, u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) r) (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u2 u3, u2, u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A B (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u3, u2, u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u3, u2, u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))))) (algebraMap.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6) r) B') -> (Exists.{1} Nat (fun (n₀ : Nat) => forall (n : Nat), (GE.ge.{0} Nat instLENat n n₀) -> (Membership.mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) (HSMul.hSMul.{u2, u3, u3} A B B (instHSMul.{u2, u3} A B (Algebra.toSMul.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6)) (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2))))) r n) x) B'))))
 Case conversion may be inaccurate. Consider using '#align algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoinₓ'. -/
 theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]
     [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
@@ -578,7 +578,7 @@ theorem adjoin_int (s : Set R) : adjoin ℤ s = subalgebraOfSubring (Subring.clo
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : Ring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2)] {s : Set.{u2} A} {x : A}, Iff (Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3)) x (Algebra.adjoin.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3 s)) (Membership.Mem.{u2, u2} A (Subring.{u2} A _inst_2) (SetLike.hasMem.{u2, u2} (Subring.{u2} A _inst_2) A (Subring.setLike.{u2} A _inst_2)) x (Subring.closure.{u2} A _inst_2 (Union.union.{u2} (Set.{u2} A) (Set.hasUnion.{u2} A) (Set.range.{u2, succ u1} A R (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 _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 _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 _inst_2))) (algebraMap.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3))) s)))
 but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : Ring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2)] {s : Set.{u2} A} {x : A}, Iff (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3)) x (Algebra.adjoin.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3 s)) (Membership.mem.{u2, u2} A (Subring.{u2} A _inst_2) (SetLike.instMembership.{u2, u2} (Subring.{u2} A _inst_2) A (Subring.instSetLikeSubring.{u2} A _inst_2)) x (Subring.closure.{u2} A _inst_2 (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) (Set.range.{u2, succ u1} A R (FunLike.coe.{max (succ u1) (succ u2), 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 _inst_2))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => A) _x) (MulHomClass.toFunLike.{max u1 u2, u1, 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 _inst_2))) R A (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A _inst_2)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, 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 _inst_2))) R A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A _inst_2))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, 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 _inst_2))) R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{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 (Ring.toSemiring.{u2} A _inst_2)))))) (algebraMap.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3))) s)))
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : Ring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2)] {s : Set.{u2} A} {x : A}, Iff (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3)) x (Algebra.adjoin.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3 s)) (Membership.mem.{u2, u2} A (Subring.{u2} A _inst_2) (SetLike.instMembership.{u2, u2} (Subring.{u2} A _inst_2) A (Subring.instSetLikeSubring.{u2} A _inst_2)) x (Subring.closure.{u2} A _inst_2 (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) (Set.range.{u2, succ u1} A R (FunLike.coe.{max (succ u1) (succ u2), 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 _inst_2))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => A) _x) (MulHomClass.toFunLike.{max u1 u2, u1, 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 _inst_2))) R A (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A _inst_2)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, 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 _inst_2))) R A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A _inst_2))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, 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 _inst_2))) R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{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 (Ring.toSemiring.{u2} A _inst_2)))))) (algebraMap.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_3))) s)))
 Case conversion may be inaccurate. Consider using '#align algebra.mem_adjoin_iff Algebra.mem_adjoin_iffₓ'. -/
 theorem mem_adjoin_iff {s : Set A} {x : A} :
     x ∈ adjoin R s ↔ x ∈ Subring.closure (Set.range (algebraMap R A) ∪ s) :=

69c6a5a1

bump to nightly-2023-05-18-03

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

b46e9d53

Diff

@@ -371,7 +371,7 @@ theorem adjoin_prod_le (s : Set A) (t : Set B) :
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] {s : Set.{u2} A} {x : A} {f : LinearMap.{u1, u1, u2, u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)}, (forall (a₁ : A) (a₂ : A), Eq.{succ u3} B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ 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 but is expected to have type
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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f a₂))) -> (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (Membership.mem.{u3, u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : A) => B) x) (Subalgebra.{u1, u3} R B _inst_1 _inst_3 _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 _inst_3 _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (LinearMap.{u1, u1, u2, u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : A) => B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u3} R R A B (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f x) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 (Set.image.{u2, u3} A B (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (LinearMap.{u1, u1, u2, u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : A) => B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u3} R R A B (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f) (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) (OfNat.ofNat.{u2} A 1 (One.toOfNat1.{u2} A (Semiring.toOne.{u2} A _inst_2))))))))
+  forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] {s : Set.{u2} A} {x : A} {f : LinearMap.{u1, u1, u2, u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)}, (forall (a₁ : A) (a₂ : A), Eq.{succ u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : 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(LinearMap.instFunLikeLinearMap.{u1, u1, u2, u3} R R A B (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) a₁ a₂)) (HMul.hMul.{u3, u3, u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : A) => B) a₁) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : 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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f a₂))) -> (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (Membership.mem.{u3, u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : A) => B) x) (Subalgebra.{u1, u3} R B _inst_1 _inst_3 _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 _inst_3 _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (LinearMap.{u1, u1, u2, u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : A) => B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u3} R R A B (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f x) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 (Set.image.{u2, u3} A B (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (LinearMap.{u1, u1, u2, u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : A) => B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u3} R R A B (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f) (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) (OfNat.ofNat.{u2} A 1 (One.toOfNat1.{u2} A (Semiring.toOne.{u2} A _inst_2))))))))
 Case conversion may be inaccurate. Consider using '#align algebra.mem_adjoin_of_map_mul Algebra.mem_adjoin_of_map_mulₓ'. -/
 theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a₂, f (a₁ * a₂) = f a₁ * f a₂)
     (h : x ∈ adjoin R s) : f x ∈ adjoin R (f '' (s ∪ {1})) :=
@@ -393,7 +393,7 @@ theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] (s : Set.{u2} A) (t : Set.{u3} B), Eq.{succ (max u2 u3)} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Algebra.adjoin.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5) (Union.union.{max u2 u3} (Set.{max u2 u3} (Prod.{u2, u3} A B)) (Set.hasUnion.{max u2 u3} (Prod.{u2, u3} A B)) (Set.image.{u2, max u2 u3} A (Prod.{u2, u3} A B) (coeFn.{max (succ u2) (succ (max u2 u3)), max (succ u2) (succ (max u2 u3))} (LinearMap.{u1, u1, u2, max u2 u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A (Prod.{u2, u3} A B) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Prod.addCommMonoid.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) (fun (_x : LinearMap.{u1, u1, u2, max u2 u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A (Prod.{u2, u3} A B) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Prod.addCommMonoid.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) => A -> (Prod.{u2, u3} A B)) (LinearMap.hasCoeToFun.{u1, u1, u2, max u2 u3} R R A (Prod.{u2, u3} A B) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Prod.addCommMonoid.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} 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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) (Union.union.{u2} (Set.{u2} A) (Set.hasUnion.{u2} A) s (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) (OfNat.ofNat.{u2} A 1 (OfNat.mk.{u2} A 1 (One.one.{u2} A (AddMonoidWithOne.toOne.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))))))) (Set.image.{u3, max u2 u3} B (Prod.{u2, u3} A B) (coeFn.{max (succ u3) (succ (max u2 u3)), max (succ u3) (succ (max u2 u3))} (LinearMap.{u1, u1, u3, max u2 u3} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) B (Prod.{u2, u3} A B) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Prod.addCommMonoid.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) (Union.union.{u3} (Set.{u3} B) (Set.hasUnion.{u3} B) t (Singleton.singleton.{u3, u3} B (Set.{u3} B) (Set.hasSingleton.{u3} B) (OfNat.ofNat.{u3} B 1 (OfNat.mk.{u3} B 1 (One.one.{u3} B (AddMonoidWithOne.toOne.{u3} B (AddCommMonoidWithOne.toAddMonoidWithOne.{u3} B (NonAssocSemiring.toAddCommMonoidWithOne.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))))))))))) (Subalgebra.prod.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 t))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] (s : Set.{u2} A) (t : Set.{u3} B), Eq.{max (succ u2) (succ u3)} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Algebra.adjoin.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5) (Union.union.{max u2 u3} (Set.{max u2 u3} (Prod.{u2, u3} A B)) (Set.instUnionSet.{max u2 u3} (Prod.{u2, u3} A B)) (Set.image.{u2, max u2 u3} A (Prod.{u2, u3} A B) (FunLike.coe.{max (succ u2) (succ u3), succ u2, max (succ u2) (succ u3)} (LinearMap.{u1, u1, u2, max u3 u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) 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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) B (fun (_x : B) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : B) => Prod.{u2, u3} A B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u3, max u2 u3} R R B (Prod.{u2, u3} A B) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Prod.instAddCommMonoidSum.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (LinearMap.inr.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) (Union.union.{u3} (Set.{u3} B) (Set.instUnionSet.{u3} B) t (Singleton.singleton.{u3, u3} B (Set.{u3} B) (Set.instSingletonSet.{u3} B) (OfNat.ofNat.{u3} B 1 (One.toOfNat1.{u3} B (Semiring.toOne.{u3} B _inst_3)))))))) (Subalgebra.prod.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 t))
+  forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] (s : Set.{u2} A) (t : Set.{u3} B), Eq.{max (succ u2) (succ u3)} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Algebra.adjoin.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5) (Union.union.{max u2 u3} (Set.{max u2 u3} (Prod.{u2, u3} A B)) (Set.instUnionSet.{max u2 u3} (Prod.{u2, u3} A B)) (Set.image.{u2, max u2 u3} A (Prod.{u2, u3} A B) (FunLike.coe.{max (succ u2) (succ u3), succ u2, max (succ u2) (succ u3)} (LinearMap.{u1, u1, u2, max u3 u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A (Prod.{u2, u3} A B) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Prod.instAddCommMonoidSum.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : A) => Prod.{u2, u3} A B) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, max u2 u3} R R A (Prod.{u2, u3} A B) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Prod.instAddCommMonoidSum.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u2} 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(Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) (OfNat.ofNat.{u2} A 1 (One.toOfNat1.{u2} A (Semiring.toOne.{u2} A _inst_2)))))) (Set.image.{u3, max u2 u3} B (Prod.{u2, u3} A B) (FunLike.coe.{max (succ u2) (succ u3), succ u3, max (succ u2) (succ u3)} (LinearMap.{u1, u1, u3, max u3 u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) B (Prod.{u2, u3} A B) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Prod.instAddCommMonoidSum.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A 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_inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Prod.instAddCommMonoidSum.{u2, u3} A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5) (Prod.module.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (LinearMap.inr.{u1, u2, u3} R A B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))) (Union.union.{u3} (Set.{u3} B) (Set.instUnionSet.{u3} B) t (Singleton.singleton.{u3, u3} B (Set.{u3} B) (Set.instSingletonSet.{u3} B) (OfNat.ofNat.{u3} B 1 (One.toOfNat1.{u3} B (Semiring.toOne.{u3} B _inst_3)))))))) (Subalgebra.prod.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 t))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_inl_union_inr_eq_prod Algebra.adjoin_inl_union_inr_eq_prodₓ'. -/
 theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     adjoin R (LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1})) =

69c6a5a1

bump to nightly-2023-05-16-16

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

9cb92e8c

Diff

@@ -57,7 +57,7 @@ theorem subset_adjoin : s ⊆ adjoin R s :=
 
 /- warning: algebra.adjoin_le -> Algebra.adjoin_le is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) S)) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S)
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) S)) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S)
 but is expected to have type
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) S)) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S)
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_le Algebra.adjoin_leₓ'. -/
@@ -77,7 +77,7 @@ theorem adjoin_eq_sInf : adjoin R s = sInf { p | s ⊆ p } :=
 
 /- warning: algebra.adjoin_le_iff -> Algebra.adjoin_le_iff is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4}, Iff (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) S))
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4}, Iff (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) S))
 but is expected to have type
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4}, Iff (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) S))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_le_iff Algebra.adjoin_le_iffₓ'. -/
@@ -87,7 +87,7 @@ theorem adjoin_le_iff {S : Subalgebra R A} : adjoin R s ≤ S ↔ s ⊆ S :=
 
 /- warning: algebra.adjoin_mono -> Algebra.adjoin_mono is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s t) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 t))
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s t) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 t))
 but is expected to have type
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s t) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 t))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_mono Algebra.adjoin_monoₓ'. -/
@@ -97,7 +97,7 @@ theorem adjoin_mono (H : s ⊆ t) : adjoin R s ≤ adjoin R t :=
 
 /- warning: algebra.adjoin_eq_of_le -> Algebra.adjoin_eq_of_le is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4), (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) S)) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) S (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S)
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4), (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) S)) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) S (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S)
 but is expected to have type
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4), (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) S)) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) S (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) -> (Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S)
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_of_le Algebra.adjoin_eq_of_leₓ'. -/
@@ -259,7 +259,7 @@ variable (R) {A} (s)
 
 /- warning: algebra.adjoin_eq_span -> Algebra.adjoin_eq_span is a dubious translation:
 lean 3 declaration is
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} 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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
 but is expected to have type
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span Algebra.adjoin_eq_spanₓ'. -/
@@ -300,7 +300,7 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
 
 /- warning: algebra.span_le_adjoin -> Algebra.span_le_adjoin is a dubious translation:
 lean 3 declaration is
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
 but is expected to have type
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
 Case conversion may be inaccurate. Consider using '#align algebra.span_le_adjoin Algebra.span_le_adjoinₓ'. -/
@@ -310,7 +310,7 @@ theorem span_le_adjoin (s : Set A) : span R s ≤ (adjoin R s).toSubmodule :=
 
 /- warning: algebra.adjoin_to_submodule_le -> Algebra.adjoin_toSubmodule_le is a dubious translation:
 lean 3 declaration is
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(Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) t))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) t) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) t))
 but is expected to have type
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Preorder.toLE.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (PartialOrder.toPreorder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (OmegaCompletePartialOrder.toPartialOrder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) t) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) t))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_to_submodule_le Algebra.adjoin_toSubmodule_leₓ'. -/
@@ -321,7 +321,7 @@ theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
 
 /- warning: algebra.adjoin_eq_span_of_subset -> Algebra.adjoin_eq_span_of_subset is a dubious translation:
 lean 3 declaration is
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
 but is expected to have type
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span_of_subset Algebra.adjoin_eq_span_of_subsetₓ'. -/
@@ -357,7 +357,7 @@ theorem adjoin_insert_adjoin (x : A) : adjoin R (insert x ↑(adjoin R s)) = adj
 
 /- warning: algebra.adjoin_prod_le -> Algebra.adjoin_prod_le is a dubious translation:
 lean 3 declaration is
-  forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] (s : Set.{u2} A) (t : Set.{u3} B), LE.le.{max u2 u3} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Preorder.toLE.{max u2 u3} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (PartialOrder.toPreorder.{max u2 u3} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (CompleteSemilatticeInf.toPartialOrder.{max u2 u3} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (CompleteLattice.toCompleteSemilatticeInf.{max u2 u3} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Algebra.Subalgebra.completeLattice.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)))))) (Algebra.adjoin.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5) (Set.prod.{u2, u3} A B s t)) (Subalgebra.prod.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 t))
+  forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] (s : Set.{u2} A) (t : Set.{u3} B), LE.le.{max u2 u3} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Preorder.toHasLe.{max u2 u3} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (PartialOrder.toPreorder.{max u2 u3} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (CompleteSemilatticeInf.toPartialOrder.{max u2 u3} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (CompleteLattice.toCompleteSemilatticeInf.{max u2 u3} (Subalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Algebra.Subalgebra.completeLattice.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)))))) (Algebra.adjoin.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5) (Set.prod.{u2, u3} A B s t)) (Subalgebra.prod.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 t))
 but is expected to have type
   forall (R : Type.{u1}) {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Semiring.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_3] (s : Set.{u2} A) (t : Set.{u3} B), LE.le.{max u2 u3} (Subalgebra.{u1, max u3 u2} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Preorder.toLE.{max u2 u3} (Subalgebra.{u1, max u3 u2} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (PartialOrder.toPreorder.{max u2 u3} (Subalgebra.{u1, max u3 u2} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{max u2 u3} (Subalgebra.{u1, max u3 u2} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{max u2 u3} (Subalgebra.{u1, max u3 u2} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)) (Algebra.instCompleteLatticeSubalgebra.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5)))))) (Algebra.adjoin.{u1, max u3 u2} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_3) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5) (Set.prod.{u2, u3} A B s t)) (Subalgebra.prod.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_4 _inst_3 _inst_5 (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 t))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_prod_le Algebra.adjoin_prod_leₓ'. -/
@@ -486,7 +486,7 @@ theorem adjoin_union_eq_adjoin_adjoin :
 
 /- warning: algebra.adjoin_union_coe_submodule -> Algebra.adjoin_union_coe_submodule is a dubious translation:
 lean 3 declaration is
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] (s : Set.{u2} A) (t : Set.{u2} A), Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A 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_inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 s)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))))) => (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] (s : Set.{u2} A) (t : Set.{u2} A), Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A 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_inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A 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A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toHasLe.{u2} 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
 but is expected to have type
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] (s : Set.{u2} A) (t : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s t))) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A 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u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 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_inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A 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(CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) 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(Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A 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(x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A 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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_union_coe_submodule Algebra.adjoin_union_coe_submoduleₓ'. -/

69c6a5a1

bump to nightly-2023-05-14-08

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

d31da313

Diff

@@ -65,15 +65,15 @@ theorem adjoin_le {S : Subalgebra R A} (H : s ⊆ S) : adjoin R s ≤ S :=
   Algebra.gc.l_le H
 #align algebra.adjoin_le Algebra.adjoin_le
 
-/- warning: algebra.adjoin_eq_Inf -> Algebra.adjoin_eq_infₛ is a dubious translation:
+/- warning: algebra.adjoin_eq_Inf -> Algebra.adjoin_eq_sInf is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (InfSet.infₛ.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toHasInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (setOf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (p : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) p))))
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (InfSet.sInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toHasInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (setOf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (p : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) p))))
 but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (InfSet.infₛ.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toInfSet.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (setOf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (p : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) p))))
-Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_Inf Algebra.adjoin_eq_infₛₓ'. -/
-theorem adjoin_eq_infₛ : adjoin R s = infₛ { p | s ⊆ p } :=
-  le_antisymm (le_infₛ fun _ h => adjoin_le h) (infₛ_le subset_adjoin)
-#align algebra.adjoin_eq_Inf Algebra.adjoin_eq_infₛ
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (InfSet.sInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toInfSet.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (setOf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (p : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) p))))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_Inf Algebra.adjoin_eq_sInfₓ'. -/
+theorem adjoin_eq_sInf : adjoin R s = sInf { p | s ⊆ p } :=
+  le_antisymm (le_sInf fun _ h => adjoin_le h) (sInf_le subset_adjoin)
+#align algebra.adjoin_eq_Inf Algebra.adjoin_eq_sInf
 
 /- warning: algebra.adjoin_le_iff -> Algebra.adjoin_le_iff is a dubious translation:
 lean 3 declaration is
@@ -111,26 +111,26 @@ theorem adjoin_eq (S : Subalgebra R A) : adjoin R ↑S = S :=
 #align algebra.adjoin_eq Algebra.adjoin_eq
 -/
 
-/- warning: algebra.adjoin_Union -> Algebra.adjoin_unionᵢ is a dubious translation:
+/- warning: algebra.adjoin_Union -> Algebra.adjoin_iUnion is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {α : Type.{u3}} (s : α -> (Set.{u2} A)), Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Set.unionᵢ.{u2, succ u3} A α s)) (supᵢ.{u2, succ u3} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toHasSup.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))) α (fun (i : α) => Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (s i)))
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {α : Type.{u3}} (s : α -> (Set.{u2} A)), Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Set.iUnion.{u2, succ u3} A α s)) (iSup.{u2, succ u3} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toHasSup.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))) α (fun (i : α) => Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (s i)))
 but is expected to have type
-  forall {R : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : Semiring.{u3} A] [_inst_4 : Algebra.{u2, u3} R A _inst_1 _inst_2] {α : Type.{u1}} (s : α -> (Set.{u3} A)), Eq.{succ u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u2, u3} R A _inst_1 _inst_2 _inst_4 (Set.unionᵢ.{u3, succ u1} A α s)) (supᵢ.{u3, succ u1} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toSupSet.{u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4))) α (fun (i : α) => Algebra.adjoin.{u2, u3} R A _inst_1 _inst_2 _inst_4 (s i)))
-Case conversion may be inaccurate. Consider using '#align algebra.adjoin_Union Algebra.adjoin_unionᵢₓ'. -/
-theorem adjoin_unionᵢ {α : Type _} (s : α → Set A) :
-    adjoin R (Set.unionᵢ s) = ⨆ i : α, adjoin R (s i) :=
-  (@Algebra.gc R A _ _ _).l_supᵢ
-#align algebra.adjoin_Union Algebra.adjoin_unionᵢ
-
-/- warning: algebra.adjoin_attach_bUnion -> Algebra.adjoin_attach_bunionᵢ is a dubious translation:
+  forall {R : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : Semiring.{u3} A] [_inst_4 : Algebra.{u2, u3} R A _inst_1 _inst_2] {α : Type.{u1}} (s : α -> (Set.{u3} A)), Eq.{succ u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u2, u3} R A _inst_1 _inst_2 _inst_4 (Set.iUnion.{u3, succ u1} A α s)) (iSup.{u3, succ u1} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toSupSet.{u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4))) α (fun (i : α) => Algebra.adjoin.{u2, u3} R A _inst_1 _inst_2 _inst_4 (s i)))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_Union Algebra.adjoin_iUnionₓ'. -/
+theorem adjoin_iUnion {α : Type _} (s : α → Set A) :
+    adjoin R (Set.iUnion s) = ⨆ i : α, adjoin R (s i) :=
+  (@Algebra.gc R A _ _ _).l_iSup
+#align algebra.adjoin_Union Algebra.adjoin_iUnion
+
+/- warning: algebra.adjoin_attach_bUnion -> Algebra.adjoin_attach_biUnion is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_6 : DecidableEq.{succ u2} A] {α : Type.{u3}} {s : Finset.{u3} α} (f : (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} α) Type.{u3} (Finset.hasCoeToSort.{u3} α) s) -> (Finset.{u2} A)), Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 ((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))) (Finset.bunionᵢ.{u3, u2} (Subtype.{succ u3} α (fun (x : α) => Membership.Mem.{u3, u3} α (Finset.{u3} α) (Finset.hasMem.{u3} α) x s)) A (fun (a : A) (b : A) => _inst_6 a b) (Finset.attach.{u3} α s) f))) (supᵢ.{u2, succ u3} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toHasSup.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} α) Type.{u3} (Finset.hasCoeToSort.{u3} α) s) (fun (x : coeSort.{succ u3, succ (succ u3)} (Finset.{u3} α) Type.{u3} (Finset.hasCoeToSort.{u3} α) s) => Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 ((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))) (f x))))
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_6 : DecidableEq.{succ u2} A] {α : Type.{u3}} {s : Finset.{u3} α} (f : (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} α) Type.{u3} (Finset.hasCoeToSort.{u3} α) s) -> (Finset.{u2} A)), Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 ((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))) (Finset.biUnion.{u3, u2} (Subtype.{succ u3} α (fun (x : α) => Membership.Mem.{u3, u3} α (Finset.{u3} α) (Finset.hasMem.{u3} α) x s)) A (fun (a : A) (b : A) => _inst_6 a b) (Finset.attach.{u3} α s) f))) (iSup.{u2, succ u3} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toHasSup.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} α) Type.{u3} (Finset.hasCoeToSort.{u3} α) s) (fun (x : coeSort.{succ u3, succ (succ u3)} (Finset.{u3} α) Type.{u3} (Finset.hasCoeToSort.{u3} α) s) => Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 ((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))) (f x))))
 but is expected to have type
-  forall {R : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : Semiring.{u3} A] [_inst_4 : Algebra.{u2, u3} R A _inst_1 _inst_2] [_inst_6 : DecidableEq.{succ u3} A] {α : Type.{u1}} {s : Finset.{u1} α} (f : (Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Finset.{u1} α) (Finset.instMembershipFinset.{u1} α) x s)) -> (Finset.{u3} A)), Eq.{succ u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u2, u3} R A _inst_1 _inst_2 _inst_4 (Finset.toSet.{u3} A (Finset.bunionᵢ.{u1, u3} (Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Finset.{u1} α) (Finset.instMembershipFinset.{u1} α) x s)) A (fun (a : A) (b : A) => _inst_6 a b) (Finset.attach.{u1} α s) f))) (supᵢ.{u3, succ u1} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toSupSet.{u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4))) (Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Finset.{u1} α) (Finset.instMembershipFinset.{u1} α) x s)) (fun (x : Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Finset.{u1} α) (Finset.instMembershipFinset.{u1} α) x s)) => Algebra.adjoin.{u2, u3} R A _inst_1 _inst_2 _inst_4 (Finset.toSet.{u3} A (f x))))
-Case conversion may be inaccurate. Consider using '#align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_bunionᵢₓ'. -/
-theorem adjoin_attach_bunionᵢ [DecidableEq A] {α : Type _} {s : Finset α} (f : s → Finset A) :
-    adjoin R (s.attach.bunionᵢ f : Set A) = ⨆ x, adjoin R (f x) := by simpa [adjoin_Union]
-#align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_bunionᵢ
+  forall {R : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : Semiring.{u3} A] [_inst_4 : Algebra.{u2, u3} R A _inst_1 _inst_2] [_inst_6 : DecidableEq.{succ u3} A] {α : Type.{u1}} {s : Finset.{u1} α} (f : (Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Finset.{u1} α) (Finset.instMembershipFinset.{u1} α) x s)) -> (Finset.{u3} A)), Eq.{succ u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u2, u3} R A _inst_1 _inst_2 _inst_4 (Finset.toSet.{u3} A (Finset.biUnion.{u1, u3} (Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Finset.{u1} α) (Finset.instMembershipFinset.{u1} α) x s)) A (fun (a : A) (b : A) => _inst_6 a b) (Finset.attach.{u1} α s) f))) (iSup.{u3, succ u1} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toSupSet.{u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u3} (Subalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u2, u3} R A _inst_1 _inst_2 _inst_4))) (Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Finset.{u1} α) (Finset.instMembershipFinset.{u1} α) x s)) (fun (x : Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Finset.{u1} α) (Finset.instMembershipFinset.{u1} α) x s)) => Algebra.adjoin.{u2, u3} R A _inst_1 _inst_2 _inst_4 (Finset.toSet.{u3} A (f x))))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_biUnionₓ'. -/
+theorem adjoin_attach_biUnion [DecidableEq A] {α : Type _} {s : Finset α} (f : s → Finset A) :
+    adjoin R (s.attach.biUnion f : Set A) = ⨆ x, adjoin R (f x) := by simpa [adjoin_Union]
+#align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_biUnion
 
 /- warning: algebra.adjoin_induction -> Algebra.adjoin_induction is a dubious translation:
 lean 3 declaration is

69c6a5a1

bump to nightly-2023-05-08-22

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

63e712c6

Diff

@@ -567,7 +567,7 @@ variable {R s t}
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (s : Set.{u1} R), Eq.{succ u1} (Subalgebra.{0, u1} Int R Int.commSemiring (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (algebraInt.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Algebra.adjoin.{0, u1} Int R Int.commSemiring (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (algebraInt.{u1} R (CommRing.toRing.{u1} R _inst_1)) s) (subalgebraOfSubring.{u1} R (CommRing.toRing.{u1} R _inst_1) (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (s : Set.{u1} R), Eq.{succ u1} (Subalgebra.{0, u1} Int R Int.instCommSemiringInt (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (algebraInt.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Algebra.adjoin.{0, u1} Int R Int.instCommSemiringInt (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (algebraInt.{u1} R (CommRing.toRing.{u1} R _inst_1)) s) (subalgebraOfSubring.{u1} R (CommRing.toRing.{u1} R _inst_1) (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (s : Set.{u1} R), Eq.{succ u1} (Subalgebra.{0, u1} Int R Int.instCommSemiringInt (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (algebraInt.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Algebra.adjoin.{0, u1} Int R Int.instCommSemiringInt (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (algebraInt.{u1} R (CommRing.toRing.{u1} R _inst_1)) s) (subalgebraOfSubring.{u1} R (CommRing.toRing.{u1} R _inst_1) (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_int Algebra.adjoin_intₓ'. -/
 theorem adjoin_int (s : Set R) : adjoin ℤ s = subalgebraOfSubring (Subring.closure s) :=
   le_antisymm (adjoin_le Subring.subset_closure)

69c6a5a1

bump to nightly-2023-04-20-10

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

fbfe5c3e

Diff

@@ -261,7 +261,7 @@ variable (R) {A} (s)
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span Algebra.adjoin_eq_spanₓ'. -/
 theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s) :=
   by
@@ -302,7 +302,7 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
 Case conversion may be inaccurate. Consider using '#align algebra.span_le_adjoin Algebra.span_le_adjoinₓ'. -/
 theorem span_le_adjoin (s : Set A) : span R s ≤ (adjoin R s).toSubmodule :=
   span_le.mpr subset_adjoin
@@ -312,7 +312,7 @@ theorem span_le_adjoin (s : Set A) : span R s ≤ (adjoin R s).toSubmodule :=
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) t) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) t))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Preorder.toLE.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (PartialOrder.toPreorder.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (OmegaCompletePartialOrder.toPartialOrder.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A 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A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) t) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) t))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Preorder.toLE.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (PartialOrder.toPreorder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (OmegaCompletePartialOrder.toPartialOrder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) t) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) t))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_to_submodule_le Algebra.adjoin_toSubmodule_leₓ'. -/
 theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
     (adjoin R s).toSubmodule ≤ t ↔ ↑(Submonoid.closure s) ⊆ (t : Set A) := by
@@ -323,7 +323,7 @@ theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A 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A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A 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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span_of_subset Algebra.adjoin_eq_span_of_subsetₓ'. -/
 theorem adjoin_eq_span_of_subset {s : Set A} (hs : ↑(Submonoid.closure s) ⊆ (span R s : Set A)) :
     (adjoin R s).toSubmodule = span R s :=
@@ -488,7 +488,7 @@ theorem adjoin_union_eq_adjoin_adjoin :
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] (s : Set.{u2} A) (t : Set.{u2} A), Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A 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(CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))))) => (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] (s : Set.{u2} A) (t : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s t))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))) (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 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(Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] (s : Set.{u2} A) (t : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s t))) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))))) (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))))) (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_union_coe_submodule Algebra.adjoin_union_coe_submoduleₓ'. -/
 theorem adjoin_union_coe_submodule :
     (adjoin R (s ∪ t)).toSubmodule = (adjoin R s).toSubmodule * (adjoin R t).toSubmodule :=

69c6a5a1

bump to nightly-2023-03-18-14

mathlib commit https://github.com/leanprover-community/mathlib/commit/02ba8949f486ebecf93fe7460f1ed0564b5e442c

73d15350

Diff

@@ -312,7 +312,7 @@ theorem span_le_adjoin (s : Set A) : span R s ≤ (adjoin R s).toSubmodule :=
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A 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(Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) t) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) t))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Preorder.toLE.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (PartialOrder.toPreorder.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (OmegaCompletePartialOrder.toPartialOrder.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) t) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.instSetLikeSubmodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) t))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {t : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)}, Iff (LE.le.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Preorder.toLE.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (PartialOrder.toPreorder.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (OmegaCompletePartialOrder.toPartialOrder.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) t) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) t))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_to_submodule_le Algebra.adjoin_toSubmodule_leₓ'. -/
 theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
     (adjoin R s).toSubmodule ≤ t ↔ ↑(Submonoid.closure s) ⊆ (t : Set A) := by
@@ -323,7 +323,7 @@ theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
 lean 3 declaration is
   forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
 but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.instSetLikeSubmodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
 Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span_of_subset Algebra.adjoin_eq_span_of_subsetₓ'. -/
 theorem adjoin_eq_span_of_subset {s : Set A} (hs : ↑(Submonoid.closure s) ⊆ (span R s : Set A)) :
     (adjoin R s).toSubmodule = span R s :=

69c6a5a1

bump to nightly-2023-03-14-02

mathlib commit https://github.com/leanprover-community/mathlib/commit/2196ab363eb097c008d4497125e0dde23fb36db2

d4b38a42

Diff

@@ -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.adjoin.basic
-! leanprover-community/mathlib commit a35ddf20601f85f78cd57e7f5b09ed528d71b7af
+! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.LinearAlgebra.Finsupp
 /-!
 # Adjoining elements to form subalgebras
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file develops the basic theory of subalgebras of an R-algebra generated
 by a set of elements. A basic interface for `adjoin` is set up.

a35ddf20

bump to nightly-2023-03-12-10

mathlib commit https://github.com/leanprover-community/mathlib/commit/f24cc2891c0e328f0ee8c57387103aa462c44b5e

7b820e3a

Diff

@@ -42,43 +42,99 @@ variable [CommSemiring R] [Semiring A] [Semiring B]
 
 variable [Algebra R A] [Algebra R B] {s t : Set A}
 
+/- warning: algebra.subset_adjoin -> Algebra.subset_adjoin is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
+Case conversion may be inaccurate. Consider using '#align algebra.subset_adjoin Algebra.subset_adjoinₓ'. -/
 theorem subset_adjoin : s ⊆ adjoin R s :=
   Algebra.gc.le_u_l s
 #align algebra.subset_adjoin Algebra.subset_adjoin
 
+/- warning: algebra.adjoin_le -> Algebra.adjoin_le is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) S)) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A} {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) S)) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) S)
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_le Algebra.adjoin_leₓ'. -/
 theorem adjoin_le {S : Subalgebra R A} (H : s ⊆ S) : adjoin R s ≤ S :=
   Algebra.gc.l_le H
 #align algebra.adjoin_le Algebra.adjoin_le
 
+/- warning: algebra.adjoin_eq_Inf -> Algebra.adjoin_eq_infₛ is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s) (InfSet.infₛ.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (ConditionallyCompleteLattice.toInfSet.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (setOf.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (p : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) p))))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_Inf Algebra.adjoin_eq_infₛₓ'. -/
 theorem adjoin_eq_infₛ : adjoin R s = infₛ { p | s ⊆ p } :=
   le_antisymm (le_infₛ fun _ h => adjoin_le h) (infₛ_le subset_adjoin)
 #align algebra.adjoin_eq_Inf Algebra.adjoin_eq_infₛ
 
+/- warning: algebra.adjoin_le_iff -> Algebra.adjoin_le_iff is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_le_iff Algebra.adjoin_le_iffₓ'. -/
 theorem adjoin_le_iff {S : Subalgebra R A} : adjoin R s ≤ S ↔ s ⊆ S :=
   Algebra.gc _ _
 #align algebra.adjoin_le_iff Algebra.adjoin_le_iff
 
+/- warning: algebra.adjoin_mono -> Algebra.adjoin_mono is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_mono Algebra.adjoin_monoₓ'. -/
 theorem adjoin_mono (H : s ⊆ t) : adjoin R s ≤ adjoin R t :=
   Algebra.gc.monotone_l H
 #align algebra.adjoin_mono Algebra.adjoin_mono
 
+/- warning: algebra.adjoin_eq_of_le -> Algebra.adjoin_eq_of_le is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_of_le Algebra.adjoin_eq_of_leₓ'. -/
 theorem adjoin_eq_of_le (S : Subalgebra R A) (h₁ : s ⊆ S) (h₂ : S ≤ adjoin R s) : adjoin R s = S :=
   le_antisymm (adjoin_le h₁) h₂
 #align algebra.adjoin_eq_of_le Algebra.adjoin_eq_of_le
 
+#print Algebra.adjoin_eq /-
 theorem adjoin_eq (S : Subalgebra R A) : adjoin R ↑S = S :=
   adjoin_eq_of_le _ (Set.Subset.refl _) subset_adjoin
 #align algebra.adjoin_eq Algebra.adjoin_eq
+-/
 
+/- warning: algebra.adjoin_Union -> Algebra.adjoin_unionᵢ is a dubious translation:
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 theorem adjoin_unionᵢ {α : Type _} (s : α → Set A) :
     adjoin R (Set.unionᵢ s) = ⨆ i : α, adjoin R (s i) :=
   (@Algebra.gc R A _ _ _).l_supᵢ
 #align algebra.adjoin_Union Algebra.adjoin_unionᵢ
 
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 theorem adjoin_attach_bunionᵢ [DecidableEq A] {α : Type _} {s : Finset α} (f : s → Finset A) :
     adjoin R (s.attach.bunionᵢ f : Set A) = ⨆ x, adjoin R (f x) := by simpa [adjoin_Union]
 #align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_bunionᵢ
 
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+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_induction Algebra.adjoin_inductionₓ'. -/
 @[elab_as_elim]
 theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : ∀ x ∈ s, p x)
     (Halg : ∀ r, p (algebraMap R A r)) (Hadd : ∀ x y, p x → p y → p (x + y))
@@ -91,6 +147,12 @@ theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : 
   adjoin_le (show s ≤ S from Hs) h
 #align algebra.adjoin_induction Algebra.adjoin_induction
 
+/- warning: algebra.adjoin_induction₂ -> Algebra.adjoin_induction₂ is a dubious translation:
+lean 3 declaration is
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(Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2)) R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2) (RingHom.instRingHomClassRingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (algebraMap.{u1, u2} R A _inst_1 _inst_2 _inst_4) r))) -> (forall (x₁ : A) (x₂ : A) (y : A), (p x₁ y) -> (p x₂ y) -> (p (HAdd.hAdd.{u2, u2, u2} A A A (instHAdd.{u2} A (Distrib.toAdd.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) x₁ x₂) y)) -> (forall (x : A) (y₁ : A) (y₂ : A), (p x y₁) -> (p x y₂) -> (p x (HAdd.hAdd.{u2, u2, u2} A A A (instHAdd.{u2} A (Distrib.toAdd.{u2} A (NonUnitalNonAssocSemiring.toDistrib.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) y₁ y₂))) -> (forall (x₁ : A) (x₂ : A) (y : A), (p x₁ y) -> (p x₂ y) -> (p (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) x₁ x₂) y)) -> (forall (x : A) (y₁ : A) (y₂ : A), (p x y₁) -> (p x y₂) -> (p x (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) y₁ y₂))) -> (p a b)
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_induction₂ Algebra.adjoin_induction₂ₓ'. -/
 /-- Induction principle for the algebra generated by a set `s`: show that `p x y` holds for any
 `x y ∈ adjoin R s` given that that it holds for `x y ∈ s` and that it satisfies a number of
 natural properties. -/
@@ -116,6 +178,12 @@ theorem adjoin_induction₂ {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin
         Hmul_left x y ((algebraMap R A) r)
 #align algebra.adjoin_induction₂ Algebra.adjoin_induction₂
 
+/- warning: algebra.adjoin_induction' -> Algebra.adjoin_induction' is a dubious translation:
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+but is expected to have type
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(Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))), p x)
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_induction' Algebra.adjoin_induction'ₓ'. -/
 /-- The difference with `algebra.adjoin_induction` is that this acts on the subtype. -/
 theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin h⟩)
     (Halg : ∀ r, p (algebraMap R _ r)) (Hadd : ∀ x y, p x → p y → p (x + y))
@@ -134,6 +202,12 @@ theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s),
           Exists.elim hy fun hy' hy => ⟨Subalgebra.mul_mem _ hx' hy', Hmul _ _ hx hy⟩
 #align algebra.adjoin_induction' Algebra.adjoin_induction'
 
+/- warning: algebra.adjoin_adjoin_coe_preimage -> Algebra.adjoin_adjoin_coe_preimage is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_adjoin_coe_preimage Algebra.adjoin_adjoin_coe_preimageₓ'. -/
 @[simp]
 theorem adjoin_adjoin_coe_preimage {s : Set A} : adjoin R ((coe : adjoin R s → A) ⁻¹' s) = ⊤ :=
   by
@@ -146,12 +220,20 @@ theorem adjoin_adjoin_coe_preimage {s : Set A} : adjoin R ((coe : adjoin R s →
   · exact Subalgebra.mul_mem _
 #align algebra.adjoin_adjoin_coe_preimage Algebra.adjoin_adjoin_coe_preimage
 
+#print Algebra.adjoin_union /-
 theorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=
   (Algebra.gc : GaloisConnection _ (coe : Subalgebra R A → Set A)).l_sup
 #align algebra.adjoin_union Algebra.adjoin_union
+-/
 
 variable (R A)
 
+/- warning: algebra.adjoin_empty -> Algebra.adjoin_empty is a dubious translation:
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+  forall (R : Type.{u1}) (A : Type.{u2}) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2], Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (EmptyCollection.emptyCollection.{u2} (Set.{u2} A) (Set.hasEmptyc.{u2} A))) (Bot.bot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toHasBot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4)))
+but is expected to have type
+  forall (R : Type.{u1}) (A : Type.{u2}) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2], Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (EmptyCollection.emptyCollection.{u2} (Set.{u2} A) (Set.instEmptyCollectionSet.{u2} A))) (Bot.bot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toBot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_empty Algebra.adjoin_emptyₓ'. -/
 @[simp]
 theorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=
   show adjoin R ⊥ = ⊥ by
@@ -159,6 +241,12 @@ theorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=
     exact Algebra.gc
 #align algebra.adjoin_empty Algebra.adjoin_empty
 
+/- warning: algebra.adjoin_univ -> Algebra.adjoin_univ is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) (A : Type.{u2}) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2], Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Set.univ.{u2} A)) (Top.top.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toHasTop.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4)))
+but is expected to have type
+  forall (R : Type.{u1}) (A : Type.{u2}) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2], Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Set.univ.{u2} A)) (Top.top.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toTop.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_univ Algebra.adjoin_univₓ'. -/
 @[simp]
 theorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=
   eq_top_iff.2 fun x => subset_adjoin <| Set.mem_univ _
@@ -166,6 +254,12 @@ theorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=
 
 variable (R) {A} (s)
 
+/- warning: algebra.adjoin_eq_span -> Algebra.adjoin_eq_span is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.setLike.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
+but is expected to have type
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span Algebra.adjoin_eq_spanₓ'. -/
 theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s) :=
   by
   apply le_antisymm
@@ -201,30 +295,53 @@ theorem adjoin_eq_span : (adjoin R s).toSubmodule = span R (Submonoid.closure s)
   exact Submonoid.closure_le.2 subset_adjoin
 #align algebra.adjoin_eq_span Algebra.adjoin_eq_span
 
+/- warning: algebra.span_le_adjoin -> Algebra.span_le_adjoin is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
+but is expected to have type
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (s : Set.{u2} A), LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))
+Case conversion may be inaccurate. Consider using '#align algebra.span_le_adjoin Algebra.span_le_adjoinₓ'. -/
 theorem span_le_adjoin (s : Set A) : span R s ≤ (adjoin R s).toSubmodule :=
   span_le.mpr subset_adjoin
 #align algebra.span_le_adjoin Algebra.span_le_adjoin
 
+/- warning: algebra.adjoin_to_submodule_le -> Algebra.adjoin_toSubmodule_le is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A 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+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_to_submodule_le Algebra.adjoin_toSubmodule_leₓ'. -/
 theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
     (adjoin R s).toSubmodule ≤ t ↔ ↑(Submonoid.closure s) ⊆ (t : Set A) := by
   rw [adjoin_eq_span, span_le]
 #align algebra.adjoin_to_submodule_le Algebra.adjoin_toSubmodule_le
 
+/- warning: algebra.adjoin_eq_span_of_subset -> Algebra.adjoin_eq_span_of_subset is a dubious translation:
+lean 3 declaration is
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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
+but is expected to have type
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) (SetLike.coe.{u2, u2} (Submonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) A (Submonoid.instSetLikeSubmonoid.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Submonoid.closure.{u2} A (MulZeroOneClass.toMulOneClass.{u2} A (NonAssocSemiring.toMulZeroOneClass.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) s)) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) A (Submodule.instSetLikeSubmodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (fun (_x : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s)) (Submodule.span.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4) s))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_eq_span_of_subset Algebra.adjoin_eq_span_of_subsetₓ'. -/
 theorem adjoin_eq_span_of_subset {s : Set A} (hs : ↑(Submonoid.closure s) ⊆ (span R s : Set A)) :
     (adjoin R s).toSubmodule = span R s :=
   le_antisymm ((adjoin_toSubmodule_le R).mpr hs) (span_le_adjoin R s)
 #align algebra.adjoin_eq_span_of_subset Algebra.adjoin_eq_span_of_subset
 
+#print Algebra.adjoin_span /-
 @[simp]
 theorem adjoin_span {s : Set A} : adjoin R (Submodule.span R s : Set A) = adjoin R s :=
   le_antisymm (adjoin_le (span_le_adjoin _ _)) (adjoin_mono Submodule.subset_span)
 #align algebra.adjoin_span Algebra.adjoin_span
+-/
 
+#print Algebra.adjoin_image /-
 theorem adjoin_image (f : A →ₐ[R] B) (s : Set A) : adjoin R (f '' s) = (adjoin R s).map f :=
   le_antisymm (adjoin_le <| Set.image_subset _ subset_adjoin) <|
     Subalgebra.map_le.2 <| adjoin_le <| Set.image_subset_iff.1 subset_adjoin
 #align algebra.adjoin_image Algebra.adjoin_image
+-/
 
+#print Algebra.adjoin_insert_adjoin /-
 @[simp]
 theorem adjoin_insert_adjoin (x : A) : adjoin R (insert x ↑(adjoin R s)) = adjoin R (insert x s) :=
   le_antisymm
@@ -233,13 +350,26 @@ theorem adjoin_insert_adjoin (x : A) : adjoin R (insert x ↑(adjoin R s)) = adj
         ⟨subset_adjoin (Set.mem_insert _ _), adjoin_mono (Set.subset_insert _ _)⟩))
     (Algebra.adjoin_mono (Set.insert_subset_insert Algebra.subset_adjoin))
 #align algebra.adjoin_insert_adjoin Algebra.adjoin_insert_adjoin
+-/
 
+/- warning: algebra.adjoin_prod_le -> Algebra.adjoin_prod_le is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_prod_le Algebra.adjoin_prod_leₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem adjoin_prod_le (s : Set A) (t : Set B) :
     adjoin R (s ×ˢ t) ≤ (adjoin R s).Prod (adjoin R t) :=
   adjoin_le <| Set.prod_mono subset_adjoin subset_adjoin
 #align algebra.adjoin_prod_le Algebra.adjoin_prod_le
 
+/- warning: algebra.mem_adjoin_of_map_mul -> Algebra.mem_adjoin_of_map_mul is a dubious translation:
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+but is expected to have type
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f) (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) (OfNat.ofNat.{u2} A 1 (One.toOfNat1.{u2} A (Semiring.toOne.{u2} A _inst_2))))))))
+Case conversion may be inaccurate. Consider using '#align algebra.mem_adjoin_of_map_mul Algebra.mem_adjoin_of_map_mulₓ'. -/
 theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a₂, f (a₁ * a₂) = f a₁ * f a₂)
     (h : x ∈ adjoin R s) : f x ∈ adjoin R (f '' (s ∪ {1})) :=
   by
@@ -256,6 +386,12 @@ theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a
   exact f.map_smul _ _
 #align algebra.mem_adjoin_of_map_mul Algebra.mem_adjoin_of_map_mul
 
+/- warning: algebra.adjoin_inl_union_inr_eq_prod -> Algebra.adjoin_inl_union_inr_eq_prod is a dubious translation:
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(Algebra.adjoin.{u1, u3} R B _inst_1 _inst_3 _inst_5 t))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_inl_union_inr_eq_prod Algebra.adjoin_inl_union_inr_eq_prodₓ'. -/
 theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     adjoin R (LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1})) =
       (adjoin R s).Prod (adjoin R t) :=
@@ -279,6 +415,12 @@ theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     simpa using Subalgebra.add_mem _ Ha Hb
 #align algebra.adjoin_inl_union_inr_eq_prod Algebra.adjoin_inl_union_inr_eq_prod
 
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+but is expected to have type
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] {s : Set.{u2} A}, (forall (a : A), (Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) a s) -> (forall (b : A), (Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) b s) -> (Eq.{succ u2} A (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) a b) (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) b a)))) -> (CommSemiring.{u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 s))))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_comm_semiring_of_comm Algebra.adjoinCommSemiringOfCommₓ'. -/
 /-- If all elements of `s : set A` commute pairwise, then `adjoin R s` is a commutative
 semiring.  -/
 def adjoinCommSemiringOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) :
@@ -296,10 +438,22 @@ def adjoinCommSemiringOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a *
           fun x₁ x₂ y₁ h₁ h₂ => by rw [mul_assoc x₂, ← h₂, ← mul_assoc x₂, ← h₁, ← mul_assoc] }
 #align algebra.adjoin_comm_semiring_of_comm Algebra.adjoinCommSemiringOfComm
 
+/- warning: algebra.adjoin_singleton_one -> Algebra.adjoin_singleton_one is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2], Eq.{succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) (OfNat.ofNat.{u2} A 1 (OfNat.mk.{u2} A 1 (One.one.{u2} A (AddMonoidWithOne.toOne.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))))))) (Bot.bot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (CompleteLattice.toHasBot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_4)))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_singleton_one Algebra.adjoin_singleton_oneₓ'. -/
 theorem adjoin_singleton_one : adjoin R ({1} : Set A) = ⊥ :=
   eq_bot_iff.2 <| adjoin_le <| Set.singleton_subset_iff.2 <| SetLike.mem_coe.2 <| one_mem _
 #align algebra.adjoin_singleton_one Algebra.adjoin_singleton_one
 
+/- warning: algebra.self_mem_adjoin_singleton -> Algebra.self_mem_adjoin_singleton is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (x : A), Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) x))
+but is expected to have type
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_4 : Algebra.{u1, u2} R A _inst_1 _inst_2] (x : A), Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_4)) x (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) x))
+Case conversion may be inaccurate. Consider using '#align algebra.self_mem_adjoin_singleton Algebra.self_mem_adjoin_singletonₓ'. -/
 theorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=
   Algebra.subset_adjoin (Set.mem_singleton_iff.mpr rfl)
 #align algebra.self_mem_adjoin_singleton Algebra.self_mem_adjoin_singleton
@@ -314,6 +468,7 @@ variable [Algebra R A] {s t : Set A}
 
 variable (R s t)
 
+#print Algebra.adjoin_union_eq_adjoin_adjoin /-
 theorem adjoin_union_eq_adjoin_adjoin :
     adjoin R (s ∪ t) = (adjoin (adjoin R s) t).restrictScalars R :=
   le_antisymm
@@ -324,7 +479,14 @@ theorem adjoin_union_eq_adjoin_adjoin :
       Set.union_subset (Set.range_subset_iff.2 fun x => adjoin_mono (Set.subset_union_left _ _) x.2)
         (Set.Subset.trans (Set.subset_union_right _ _) subset_adjoin))
 #align algebra.adjoin_union_eq_adjoin_adjoin Algebra.adjoin_union_eq_adjoin_adjoin
+-/
 
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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
+but is expected to have type
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] (s : Set.{u2} A) (t : Set.{u2} A), Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s t))) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 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(CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) _x) (EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))) (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A 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(CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Algebra.adjoin.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3 t)))
+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_union_coe_submodule Algebra.adjoin_union_coe_submoduleₓ'. -/
 theorem adjoin_union_coe_submodule :
     (adjoin R (s ∪ t)).toSubmodule = (adjoin R s).toSubmodule * (adjoin R t).toSubmodule :=
   by
@@ -332,6 +494,7 @@ theorem adjoin_union_coe_submodule :
   congr 1 with z; simp [Submonoid.closure_union, Submonoid.mem_sup, Set.mem_mul]
 #align algebra.adjoin_union_coe_submodule Algebra.adjoin_union_coe_submodule
 
+#print Algebra.adjoin_adjoin_of_tower /-
 theorem adjoin_adjoin_of_tower [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
     (s : Set B) : adjoin A (adjoin R s : Set B) = adjoin A s :=
   by
@@ -341,9 +504,16 @@ theorem adjoin_adjoin_of_tower [Semiring B] [Algebra R B] [Algebra A B] [IsScala
     refine' adjoin_le _
     exact subset_adjoin
 #align algebra.adjoin_adjoin_of_tower Algebra.adjoin_adjoin_of_tower
+-/
 
 variable {R}
 
+/- warning: algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin -> Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] [_inst_4 : CommSemiring.{u3} B] [_inst_5 : Algebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_6 : Algebra.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_7 : 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 (CommSemiring.toSemiring.{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 _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 (CommSemiring.toSemiring.{u2} A _inst_2))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{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 (CommSemiring.toSemiring.{u2} A _inst_2))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3))))) (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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (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 _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 (CommSemiring.toSemiring.{u3} B _inst_4))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A 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_inst_4))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) (algebraMap.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6) r) B') -> (Exists.{1} Nat (fun (n₀ : Nat) => forall (n : Nat), (GE.ge.{0} Nat Nat.hasLe n n₀) -> (Membership.Mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.hasMem.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.setLike.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) (SMul.smul.{u2, u3} A B (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 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_inst_4))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))) (Algebra.toModule.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6))))) (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2))))) r n) x) B'))))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] [_inst_4 : CommSemiring.{u3} B] [_inst_5 : Algebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_6 : Algebra.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4)] [_inst_7 : IsScalarTower.{u1, u2, u3} R A B (Algebra.toSMul.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Algebra.toSMul.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6) (Algebra.toSMul.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)] (r : A) (s : Set.{u3} B) (B' : Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5), (HasSubset.Subset.{u3} (Set.{u3} B) (Set.instHasSubsetSet.{u3} B) (HSMul.hSMul.{u2, u3, u3} A (Set.{u3} B) (Set.{u3} B) (instHSMul.{u2, u3} A (Set.{u3} B) (Set.smulSet.{u2, u3} A B (Algebra.toSMul.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6))) r s) (SetLike.coe.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B')) -> (forall {x : B}, (Membership.mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) x (Algebra.adjoin.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5 s)) -> (Membership.mem.{u3, u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : A) => B) r) (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : A) => B) _x) (MulHomClass.toFunLike.{max u2 u3, u2, u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A B (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u3, u2, u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u3, u2, u3} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4))) A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)) (Semiring.toNonAssocSemiring.{u3} B (CommSemiring.toSemiring.{u3} B _inst_4)))))) (algebraMap.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6) r) B') -> (Exists.{1} Nat (fun (n₀ : Nat) => forall (n : Nat), (GE.ge.{0} Nat instLENat n n₀) -> (Membership.mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_5)) (HSMul.hSMul.{u2, u3, u3} A B B (instHSMul.{u2, u3} A B (Algebra.toSMul.{u2, u3} A B _inst_2 (CommSemiring.toSemiring.{u3} B _inst_4) _inst_6)) (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2))))) r n) x) B'))))
+Case conversion may be inaccurate. Consider using '#align algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoinₓ'. -/
 theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]
     [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
     (hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B' :=
@@ -369,6 +539,12 @@ theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B]
   apply Submonoid.closure_mono hs (n₂ a)
 #align algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin
 
+/- warning: algebra.pow_smul_mem_adjoin_smul -> Algebra.pow_smul_mem_adjoin_smul is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align algebra.pow_smul_mem_adjoin_smul Algebra.pow_smul_mem_adjoin_smulₓ'. -/
 theorem pow_smul_mem_adjoin_smul (r : R) (s : Set A) {x : A} (hx : x ∈ adjoin R s) :
     ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ adjoin R (r • s) :=
   pow_smul_mem_of_smul_subset_of_mem_adjoin r s _ subset_adjoin hx (Subalgebra.algebraMap_mem _ _)
@@ -384,11 +560,23 @@ variable [Algebra R A] {s t : Set A}
 
 variable {R s t}
 
+/- warning: algebra.adjoin_int -> Algebra.adjoin_int is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_int Algebra.adjoin_intₓ'. -/
 theorem adjoin_int (s : Set R) : adjoin ℤ s = subalgebraOfSubring (Subring.closure s) :=
   le_antisymm (adjoin_le Subring.subset_closure)
     (Subring.closure_le.2 subset_adjoin : Subring.closure s ≤ (adjoin ℤ s).toSubring)
 #align algebra.adjoin_int Algebra.adjoin_int
 
+/- warning: algebra.mem_adjoin_iff -> Algebra.mem_adjoin_iff is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align algebra.mem_adjoin_iff Algebra.mem_adjoin_iffₓ'. -/
 theorem mem_adjoin_iff {s : Set A} {x : A} :
     x ∈ adjoin R s ↔ x ∈ Subring.closure (Set.range (algebraMap R A) ∪ s) :=
   ⟨fun hx =>
@@ -398,13 +586,21 @@ theorem mem_adjoin_iff {s : Set A} {x : A} :
     Subring.closure_le.2 Subsemiring.subset_closure⟩
 #align algebra.mem_adjoin_iff Algebra.mem_adjoin_iff
 
+#print Algebra.adjoin_eq_ring_closure /-
 theorem adjoin_eq_ring_closure (s : Set A) :
     (adjoin R s).toSubring = Subring.closure (Set.range (algebraMap R A) ∪ s) :=
   Subring.ext fun x => mem_adjoin_iff
 #align algebra.adjoin_eq_ring_closure Algebra.adjoin_eq_ring_closure
+-/
 
 variable (R)
 
+/- warning: algebra.adjoin_comm_ring_of_comm -> Algebra.adjoinCommRingOfComm is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align algebra.adjoin_comm_ring_of_comm Algebra.adjoinCommRingOfCommₓ'. -/
 /-- If all elements of `s : set A` commute pairwise, then `adjoin R s` is a commutative
 ring.  -/
 def adjoinCommRingOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) :
@@ -422,15 +618,25 @@ namespace AlgHom
 
 variable [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
 
+#print AlgHom.map_adjoin /-
 theorem map_adjoin (φ : A →ₐ[R] B) (s : Set A) : (adjoin R s).map φ = adjoin R (φ '' s) :=
   (adjoin_image _ _ _).symm
 #align alg_hom.map_adjoin AlgHom.map_adjoin
+-/
 
+#print AlgHom.adjoin_le_equalizer /-
 theorem adjoin_le_equalizer (φ₁ φ₂ : A →ₐ[R] B) {s : Set A} (h : s.EqOn φ₁ φ₂) :
     adjoin R s ≤ φ₁.equalizer φ₂ :=
   adjoin_le h
 #align alg_hom.adjoin_le_equalizer AlgHom.adjoin_le_equalizer
+-/
 
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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A 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_inst_2 _inst_3 _inst_4 _inst_5 (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) (AlgHom.algHomClass.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5))))) φ₁) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) _x) (SMulHomClass.toFunLike.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (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 _inst_2))))) (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 _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4))))) (SMulZeroClass.toSMul.{u1, u3} R B (AddMonoid.toZero.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))))) (DistribSMul.toSMulZeroClass.{u1, u3} R B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))))) (DistribMulAction.toDistribSMul.{u1, u3} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)))) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u2 u3, u1, u2, u3} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3)) (Module.toDistribMulAction.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_4)) (Module.toDistribMulAction.{u1, u3} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B _inst_3))) (Algebra.toModule.{u1, u3} R B _inst_1 _inst_3 _inst_5)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u2, u3, max u2 u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) (AlgHom.algHomClass.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5))))) φ₂) s) -> (Eq.{max (succ u2) (succ u3)} (AlgHom.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) φ₁ φ₂))
+Case conversion may be inaccurate. Consider using '#align alg_hom.ext_of_adjoin_eq_top AlgHom.ext_of_adjoin_eq_topₓ'. -/
 theorem ext_of_adjoin_eq_top {s : Set A} (h : adjoin R s = ⊤) ⦃φ₁ φ₂ : A →ₐ[R] B⦄
     (hs : s.EqOn φ₁ φ₂) : φ₁ = φ₂ :=
   ext fun x => adjoin_le_equalizer φ₁ φ₂ hs <| h.symm ▸ trivial

a35ddf20

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

a35ddf20

chore(Algebra/Algebra): split Subalgebra.Basic (#12267)

This PR was supposed to be simultaneous with #12090 but I got ill last week.

This is based on seeing the import Algebra.Algebra.Subalgebra.Basic → RingTheory.Ideal.Operations on the longest pole. It feels like Ideal.Operations should not be needed to define the notion of subalgebra, only to construct some interesting examples. So I removed the import and split off anything that wouldn't fit.

The following results and their corollaries were split off:

Subalgebra.prod
Subalgebra.iSupLift
AlgHom.ker_rangeRestrict
Subalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

4e9bd1d4

Diff

@@ -4,10 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 -/
 import Mathlib.Algebra.Algebra.Operations
+import Mathlib.Algebra.Algebra.Subalgebra.Prod
 import Mathlib.Algebra.Algebra.Subalgebra.Tower
 import Mathlib.LinearAlgebra.Basis
 import Mathlib.LinearAlgebra.Prod
 import Mathlib.LinearAlgebra.Finsupp
+import Mathlib.LinearAlgebra.Prod
 
 #align_import ring_theory.adjoin.basic from "leanprover-community/mathlib"@"a35ddf20601f85f78cd57e7f5b09ed528d71b7af"

a35ddf20

chore: small splits of RingTheory.Ideal.Operations; clean imports (#12090)

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

The following results and their corollaries were split off:

Ideal.basisSpanSingleton
Basis.mem_ideal_iff
Ideal.colon

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

46c2efe1

Diff

@@ -5,6 +5,7 @@ Authors: Kenny Lau
 -/
 import Mathlib.Algebra.Algebra.Operations
 import Mathlib.Algebra.Algebra.Subalgebra.Tower
+import Mathlib.LinearAlgebra.Basis
 import Mathlib.LinearAlgebra.Prod
 import Mathlib.LinearAlgebra.Finsupp

a35ddf20

feat: add induction principle for elementalStarAlgebra (#12039)

61447777

Diff

@@ -127,6 +127,16 @@ theorem adjoin_induction' {p : adjoin R s → Prop} (mem : ∀ (x) (h : x ∈ s)
           Exists.elim hy fun hy' hy => ⟨Subalgebra.mul_mem _ hx' hy', mul _ _ hx hy⟩
 #align algebra.adjoin_induction' Algebra.adjoin_induction'
 
+@[elab_as_elim]
+theorem adjoin_induction'' {x : A} (hx : x ∈ adjoin R s)
+    {p : (x : A) → x ∈ adjoin R s → Prop} (mem : ∀ x (h : x ∈ s), p x (subset_adjoin h))
+    (algebraMap : ∀ (r : R), p (algebraMap R A r) (algebraMap_mem _ r))
+    (add : ∀ x hx y hy, p x hx → p y hy → p (x + y) (add_mem hx hy))
+    (mul : ∀ x hx y hy, p x hx → p y hy → p (x * y) (mul_mem hx hy)) :
+    p x hx := by
+  refine adjoin_induction' mem algebraMap ?_ ?_ ⟨x, hx⟩ (p := fun x : adjoin R s ↦ p x.1 x.2)
+  exacts [fun x y ↦ add x.1 x.2 y.1 y.2, fun x y ↦ mul x.1 x.2 y.1 y.2]
+
 @[simp]
 theorem adjoin_adjoin_coe_preimage {s : Set A} : adjoin R (((↑) : adjoin R s → A) ⁻¹' s) = ⊤ := by
   refine eq_top_iff.2 fun x ↦

a35ddf20

chore: resolve a few porting notes could not infer motive (#11302)

These are not all; most of these porting notes are still real.

aad754f5

Diff

@@ -129,11 +129,8 @@ theorem adjoin_induction' {p : adjoin R s → Prop} (mem : ∀ (x) (h : x ∈ s)
 
 @[simp]
 theorem adjoin_adjoin_coe_preimage {s : Set A} : adjoin R (((↑) : adjoin R s → A) ⁻¹' s) = ⊤ := by
-  refine'
-    eq_top_iff.2 fun x =>
-      -- Porting note: Lean could no longer infer the motive
-      adjoin_induction' (p := fun y => y ∈ adjoin R (((↑) : adjoin R s → A) ⁻¹' s))
-      (fun a ha => _) (fun r => _) (fun _ _ => _) (fun _ _ => _) x
+  refine eq_top_iff.2 fun x ↦
+      adjoin_induction' (fun a ha ↦ ?_) (fun r ↦ ?_) (fun _ _ ↦ ?_) (fun _ _ ↦ ?_) x
   · exact subset_adjoin ha
   · exact Subalgebra.algebraMap_mem _ r
   · exact Subalgebra.add_mem _

a35ddf20

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

Empty lines were removed by executing the following Python script twice

import os
import re


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

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

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

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

75c86f04

Diff

@@ -36,7 +36,6 @@ namespace Algebra
 section Semiring
 
 variable [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]
-
 variable [Algebra R S] [Algebra R A] [Algebra S A] [Algebra R B] [IsScalarTower R S A]
 variable {s t : Set A}
 
@@ -342,9 +341,7 @@ end Semiring
 section CommSemiring
 
 variable [CommSemiring R] [CommSemiring A]
-
 variable [Algebra R A] {s t : Set A}
-
 variable (R s t)
 
 theorem adjoin_union_eq_adjoin_adjoin :
@@ -395,7 +392,6 @@ end CommSemiring
 section Ring
 
 variable [CommRing R] [Ring A]
-
 variable [Algebra R A] {s t : Set A}
 
 theorem mem_adjoin_iff {s : Set A} {x : A} :

a35ddf20

chore(Algebra): improve argument names to induction principles (#11439)

These are the case names used by the induction tactic after the with.

This replaces H0, H1, Hmul etc with zero, one, mul.

This PR does not touch Submonoid or Subgroup, as to_additive does not know how to rename the argument names. There are ways to work around this, but I'd prefer to leave them to a later PR.

This also leaves the closure_induction₂ variants alone, as renaming the arguments is more work for less gain.

9d3f2bf8

Diff

@@ -79,15 +79,15 @@ theorem adjoin_attach_biUnion [DecidableEq A] {α : Type*} {s : Finset α} (f :
 #align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_biUnion
 
 @[elab_as_elim]
-theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : ∀ x ∈ s, p x)
-    (Halg : ∀ r, p (algebraMap R A r)) (Hadd : ∀ x y, p x → p y → p (x + y))
-    (Hmul : ∀ x y, p x → p y → p (x * y)) : p x :=
+theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (mem : ∀ x ∈ s, p x)
+    (algebraMap : ∀ r, p (algebraMap R A r)) (add : ∀ x y, p x → p y → p (x + y))
+    (mul : ∀ x y, p x → p y → p (x * y)) : p x :=
   let S : Subalgebra R A :=
     { carrier := p
-      mul_mem' := Hmul _ _
-      add_mem' := Hadd _ _
-      algebraMap_mem' := Halg }
-  adjoin_le (show s ≤ S from Hs) h
+      mul_mem' := mul _ _
+      add_mem' := add _ _
+      algebraMap_mem' := algebraMap }
+  adjoin_le (show s ≤ S from mem) h
 #align algebra.adjoin_induction Algebra.adjoin_induction
 
 /-- Induction principle for the algebra generated by a set `s`: show that `p x y` holds for any
@@ -113,19 +113,19 @@ theorem adjoin_induction₂ {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin
 
 /-- The difference with `Algebra.adjoin_induction` is that this acts on the subtype. -/
 @[elab_as_elim]
-theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin h⟩)
-    (Halg : ∀ r, p (algebraMap R _ r)) (Hadd : ∀ x y, p x → p y → p (x + y))
-    (Hmul : ∀ x y, p x → p y → p (x * y)) (x : adjoin R s) : p x :=
+theorem adjoin_induction' {p : adjoin R s → Prop} (mem : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin h⟩)
+    (algebraMap : ∀ r, p (algebraMap R _ r)) (add : ∀ x y, p x → p y → p (x + y))
+    (mul : ∀ x y, p x → p y → p (x * y)) (x : adjoin R s) : p x :=
   Subtype.recOn x fun x hx => by
     refine' Exists.elim _ fun (hx : x ∈ adjoin R s) (hc : p ⟨x, hx⟩) => hc
-    exact adjoin_induction hx (fun x hx => ⟨subset_adjoin hx, Hs x hx⟩)
-      (fun r => ⟨Subalgebra.algebraMap_mem _ r, Halg r⟩)
+    exact adjoin_induction hx (fun x hx => ⟨subset_adjoin hx, mem x hx⟩)
+      (fun r => ⟨Subalgebra.algebraMap_mem _ r, algebraMap r⟩)
       (fun x y hx hy =>
         Exists.elim hx fun hx' hx =>
-          Exists.elim hy fun hy' hy => ⟨Subalgebra.add_mem _ hx' hy', Hadd _ _ hx hy⟩)
+          Exists.elim hy fun hy' hy => ⟨Subalgebra.add_mem _ hx' hy', add _ _ hx hy⟩)
       fun x y hx hy =>
         Exists.elim hx fun hx' hx =>
-          Exists.elim hy fun hy' hy => ⟨Subalgebra.mul_mem _ hx' hy', Hmul _ _ hx hy⟩
+          Exists.elim hy fun hy' hy => ⟨Subalgebra.mul_mem _ hx' hy', mul _ _ hx hy⟩
 #align algebra.adjoin_induction' Algebra.adjoin_induction'
 
 @[simp]

a35ddf20

style: homogenise porting notes (#11145)

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".

8be0b69c

Diff

@@ -132,7 +132,7 @@ theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s),
 theorem adjoin_adjoin_coe_preimage {s : Set A} : adjoin R (((↑) : adjoin R s → A) ⁻¹' s) = ⊤ := by
   refine'
     eq_top_iff.2 fun x =>
-      -- porting note: Lean could no longer infer the motive
+      -- Porting note: Lean could no longer infer the motive
       adjoin_induction' (p := fun y => y ∈ adjoin R (((↑) : adjoin R s → A) ⁻¹' s))
       (fun a ha => _) (fun r => _) (fun _ _ => _) (fun _ _ => _) x
   · exact subset_adjoin ha
@@ -215,7 +215,7 @@ theorem adjoin_span {s : Set A} : adjoin R (Submodule.span R s : Set A) = adjoin
 theorem adjoin_image (f : A →ₐ[R] B) (s : Set A) : adjoin R (f '' s) = (adjoin R s).map f :=
   le_antisymm (adjoin_le <| Set.image_subset _ subset_adjoin) <|
     Subalgebra.map_le.2 <| adjoin_le <| Set.image_subset_iff.1 <| by
-      -- porting note: I don't understand how this worked in Lean 3 with just `subset_adjoin`
+      -- Porting note: I don't understand how this worked in Lean 3 with just `subset_adjoin`
       simp only [Set.image_id', coe_carrier_toSubmonoid, Subalgebra.coe_toSubsemiring,
         Subalgebra.coe_comap]
       exact fun x hx => subset_adjoin ⟨x, hx, rfl⟩
@@ -317,7 +317,7 @@ theorem adjoin_adjoin_of_tower (s : Set A) : adjoin S (adjoin R s : Set A) = adj
   · exact adjoin_mono subset_adjoin
   · change adjoin R s ≤ (adjoin S s).restrictScalars R
     refine' adjoin_le _
-    -- porting note: unclear why this was broken
+    -- Porting note: unclear why this was broken
     have : (Subalgebra.restrictScalars R (adjoin S s) : Set A) = adjoin S s := rfl
     rw [this]
     exact subset_adjoin
@@ -405,7 +405,7 @@ theorem mem_adjoin_iff {s : Set A} {x : A} :
       (fun _ _ => Subring.add_mem _) fun _ _ => Subring.mul_mem _,
     suffices Subring.closure (Set.range (algebraMap R A) ∪ s) ≤ (adjoin R s).toSubring
       from (show (_ : Set A) ⊆ _ from this) (a := x)
-    -- porting note: Lean doesn't seem to recognize the defeq between the order on subobjects and
+    -- Porting note: Lean doesn't seem to recognize the defeq between the order on subobjects and
     -- subsets of their coercions to sets as easily as in Lean 3
     Subring.closure_le.2 Subsemiring.subset_closure⟩
 #align algebra.mem_adjoin_iff Algebra.mem_adjoin_iff

a35ddf20

chore: more backporting of simp changes from #10995 (#11001)

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

d9589c14

Diff

@@ -267,7 +267,7 @@ theorem adjoin_inl_union_inr_eq_prod (s) (t) :
       mem_adjoin_of_map_mul R LinearMap.inr_map_mul hb
     replace Ha : (a, (0 : B)) ∈ P := adjoin_mono (Set.subset_union_left _ _) Ha
     replace Hb : ((0 : A), b) ∈ P := adjoin_mono (Set.subset_union_right _ _) Hb
-    simpa using Subalgebra.add_mem _ Ha Hb
+    simpa [P] using Subalgebra.add_mem _ Ha Hb
 #align algebra.adjoin_inl_union_inr_eq_prod Algebra.adjoin_inl_union_inr_eq_prod
 
 /-- If all elements of `s : Set A` commute pairwise, then `adjoin R s` is a commutative

a35ddf20

chore: bump aesop; update syntax (#10955)

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

46974e92

Diff

@@ -40,7 +40,7 @@ variable [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]
 variable [Algebra R S] [Algebra R A] [Algebra S A] [Algebra R B] [IsScalarTower R S A]
 variable {s t : Set A}
 
-@[aesop safe 20 apply (rule_sets [SetLike])]
+@[aesop safe 20 apply (rule_sets := [SetLike])]
 theorem subset_adjoin : s ⊆ adjoin R s :=
   Algebra.gc.le_u_l s
 #align algebra.subset_adjoin Algebra.subset_adjoin

a35ddf20

feat({Ring|Field}Theory/Adjoin): add various results on adjoin (#9790)

Main changes:

for intermediate fields: add extendScalars_adjoin, restrictScalars_adjoin_of_algEquiv, sup_toSubalgebra_of_isAlgebraic, adjoin_toSubalgebra_of_isAlgebraic, adjoin_rank_le_of_isAlgebraic, and golf sup_toSubalgebra
for subalgebras: add adjoin_eq_span_of_eq_span, adjoin_eq_span_basis, restrictScalars_adjoin, restrictScalars_adjoin_of_algEquiv, adjoin_rank_le

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

50388bcc

Diff

@@ -480,4 +480,52 @@ def Subring.closureEquivAdjoinInt {R : Type*} [Ring R] (s : Set R) :
     Subring.closure s ≃ₐ[ℤ] Algebra.adjoin ℤ s :=
   Subalgebra.equivOfEq (subalgebraOfSubring <| Subring.closure s) _ (adjoin_int s).symm
 
-  end NatInt
+end NatInt
+
+section
+
+variable (F E : Type*) {K : Type*} [CommSemiring E] [Semiring K] [SMul F E] [Algebra E K]
+
+/-- If `K / E / F` is a ring extension tower, `L` is a submonoid of `K / F` which is generated by
+`S` as an `F`-module, then `E[L]` is generated by `S` as an `E`-module. -/
+theorem Submonoid.adjoin_eq_span_of_eq_span [Semiring F] [Module F K] [IsScalarTower F E K]
+    (L : Submonoid K) {S : Set K} (h : (L : Set K) = span F S) :
+    toSubmodule (adjoin E (L : Set K)) = span E S := by
+  rw [adjoin_eq_span, L.closure_eq, h]
+  exact (span_le.mpr <| span_subset_span _ _ _).antisymm (span_mono subset_span)
+
+variable [CommSemiring F] [Algebra F K] [IsScalarTower F E K] (L : Subalgebra F K) {F}
+
+/-- If `K / E / F` is a ring extension tower, `L` is a subalgebra of `K / F` which is generated by
+`S` as an `F`-module, then `E[L]` is generated by `S` as an `E`-module. -/
+theorem Subalgebra.adjoin_eq_span_of_eq_span {S : Set K} (h : toSubmodule L = span F S) :
+    toSubmodule (adjoin E (L : Set K)) = span E S :=
+  L.toSubmonoid.adjoin_eq_span_of_eq_span F E (congr_arg ((↑) : _ → Set K) h)
+
+/-- If `K / E / F` is a ring extension tower, `L` is a subalgebra of `K / F`,
+then `E[L]` is generated by any basis of `L / F` as an `E`-module. -/
+theorem Subalgebra.adjoin_eq_span_basis {ι : Type*} (bL : Basis ι F L) :
+    toSubmodule (adjoin E (L : Set K)) = span E (Set.range fun i : ι ↦ (bL i).1) :=
+  L.adjoin_eq_span_of_eq_span E <| by
+    simpa only [← L.range_val, Submodule.map_span, Submodule.map_top, ← Set.range_comp]
+      using congr_arg (Submodule.map L.val) bL.span_eq.symm
+
+theorem Algebra.restrictScalars_adjoin (F : Type*) [CommSemiring F] {E : Type*} [CommSemiring E]
+    [Algebra F E] (K : Subalgebra F E) (S : Set E) :
+    (Algebra.adjoin K S).restrictScalars F = Algebra.adjoin F (K ∪ S) := by
+  conv_lhs => rw [← Algebra.adjoin_eq K, ← Algebra.adjoin_union_eq_adjoin_adjoin]
+
+/-- If `E / L / F` and `E / L' / F` are two ring extension towers, `L ≃ₐ[F] L'` is an isomorphism
+compatible with `E / L` and `E / L'`, then for any subset `S` of `E`, `L[S]` and `L'[S]` are
+equal as subalgebras of `E / F`. -/
+theorem Algebra.restrictScalars_adjoin_of_algEquiv
+    {F E L L' : Type*} [CommSemiring F] [CommSemiring L] [CommSemiring L'] [Semiring E]
+    [Algebra F L] [Algebra L E] [Algebra F L'] [Algebra L' E] [Algebra F E]
+    [IsScalarTower F L E] [IsScalarTower F L' E] (i : L ≃ₐ[F] L')
+    (hi : algebraMap L E = (algebraMap L' E) ∘ i) (S : Set E) :
+    (Algebra.adjoin L S).restrictScalars F = (Algebra.adjoin L' S).restrictScalars F := by
+  apply_fun Subalgebra.toSubsemiring using fun K K' h ↦ by rwa [SetLike.ext'_iff] at h ⊢
+  change Subsemiring.closure _ = Subsemiring.closure _
+  erw [hi, Set.range_comp, i.toEquiv.range_eq_univ, Set.image_univ]
+
+end

a35ddf20

chore: discharge porting note about 'change ... at' (#9018)

RingTheory.Adjoin.Basic was ported in #2817. change ... at was added in #2836.

bb973d5f

Diff

@@ -364,8 +364,7 @@ variable {R}
 theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]
     [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
     (hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B' := by
-  -- porting note: use `replace` because we don't have `change ... at` yet
-  replace hx : x ∈ Subalgebra.toSubmodule (adjoin R s) := hx
+  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
   rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
   rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
   choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop

a35ddf20

feat: Add AlgHom.map_adjoin_singleton. (#8655)

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

313ebb43

Diff

@@ -439,6 +439,11 @@ theorem map_adjoin (φ : A →ₐ[R] B) (s : Set A) : (adjoin R s).map φ = adjo
   (adjoin_image _ _ _).symm
 #align alg_hom.map_adjoin AlgHom.map_adjoin
 
+@[simp]
+theorem map_adjoin_singleton (e : A →ₐ[R] B) (x : A) :
+    (adjoin R {x}).map e = adjoin R {e x} := by
+  rw [map_adjoin, Set.image_singleton]
+
 theorem adjoin_le_equalizer (φ₁ φ₂ : A →ₐ[R] B) {s : Set A} (h : s.EqOn φ₁ φ₂) :
     adjoin R s ≤ φ₁.equalizer φ₂ :=
   adjoin_le h

a35ddf20

feat: add a SetLike default rule set for aesop (#7111)

This creates a new aesop rule set called SetLike to house lemmas about membership in subobjects.

Lemmas like pow_mem should be included in the rule set:

@[to_additive (attr := aesop safe apply (rule_sets [SetLike]))]
theorem pow_mem {M A} [Monoid M] [SetLike A M] [SubmonoidClass A M] {S : A} {x : M}
(hx : x ∈ S) : ∀ n : ℕ, x ^ n ∈ S

Lemmas about closures, like AddSubmonoid.closure should be included in the rule set, but they should be assigned a penalty (here we choose 20 throughout) so that they are not attempted before the general purpose ones like pow_mem.

@[to_additive (attr := simp, aesop safe 20 apply (rule_sets [SetLike]))
  "The `AddSubmonoid` generated by a set includes the set."]
theorem subset_closure : s ⊆ closure s := fun _ hx => mem_closure.2 fun _ hS => hS hx

In order for aesop to make effective use of AddSubmonoid.closure it needs the following new lemma.

@[aesop 5% apply (rule_sets [SetLike])]
lemma mem_of_subset {s : Set B} (hp : s ⊆ p) {x : B} (hx : x ∈ s) : x ∈ p := hp hx

Note: this lemma is marked as very unsafe (5%) because it will apply whenever the goal is of the form x ∈ p where p is any term of a SetLike instance; and moreover, it will create s as a metavariable, which is in general a terrible idea, but necessary for the reason mentioned above.

214e72fd

Diff

@@ -40,6 +40,7 @@ variable [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]
 variable [Algebra R S] [Algebra R A] [Algebra S A] [Algebra R B] [IsScalarTower R S A]
 variable {s t : Set A}
 
+@[aesop safe 20 apply (rule_sets [SetLike])]
 theorem subset_adjoin : s ⊆ adjoin R s :=
   Algebra.gc.le_u_l s
 #align algebra.subset_adjoin Algebra.subset_adjoin

a35ddf20

chore: remove many Type _ before the colon (#7718)

We have turned to Type* instead of Type _, but many of them remained in mathlib because the straight replacement did not work. In general, having Type _ before the colon is a code smell, though, as it hides which types should be in the same universe and which shouldn't, and is not very robust.

This PR replaces most of the remaining Type _ before the colon (except those in category theory) by Type* or Type u. This has uncovered a few bugs (where declarations were not as polymorphic as they should be).

I had to increase heartbeats at two places when replacing Type _ by Type*, but I think it's worth it as it's really more robust.

fe9572d2

Diff

@@ -452,12 +452,12 @@ end AlgHom
 
 section NatInt
 
-theorem Algebra.adjoin_nat {R : Type _} [Semiring R] (s : Set R) :
+theorem Algebra.adjoin_nat {R : Type*} [Semiring R] (s : Set R) :
     adjoin ℕ s = subalgebraOfSubsemiring (Subsemiring.closure s) :=
   le_antisymm (adjoin_le Subsemiring.subset_closure)
     (Subsemiring.closure_le.2 subset_adjoin : Subsemiring.closure s ≤ (adjoin ℕ s).toSubsemiring)
 
-theorem Algebra.adjoin_int {R : Type _} [Ring R] (s : Set R) :
+theorem Algebra.adjoin_int {R : Type*} [Ring R] (s : Set R) :
     adjoin ℤ s = subalgebraOfSubring (Subring.closure s) :=
   le_antisymm (adjoin_le Subring.subset_closure)
     (Subring.closure_le.2 subset_adjoin : Subring.closure s ≤ (adjoin ℤ s).toSubring)

a35ddf20

feat: maps from the unitization of non-unital subobjects of unital algebras (#6372)

If S is non-unital subalgebra of a unital R-algebra A, there is a natural map Unitization R S →ₐ[R] A whose range is Algebra.adjoin R (S : Set A). When 1 ∉ S and R is a field, this map is injective, and so we can restrict the codomain to Algebra.adjoin R (S : Set A) and turn it into an AlgEquiv.

We specialize this to the ℕ-unitization of a non-unital subsemiring and its Subsemiring.closure, as well as the ℤ-unitization of a non-unital subring and its Subring.closure. We also extend the above map to a StarAlgHom in the case of NonUnitalStarSubalgebras.

This continues the non-unital-ization of mathlib.

Co-authored-by: Anatole Dedecker <anatolededecker@gmail.com>

da2f8a8b

Diff

@@ -398,11 +398,6 @@ variable [CommRing R] [Ring A]
 
 variable [Algebra R A] {s t : Set A}
 
-theorem adjoin_int (s : Set R) : adjoin ℤ s = subalgebraOfSubring (Subring.closure s) :=
-  le_antisymm (adjoin_le Subring.subset_closure)
-    (Subring.closure_le.2 subset_adjoin : Subring.closure s ≤ (adjoin ℤ s).toSubring)
-#align algebra.adjoin_int Algebra.adjoin_int
-
 theorem mem_adjoin_iff {s : Set A} {x : A} :
     x ∈ adjoin R s ↔ x ∈ Subring.closure (Set.range (algebraMap R A) ∪ s) :=
   ⟨fun hx =>
@@ -454,3 +449,30 @@ theorem ext_of_adjoin_eq_top {s : Set A} (h : adjoin R s = ⊤) ⦃φ₁ φ₂ :
 #align alg_hom.ext_of_adjoin_eq_top AlgHom.ext_of_adjoin_eq_top
 
 end AlgHom
+
+section NatInt
+
+theorem Algebra.adjoin_nat {R : Type _} [Semiring R] (s : Set R) :
+    adjoin ℕ s = subalgebraOfSubsemiring (Subsemiring.closure s) :=
+  le_antisymm (adjoin_le Subsemiring.subset_closure)
+    (Subsemiring.closure_le.2 subset_adjoin : Subsemiring.closure s ≤ (adjoin ℕ s).toSubsemiring)
+
+theorem Algebra.adjoin_int {R : Type _} [Ring R] (s : Set R) :
+    adjoin ℤ s = subalgebraOfSubring (Subring.closure s) :=
+  le_antisymm (adjoin_le Subring.subset_closure)
+    (Subring.closure_le.2 subset_adjoin : Subring.closure s ≤ (adjoin ℤ s).toSubring)
+#align algebra.adjoin_int Algebra.adjoin_int
+
+/-- The `ℕ`-algebra equivalence between `Subsemiring.closure s` and `Algebra.adjoin ℕ s` given
+by the identity map. -/
+def Subsemiring.closureEquivAdjoinNat {R : Type*} [Semiring R] (s : Set R) :
+    Subsemiring.closure s ≃ₐ[ℕ] Algebra.adjoin ℕ s :=
+  Subalgebra.equivOfEq (subalgebraOfSubsemiring <| Subsemiring.closure s) _ (adjoin_nat s).symm
+
+/-- The `ℤ`-algebra equivalence between `Subring.closure s` and `Algebra.adjoin ℤ s` given by
+the identity map. -/
+def Subring.closureEquivAdjoinInt {R : Type*} [Ring R] (s : Set R) :
+    Subring.closure s ≃ₐ[ℤ] Algebra.adjoin ℤ s :=
+  Subalgebra.equivOfEq (subalgebraOfSubring <| Subring.closure s) _ (adjoin_int s).symm
+
+  end NatInt

a35ddf20

feat(RingTheory/FiniteType): generalize some results to non-commutative rings (#6681)

I was hoping to use this in combination with #6680 to show the TensorAlgebra is finitely generated, where I needed to generalize FiniteType.equiv; but it turns out that the FiniteType instance on MonoidAlgebra also isn't generalized!

The summary here is:

Move Algebra.adjoin_algebraMap from Mathlib/RingTheory/Adjoin/Tower.lean to Mathlib/RingTheory/Adjoin/Basic.lean and golf the proof to oblivion
Provide an alternative statement of adjoin_union_eq_adjoin_adjoin, adjoin_algebraMap_image_union_eq_adjoin_adjoin, which works in non-commutative rings, and use it along with a new adjoin_top lemma to prove Algebra.fg_trans' more generally.
Introduce a new S variable throughout, with the convention that R and S are commutative, A and B remain not-necessarily-commutative, and A/S/R is a tower of algebras.
Apply some zero-effort generalizations to semirings.

34d86d99

Diff

@@ -23,21 +23,22 @@ adjoin, algebra
 -/
 
 
-universe u v w
+universe uR uS uA uB
 
 open Pointwise
 
 open Submodule Subsemiring
 
-variable {R : Type u} {A : Type v} {B : Type w}
+variable {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB}
 
 namespace Algebra
 
 section Semiring
 
-variable [CommSemiring R] [Semiring A] [Semiring B]
+variable [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]
 
-variable [Algebra R A] [Algebra R B] {s t : Set A}
+variable [Algebra R S] [Algebra R A] [Algebra S A] [Algebra R B] [IsScalarTower R S A]
+variable {s t : Set A}
 
 theorem subset_adjoin : s ⊆ adjoin R s :=
   Algebra.gc.le_u_l s
@@ -292,6 +293,49 @@ theorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=
   Algebra.subset_adjoin (Set.mem_singleton_iff.mpr rfl)
 #align algebra.self_mem_adjoin_singleton Algebra.self_mem_adjoin_singleton
 
+variable (A) in
+theorem adjoin_algebraMap (s : Set S) :
+    adjoin R (algebraMap S A '' s) = (adjoin R s).map (IsScalarTower.toAlgHom R S A) :=
+  adjoin_image R (IsScalarTower.toAlgHom R S A) s
+#align algebra.adjoin_algebra_map Algebra.adjoin_algebraMap
+
+theorem adjoin_algebraMap_image_union_eq_adjoin_adjoin (s : Set S) (t : Set A) :
+    adjoin R (algebraMap S A '' s ∪ t) = (adjoin (adjoin R s) t).restrictScalars R :=
+  le_antisymm
+    (closure_mono <|
+      Set.union_subset (Set.range_subset_iff.2 fun r => Or.inl ⟨algebraMap R (adjoin R s) r,
+        (IsScalarTower.algebraMap_apply _ _ _ _).symm⟩)
+        (Set.union_subset_union_left _ fun _ ⟨_x, hx, hxs⟩ => hxs ▸ ⟨⟨_, subset_adjoin hx⟩, rfl⟩))
+    (closure_le.2 <|
+      Set.union_subset (Set.range_subset_iff.2 fun x => adjoin_mono (Set.subset_union_left _ _) <|
+        Algebra.adjoin_algebraMap R A s ▸ ⟨x, x.prop, rfl⟩)
+        (Set.Subset.trans (Set.subset_union_right _ _) subset_adjoin))
+
+theorem adjoin_adjoin_of_tower (s : Set A) : adjoin S (adjoin R s : Set A) = adjoin S s := by
+  apply le_antisymm (adjoin_le _)
+  · exact adjoin_mono subset_adjoin
+  · change adjoin R s ≤ (adjoin S s).restrictScalars R
+    refine' adjoin_le _
+    -- porting note: unclear why this was broken
+    have : (Subalgebra.restrictScalars R (adjoin S s) : Set A) = adjoin S s := rfl
+    rw [this]
+    exact subset_adjoin
+#align algebra.adjoin_adjoin_of_tower Algebra.adjoin_adjoin_of_tower
+
+@[simp]
+theorem adjoin_top :
+    adjoin (⊤ : Subalgebra R S) t = (adjoin S t).restrictScalars (⊤ : Subalgebra R S) :=
+  let equivTop : Subalgebra (⊤ : Subalgebra R S) A ≃o Subalgebra S A :=
+    { toFun := fun s => { s with algebraMap_mem' := fun r => s.algebraMap_mem ⟨r, trivial⟩ }
+      invFun := fun s => s.restrictScalars _
+      left_inv := fun _ => SetLike.coe_injective rfl
+      right_inv := fun _ => SetLike.coe_injective rfl
+      map_rel_iff' := @fun _ _ => Iff.rfl }
+  le_antisymm
+    (adjoin_le <| show t ⊆ adjoin S t from subset_adjoin)
+    (equivTop.symm_apply_le.mpr <|
+      adjoin_le <| show t ⊆ adjoin (⊤ : Subalgebra R S) t from subset_adjoin)
+
 end Semiring
 
 section CommSemiring
@@ -303,14 +347,8 @@ variable [Algebra R A] {s t : Set A}
 variable (R s t)
 
 theorem adjoin_union_eq_adjoin_adjoin :
-    adjoin R (s ∪ t) = (adjoin (adjoin R s) t).restrictScalars R :=
-  le_antisymm
-    (closure_mono <|
-      Set.union_subset (Set.range_subset_iff.2 fun r => Or.inl ⟨algebraMap R (adjoin R s) r, rfl⟩)
-        (Set.union_subset_union_left _ fun _x hxs => ⟨⟨_, subset_adjoin hxs⟩, rfl⟩))
-    (closure_le.2 <|
-      Set.union_subset (Set.range_subset_iff.2 fun x => adjoin_mono (Set.subset_union_left _ _) x.2)
-        (Set.Subset.trans (Set.subset_union_right _ _) subset_adjoin))
+    adjoin R (s ∪ t) = (adjoin (adjoin R s) t).restrictScalars R := by
+  simpa using adjoin_algebraMap_image_union_eq_adjoin_adjoin R s t
 #align algebra.adjoin_union_eq_adjoin_adjoin Algebra.adjoin_union_eq_adjoin_adjoin
 
 theorem adjoin_union_coe_submodule :
@@ -320,18 +358,6 @@ theorem adjoin_union_coe_submodule :
   congr 1 with z; simp [Submonoid.closure_union, Submonoid.mem_sup, Set.mem_mul]
 #align algebra.adjoin_union_coe_submodule Algebra.adjoin_union_coe_submodule
 
-theorem adjoin_adjoin_of_tower [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
-    (s : Set B) : adjoin A (adjoin R s : Set B) = adjoin A s := by
-  apply le_antisymm (adjoin_le _)
-  · exact adjoin_mono subset_adjoin
-  · change adjoin R s ≤ (adjoin A s).restrictScalars R
-    refine' adjoin_le _
-    -- porting note: unclear why this was broken
-    have : (Subalgebra.restrictScalars R (adjoin A s) : Set B) = adjoin A s := rfl
-    rw [this]
-    exact subset_adjoin
-#align algebra.adjoin_adjoin_of_tower Algebra.adjoin_adjoin_of_tower
-
 variable {R}
 
 theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]

a35ddf20

revert: #5602 (#6717)

This reverts commit caa9fe6b.

"feat: maps between the unitization of a non-unital subalgebra and its Algebra.adjoin (#5602)"

This revert exists because the PR was merged before it was finished, and because it will make the diff with the new changes simpler.

46eac943

Diff

@@ -428,23 +428,3 @@ theorem ext_of_adjoin_eq_top {s : Set A} (h : adjoin R s = ⊤) ⦃φ₁ φ₂ :
 #align alg_hom.ext_of_adjoin_eq_top AlgHom.ext_of_adjoin_eq_top
 
 end AlgHom
-
-section ClosureEquivAdjoin
-
-/-- The `ℕ`-algebra equivalence between `Subsemiring.closure s` and `Algebra.adjoin ℕ s` given
-by the identity map. -/
-def Subsemiring.closureEquivAdjoinNat {R : Type*} [Semiring R] (s : Set R) :
-    Subsemiring.closure s ≃ₐ[ℕ] Algebra.adjoin ℕ s :=
-  Subalgebra.equivOfEq (subalgebraOfSubsemiring <| Subsemiring.closure s) _ <|
-    le_antisymm (closure_le.mpr subset_adjoin) (adjoin_le subset_closure)
-
-/-- The `ℤ`-algebra equivalence between `Subring.closure s` and `Algebra.adjoin ℤ s` given by
-the identity map. -/
-def Subring.closureEquivAdjoinInt {R : Type*} [Ring R] (s : Set R) :
-    Subring.closure s ≃ₐ[ℤ] Algebra.adjoin ℤ s :=
-  Subalgebra.equivOfEq (subalgebraOfSubring <| Subring.closure s) _ <|
-    -- Lean is less smart here, probably because of the `with` definition of `subalgebraOfSubring`
-    le_antisymm (closure_le.mpr subset_adjoin : _ ≤ (adjoin ℤ s).toSubring)
-      (adjoin_le subset_closure)
-
-  end ClosureEquivAdjoin

a35ddf20

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

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

This has nice performance benefits.

32de3f67

Diff

@@ -67,12 +67,12 @@ theorem adjoin_eq (S : Subalgebra R A) : adjoin R ↑S = S :=
   adjoin_eq_of_le _ (Set.Subset.refl _) subset_adjoin
 #align algebra.adjoin_eq Algebra.adjoin_eq
 
-theorem adjoin_iUnion {α : Type _} (s : α → Set A) :
+theorem adjoin_iUnion {α : Type*} (s : α → Set A) :
     adjoin R (Set.iUnion s) = ⨆ i : α, adjoin R (s i) :=
   (@Algebra.gc R A _ _ _).l_iSup
 #align algebra.adjoin_Union Algebra.adjoin_iUnion
 
-theorem adjoin_attach_biUnion [DecidableEq A] {α : Type _} {s : Finset α} (f : s → Finset A) :
+theorem adjoin_attach_biUnion [DecidableEq A] {α : Type*} {s : Finset α} (f : s → Finset A) :
     adjoin R (s.attach.biUnion f : Set A) = ⨆ x, adjoin R (f x) := by simp [adjoin_iUnion]
 #align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_biUnion
 
@@ -433,14 +433,14 @@ section ClosureEquivAdjoin
 
 /-- The `ℕ`-algebra equivalence between `Subsemiring.closure s` and `Algebra.adjoin ℕ s` given
 by the identity map. -/
-def Subsemiring.closureEquivAdjoinNat {R : Type _} [Semiring R] (s : Set R) :
+def Subsemiring.closureEquivAdjoinNat {R : Type*} [Semiring R] (s : Set R) :
     Subsemiring.closure s ≃ₐ[ℕ] Algebra.adjoin ℕ s :=
   Subalgebra.equivOfEq (subalgebraOfSubsemiring <| Subsemiring.closure s) _ <|
     le_antisymm (closure_le.mpr subset_adjoin) (adjoin_le subset_closure)
 
 /-- The `ℤ`-algebra equivalence between `Subring.closure s` and `Algebra.adjoin ℤ s` given by
 the identity map. -/
-def Subring.closureEquivAdjoinInt {R : Type _} [Ring R] (s : Set R) :
+def Subring.closureEquivAdjoinInt {R : Type*} [Ring R] (s : Set R) :
     Subring.closure s ≃ₐ[ℤ] Algebra.adjoin ℤ s :=
   Subalgebra.equivOfEq (subalgebraOfSubring <| Subring.closure s) _ <|
     -- Lean is less smart here, probably because of the `with` definition of `subalgebraOfSubring`

a35ddf20

feat: maps between the unitization of a non-unital subalgebra and its Algebra.adjoin (#5602)

If S is non-unital subalgebra of a unital R-algebra A, there is a natural surjective map Unitization R S →ₐ[R] Algebra.adjoin R (S : Set A). When 1 ∉ S and R is a field, this becomes and AlgEquiv.

This continues the non-unital-ization of mathlib.

depends on: #5151
depends on: #5512
depends on: #5537

d32630e8

Diff

@@ -428,3 +428,23 @@ theorem ext_of_adjoin_eq_top {s : Set A} (h : adjoin R s = ⊤) ⦃φ₁ φ₂ :
 #align alg_hom.ext_of_adjoin_eq_top AlgHom.ext_of_adjoin_eq_top
 
 end AlgHom
+
+section ClosureEquivAdjoin
+
+/-- The `ℕ`-algebra equivalence between `Subsemiring.closure s` and `Algebra.adjoin ℕ s` given
+by the identity map. -/
+def Subsemiring.closureEquivAdjoinNat {R : Type _} [Semiring R] (s : Set R) :
+    Subsemiring.closure s ≃ₐ[ℕ] Algebra.adjoin ℕ s :=
+  Subalgebra.equivOfEq (subalgebraOfSubsemiring <| Subsemiring.closure s) _ <|
+    le_antisymm (closure_le.mpr subset_adjoin) (adjoin_le subset_closure)
+
+/-- The `ℤ`-algebra equivalence between `Subring.closure s` and `Algebra.adjoin ℤ s` given by
+the identity map. -/
+def Subring.closureEquivAdjoinInt {R : Type _} [Ring R] (s : Set R) :
+    Subring.closure s ≃ₐ[ℤ] Algebra.adjoin ℤ s :=
+  Subalgebra.equivOfEq (subalgebraOfSubring <| Subring.closure s) _ <|
+    -- Lean is less smart here, probably because of the `with` definition of `subalgebraOfSubring`
+    le_antisymm (closure_le.mpr subset_adjoin : _ ≤ (adjoin ℤ s).toSubring)
+      (adjoin_le subset_closure)
+
+  end ClosureEquivAdjoin

a35ddf20

chore: fix grammar mistakes (#6121)

a16538af

Diff

@@ -89,7 +89,7 @@ theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : 
 #align algebra.adjoin_induction Algebra.adjoin_induction
 
 /-- Induction principle for the algebra generated by a set `s`: show that `p x y` holds for any
-`x y ∈ adjoin R s` given that that it holds for `x y ∈ s` and that it satisfies a number of
+`x y ∈ adjoin R s` given that it holds for `x y ∈ s` and that it satisfies a number of
 natural properties. -/
 @[elab_as_elim]
 theorem adjoin_induction₂ {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s) (hb : b ∈ adjoin R s)

a35ddf20

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

depends on: #5966

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

89aa3a3b

Diff

@@ -2,17 +2,14 @@
 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.adjoin.basic
-! leanprover-community/mathlib commit a35ddf20601f85f78cd57e7f5b09ed528d71b7af
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.Algebra.Operations
 import Mathlib.Algebra.Algebra.Subalgebra.Tower
 import Mathlib.LinearAlgebra.Prod
 import Mathlib.LinearAlgebra.Finsupp
 
+#align_import ring_theory.adjoin.basic from "leanprover-community/mathlib"@"a35ddf20601f85f78cd57e7f5b09ed528d71b7af"
+
 /-!
 # Adjoining elements to form subalgebras

a35ddf20

chore: fix focusing dots (#5708)

This PR is the result of running

find . -type f -name "*.lean" -exec sed -i -E 's/^( +)\. /\1· /' {} \;
find . -type f -name "*.lean" -exec sed -i -E 'N;s/^( +·)\n +(.*)$/\1 \2/;P;D' {} \;

which firstly replaces . focusing dots with · and secondly removes isolated instances of such dots, unifying them with the following line. A new rule is placed in the style linter to verify this.

cb02d09e

Diff

@@ -254,8 +254,7 @@ theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     adjoin R (LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1})) =
       (adjoin R s).prod (adjoin R t) := by
   apply le_antisymm
-  ·
-    simp only [adjoin_le_iff, Set.insert_subset_iff, Subalgebra.zero_mem, Subalgebra.one_mem,
+  · simp only [adjoin_le_iff, Set.insert_subset_iff, Subalgebra.zero_mem, Subalgebra.one_mem,
       subset_adjoin,-- the rest comes from `squeeze_simp`
       Set.union_subset_iff,
       LinearMap.coe_inl, Set.mk_preimage_prod_right, Set.image_subset_iff, SetLike.mem_coe,

a35ddf20

feat(Data.Set.Basic/Data.Finset.Basic): rename insert_subset (#5450)

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.

d8b62864

Diff

@@ -226,7 +226,7 @@ theorem adjoin_image (f : A →ₐ[R] B) (s : Set A) : adjoin R (f '' s) = (adjo
 theorem adjoin_insert_adjoin (x : A) : adjoin R (insert x ↑(adjoin R s)) = adjoin R (insert x s) :=
   le_antisymm
     (adjoin_le
-      (Set.insert_subset.mpr
+      (Set.insert_subset_iff.mpr
         ⟨subset_adjoin (Set.mem_insert _ _), adjoin_mono (Set.subset_insert _ _)⟩))
     (Algebra.adjoin_mono (Set.insert_subset_insert Algebra.subset_adjoin))
 #align algebra.adjoin_insert_adjoin Algebra.adjoin_insert_adjoin
@@ -255,7 +255,7 @@ theorem adjoin_inl_union_inr_eq_prod (s) (t) :
       (adjoin R s).prod (adjoin R t) := by
   apply le_antisymm
   ·
-    simp only [adjoin_le_iff, Set.insert_subset, Subalgebra.zero_mem, Subalgebra.one_mem,
+    simp only [adjoin_le_iff, Set.insert_subset_iff, Subalgebra.zero_mem, Subalgebra.one_mem,
       subset_adjoin,-- the rest comes from `squeeze_simp`
       Set.union_subset_iff,
       LinearMap.coe_inl, Set.mk_preimage_prod_right, Set.image_subset_iff, SetLike.mem_coe,

a35ddf20

feat: port NumberTheory.Cyclotomic.Basic (#5335)

Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

5a55608d

Diff

@@ -113,6 +113,7 @@ theorem adjoin_induction₂ {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin
 #align algebra.adjoin_induction₂ Algebra.adjoin_induction₂
 
 /-- The difference with `Algebra.adjoin_induction` is that this acts on the subtype. -/
+@[elab_as_elim]
 theorem adjoin_induction' {p : adjoin R s → Prop} (Hs : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin h⟩)
     (Halg : ∀ r, p (algebraMap R _ r)) (Hadd : ∀ x y, p x → p y → p (x + y))
     (Hmul : ∀ x y, p x → p y → p (x * y)) (x : adjoin R s) : p x :=

a35ddf20

chore: Rename to sSup/iSup (#3938)

As discussed on Zulip

Renames

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

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

d1eb6264

Diff

@@ -50,9 +50,9 @@ theorem adjoin_le {S : Subalgebra R A} (H : s ⊆ S) : adjoin R s ≤ S :=
   Algebra.gc.l_le H
 #align algebra.adjoin_le Algebra.adjoin_le
 
-theorem adjoin_eq_infₛ : adjoin R s = infₛ { p : Subalgebra R A | s ⊆ p } :=
-  le_antisymm (le_infₛ fun _ h => adjoin_le h) (infₛ_le subset_adjoin)
-#align algebra.adjoin_eq_Inf Algebra.adjoin_eq_infₛ
+theorem adjoin_eq_sInf : adjoin R s = sInf { p : Subalgebra R A | s ⊆ p } :=
+  le_antisymm (le_sInf fun _ h => adjoin_le h) (sInf_le subset_adjoin)
+#align algebra.adjoin_eq_Inf Algebra.adjoin_eq_sInf
 
 theorem adjoin_le_iff {S : Subalgebra R A} : adjoin R s ≤ S ↔ s ⊆ S :=
   Algebra.gc _ _
@@ -70,14 +70,14 @@ theorem adjoin_eq (S : Subalgebra R A) : adjoin R ↑S = S :=
   adjoin_eq_of_le _ (Set.Subset.refl _) subset_adjoin
 #align algebra.adjoin_eq Algebra.adjoin_eq
 
-theorem adjoin_unionᵢ {α : Type _} (s : α → Set A) :
-    adjoin R (Set.unionᵢ s) = ⨆ i : α, adjoin R (s i) :=
-  (@Algebra.gc R A _ _ _).l_supᵢ
-#align algebra.adjoin_Union Algebra.adjoin_unionᵢ
+theorem adjoin_iUnion {α : Type _} (s : α → Set A) :
+    adjoin R (Set.iUnion s) = ⨆ i : α, adjoin R (s i) :=
+  (@Algebra.gc R A _ _ _).l_iSup
+#align algebra.adjoin_Union Algebra.adjoin_iUnion
 
-theorem adjoin_attach_bunionᵢ [DecidableEq A] {α : Type _} {s : Finset α} (f : s → Finset A) :
-    adjoin R (s.attach.bunionᵢ f : Set A) = ⨆ x, adjoin R (f x) := by simp [adjoin_unionᵢ]
-#align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_bunionᵢ
+theorem adjoin_attach_biUnion [DecidableEq A] {α : Type _} {s : Finset α} (f : s → Finset A) :
+    adjoin R (s.attach.biUnion f : Set A) = ⨆ x, adjoin R (f x) := by simp [adjoin_iUnion]
+#align algebra.adjoin_attach_bUnion Algebra.adjoin_attach_biUnion
 
 @[elab_as_elim]
 theorem adjoin_induction {p : A → Prop} {x : A} (h : x ∈ adjoin R s) (Hs : ∀ x ∈ s, p x)

a35ddf20

chore: tidy various files (#3110)

0bcbc985

Diff

@@ -340,10 +340,8 @@ variable {R}
 theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]
     [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
     (hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B' := by
-  -- porting note: use `revert` and `intro` because we don't have `change ... at` yet
-  revert hx
-  change x ∈ Subalgebra.toSubmodule (adjoin R s) → _
-  intro hx
+  -- porting note: use `replace` because we don't have `change ... at` yet
+  replace hx : x ∈ Subalgebra.toSubmodule (adjoin R s) := hx
   rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
   rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
   choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop

a35ddf20

feat: port RingTheory.Adjoin.Basic (#2817)

48c5b56d

`ring_theory.adjoin.basic` ⟷ `Mathlib.RingTheory.Adjoin.Basic`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Renames

Dependencies 8 + 455

ring_theory.adjoin.basic ⟷ Mathlib.RingTheory.Adjoin.Basic

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Renames

Dependencies 8 + 455

`ring_theory.adjoin.basic` ⟷ `Mathlib.RingTheory.Adjoin.Basic`