model_theory.semantics | 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

d565b3df

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

6cf59007

bump to nightly-2024-03-17-08

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

22f01ce4

Diff

@@ -154,7 +154,7 @@ theorem realize_restrictVar [DecidableEq α] {t : L.term α} {s : Set α} (h : 
   by
   induction' t with _ _ _ _ ih
   · rfl
-  · simp_rw [var_finset, Finset.coe_biUnion, Set.iUnion_subset_iff] at h 
+  · simp_rw [var_finset, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var FirstOrder.Language.Term.realize_restrictVar
 -/
@@ -168,7 +168,7 @@ theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.term (Sum
   by
   induction' t with a _ _ _ ih
   · cases a <;> rfl
-  · simp_rw [var_finset_left, Finset.coe_biUnion, Set.iUnion_subset_iff] at h 
+  · simp_rw [var_finset_left, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var_left FirstOrder.Language.Term.realize_restrictVarLeft
 -/
@@ -1103,7 +1103,7 @@ theorem realize_exs {φ : L.BoundedFormula α n} {v : α → M} :
     · rintro ⟨xs, x, h⟩
       exact ⟨_, h⟩
     · rintro ⟨xs, h⟩
-      rw [← Fin.snoc_init_self xs] at h 
+      rw [← Fin.snoc_init_self xs] at h
       exact ⟨_, _, h⟩
 #align first_order.language.bounded_formula.realize_exs FirstOrder.Language.BoundedFormula.realize_exs
 -/
@@ -1123,7 +1123,7 @@ theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M
   · rw [to_formula, formula.realize, realize_all, realize_all]
     refine' forall_congr' fun a => _
     have h := ih3 (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a))
-    simp only [Sum.elim_comp_inl, Sum.elim_comp_inr] at h 
+    simp only [Sum.elim_comp_inl, Sum.elim_comp_inr] at h
     rw [← h, realize_relabel, formula.realize]
     rcongr
     · cases x
@@ -1157,11 +1157,11 @@ theorem realize_boundedFormula (g : M ≃[L] N) (φ : L.BoundedFormula α n) {v
     constructor
     · intro h a
       have h' := h (g a)
-      rw [← Fin.comp_snoc, ih3] at h' 
+      rw [← Fin.comp_snoc, ih3] at h'
       exact h'
     · intro h a
       have h' := h (g.symm a)
-      rw [← ih3, Fin.comp_snoc, g.apply_symm_apply] at h' 
+      rw [← ih3, Fin.comp_snoc, g.apply_symm_apply] at h'
       exact h'
 #align first_order.language.equiv.realize_bounded_formula FirstOrder.Language.Equiv.realize_boundedFormula
 -/
@@ -1280,7 +1280,7 @@ theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ (#M
     contrapose! ij
     have hij := h _ i j ij rfl
     simp only [bounded_formula.realize_not, term.realize, bounded_formula.realize_bd_equal,
-      Sum.elim_inr] at hij 
+      Sum.elim_inr] at hij
     exact hij
   · rintro _ i j ij rfl
     simp [ij]
@@ -1329,7 +1329,7 @@ theorem model_distinctConstantsTheory {M : Type w} [L[[α]].Structure M] (s : Se
   · contrapose! ab
     have h' := h _ a b ⟨⟨as, bs⟩, ab⟩ rfl
     simp only [sentence.realize, formula.realize_not, formula.realize_equal,
-      term.realize_constants] at h' 
+      term.realize_constants] at h'
     exact h'
   · rintro h φ a b ⟨⟨as, bs⟩, ab⟩ rfl
     simp only [sentence.realize, formula.realize_not, formula.realize_equal, term.realize_constants]

6cf59007

bump to nightly-2024-01-17-06

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

8ba88f37

Diff

@@ -422,7 +422,7 @@ theorem realize_castLE_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.Boun
     (φ.castLEEmb h').realize v xs ↔ φ.realize v (xs ∘ Fin.castIso h) :=
   by
   subst h
-  simp only [cast_le_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]
+  simp only [cast_le_rfl, cast_refl, OrderIso.coe_refl, Function.comp_id]
 #align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLE_of_eq
 -/
 
@@ -806,7 +806,7 @@ theorem realize_relabel {φ : L.Formula α} {g : α → β} {v : β → M} :
 theorem realize_relabel_sum_inr (φ : L.Formula (Fin n)) {v : Empty → M} {x : Fin n → M} :
     (BoundedFormula.relabel Sum.inr φ).realize v x ↔ φ.realize x := by
   rw [bounded_formula.realize_relabel, formula.realize, Sum.elim_comp_inr, Fin.castAdd_zero,
-    cast_refl, OrderIso.coe_refl, Function.comp.right_id,
+    cast_refl, OrderIso.coe_refl, Function.comp_id,
     Subsingleton.elim (x ∘ (nat_add n : Fin 0 → Fin n)) default]
 #align first_order.language.formula.realize_relabel_sum_inr FirstOrder.Language.Formula.realize_relabel_sum_inr
 -/

6cf59007

bump to nightly-2023-10-21-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/b1abe23ae96fef89ad30d9f4362c307f72a55010

a2dc86f8

Diff

@@ -404,7 +404,7 @@ theorem realize_all : (all θ).realize v xs ↔ ∀ a : M, θ.realize v (Fin.sno
 @[simp]
 theorem realize_ex : θ.ex.realize v xs ↔ ∃ a : M, θ.realize v (Fin.snoc xs a) :=
   by
-  rw [bounded_formula.ex, realize_not, realize_all, not_forall]
+  rw [bounded_formula.ex, realize_not, realize_all, Classical.not_forall]
   simp_rw [realize_not, Classical.not_not]
 #align first_order.language.bounded_formula.realize_ex FirstOrder.Language.BoundedFormula.realize_ex
 -/
@@ -645,7 +645,7 @@ theorem realize_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ)
     · by_cases ψ.realize v xs
       · inhabit M
         exact ⟨default, fun h'' => h⟩
-      · obtain ⟨a, ha⟩ := not_forall.1 (h ∘ h')
+      · obtain ⟨a, ha⟩ := Classical.not_forall.1 (h ∘ h')
         exact ⟨a, fun h => (ha h).elim⟩
   · refine' trans (forall_congr' fun _ => ih hψ.lift_at) _
     simp

6cf59007

bump to nightly-2023-09-24-06

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

5655435f

Diff

@@ -3,8 +3,8 @@ Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All righ
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
 -/
-import Mathbin.Data.Finset.Basic
-import Mathbin.ModelTheory.Syntax
+import Data.Finset.Basic
+import ModelTheory.Syntax
 
 #align_import model_theory.semantics from "leanprover-community/mathlib"@"6cf5900728239efa287df7761ec2a1ac9cf39b29"

6cf59007

bump to nightly-2023-08-07-11

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

d9d9267a

Diff

@@ -416,14 +416,14 @@ theorem realize_iff : (φ.Iff ψ).realize v xs ↔ (φ.realize v xs ↔ ψ.reali
 #align first_order.language.bounded_formula.realize_iff FirstOrder.Language.BoundedFormula.realize_iff
 -/
 
-#print FirstOrder.Language.BoundedFormula.realize_castLe_of_eq /-
-theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
+#print FirstOrder.Language.BoundedFormula.realize_castLE_of_eq /-
+theorem realize_castLE_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
     {v : α → M} {xs : Fin n → M} :
     (φ.castLEEmb h').realize v xs ↔ φ.realize v (xs ∘ Fin.castIso h) :=
   by
   subst h
   simp only [cast_le_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]
-#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eq
+#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLE_of_eq
 -/
 
 #print FirstOrder.Language.BoundedFormula.realize_mapTermRel_id /-

6cf59007

bump to nightly-2023-07-20-09

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

9c56d0dd

Diff

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-
-! This file was ported from Lean 3 source module model_theory.semantics
-! leanprover-community/mathlib commit 6cf5900728239efa287df7761ec2a1ac9cf39b29
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Finset.Basic
 import Mathbin.ModelTheory.Syntax
 
+#align_import model_theory.semantics from "leanprover-community/mathlib"@"6cf5900728239efa287df7761ec2a1ac9cf39b29"
+
 /-!
 # Basics on First-Order Semantics

6cf59007

bump to nightly-2023-07-16-17

mathlib commit https://github.com/leanprover-community/mathlib/commit/2fe465deb81bcd7ccafa065bb686888a82f15372

6db63fa6

Diff

@@ -97,7 +97,8 @@ theorem realize_relabel {t : L.term α} {g : α → β} {v : β → M} :
 @[simp]
 theorem realize_liftAt {n n' m : ℕ} {t : L.term (Sum α (Fin n))} {v : Sum α (Fin (n + n')) → M} :
     (t.liftAt n' m).realize v =
-      t.realize (v ∘ Sum.map id fun i => if ↑i < m then Fin.castAdd n' i else Fin.addNat n' i) :=
+      t.realize
+        (v ∘ Sum.map id fun i => if ↑i < m then Fin.castAddEmb n' i else Fin.addNatEmb n' i) :=
   realize_relabel
 #align first_order.language.term.realize_lift_at FirstOrder.Language.Term.realize_liftAt
 -/
@@ -420,7 +421,8 @@ theorem realize_iff : (φ.Iff ψ).realize v xs ↔ (φ.realize v xs ↔ ψ.reali
 
 #print FirstOrder.Language.BoundedFormula.realize_castLe_of_eq /-
 theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
-    {v : α → M} {xs : Fin n → M} : (φ.castLE h').realize v xs ↔ φ.realize v (xs ∘ Fin.castIso h) :=
+    {v : α → M} {xs : Fin n → M} :
+    (φ.castLEEmb h').realize v xs ↔ φ.realize v (xs ∘ Fin.castIso h) :=
   by
   subst h
   simp only [cast_le_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]
@@ -454,11 +456,11 @@ theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
     (v : ∀ {n}, (Fin (k + n) → M) → α → M) {v' : β → M} (xs : Fin (k + n) → M)
     (h1 :
       ∀ (n) (t : L.term (Sum α (Fin n))) (xs' : Fin (k + n) → M),
-        (ft n t).realize (Sum.elim v' xs') = t.realize (Sum.elim (v xs') (xs' ∘ Fin.natAdd _)))
+        (ft n t).realize (Sum.elim v' xs') = t.realize (Sum.elim (v xs') (xs' ∘ Fin.natAddEmb _)))
     (h2 : ∀ (n) (R : L.Relations n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x)
     (hv : ∀ (n) (xs : Fin (k + n) → M) (x : M), @v (n + 1) (snoc xs x : Fin _ → M) = v xs) :
     (φ.mapTermRel ft fr fun n => castLE (add_assoc _ _ _).symm.le).realize v' xs ↔
-      φ.realize (v xs) (xs ∘ Fin.natAdd _) :=
+      φ.realize (v xs) (xs ∘ Fin.natAddEmb _) :=
   by
   induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih
   · rfl
@@ -473,7 +475,7 @@ theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
 theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum β (Fin m)} {v : β → M}
     {xs : Fin (m + n) → M} :
     (φ.relabel g).realize v xs ↔
-      φ.realize (Sum.elim v (xs ∘ Fin.castAdd n) ∘ g) (xs ∘ Fin.natAdd m) :=
+      φ.realize (Sum.elim v (xs ∘ Fin.castAddEmb n) ∘ g) (xs ∘ Fin.natAddEmb m) :=
   by rw [relabel, realize_map_term_rel_add_cast_le] <;> intros <;> simp
 #align first_order.language.bounded_formula.realize_relabel FirstOrder.Language.BoundedFormula.realize_relabel
 -/
@@ -482,7 +484,7 @@ theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum
 theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + n') → M}
     (hmn : m + n' ≤ n + 1) :
     (φ.liftAt n' m).realize v xs ↔
-      φ.realize v (xs ∘ fun i => if ↑i < m then Fin.castAdd n' i else Fin.addNat n' i) :=
+      φ.realize v (xs ∘ fun i => if ↑i < m then Fin.castAddEmb n' i else Fin.addNatEmb n' i) :=
   by
   rw [lift_at]
   induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 k _ ih3
@@ -504,7 +506,7 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
       · rw [if_neg h]
         refine' (congr rfl (ext _)).trans (snoc_last _ _)
         simp
-    · simp only [Function.comp_apply, Fin.snoc_castSuccEmb]
+    · simp only [Function.comp_apply, Fin.snoc_castSucc]
       refine' (congr rfl (ext _)).trans (snoc_cast_succ _ _ _)
       simp only [cast_refl, coe_cast_succ, OrderIso.coe_refl, id.def]
       split_ifs <;> simp

6cf59007

bump to nightly-2023-07-06-11

mathlib commit https://github.com/leanprover-community/mathlib/commit/5dc6092d09e5e489106865241986f7f2ad28d4c8

a2f50eda

Diff

@@ -504,7 +504,7 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
       · rw [if_neg h]
         refine' (congr rfl (ext _)).trans (snoc_last _ _)
         simp
-    · simp only [Function.comp_apply, Fin.snoc_castSucc]
+    · simp only [Function.comp_apply, Fin.snoc_castSuccEmb]
       refine' (congr rfl (ext _)).trans (snoc_cast_succ _ _ _)
       simp only [cast_refl, coe_cast_succ, OrderIso.coe_refl, id.def]
       split_ifs <;> simp
@@ -515,7 +515,7 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
 theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + 1) → M}
     (hmn : m ≤ n) :
     (φ.liftAt 1 m).realize v xs ↔
-      φ.realize v (xs ∘ fun i => if ↑i < m then castSucc i else i.succ) :=
+      φ.realize v (xs ∘ fun i => if ↑i < m then castSuccEmb i else i.succ) :=
   by simp_rw [realize_lift_at (add_le_add_right hmn 1), cast_succ, add_nat_one]
 #align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_one
 -/
@@ -523,7 +523,7 @@ theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α →
 #print FirstOrder.Language.BoundedFormula.realize_liftAt_one_self /-
 @[simp]
 theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α → M}
-    {xs : Fin (n + 1) → M} : (φ.liftAt 1 n).realize v xs ↔ φ.realize v (xs ∘ castSucc) :=
+    {xs : Fin (n + 1) → M} : (φ.liftAt 1 n).realize v xs ↔ φ.realize v (xs ∘ castSuccEmb) :=
   by
   rw [realize_lift_at_one (refl n), iff_eq_eq]
   refine' congr rfl (congr rfl (funext fun i => _))

6cf59007

bump to nightly-2023-07-04-19

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

4217d5f4

Diff

@@ -420,7 +420,7 @@ theorem realize_iff : (φ.Iff ψ).realize v xs ↔ (φ.realize v xs ↔ ψ.reali
 
 #print FirstOrder.Language.BoundedFormula.realize_castLe_of_eq /-
 theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
-    {v : α → M} {xs : Fin n → M} : (φ.castLE h').realize v xs ↔ φ.realize v (xs ∘ Fin.cast h) :=
+    {v : α → M} {xs : Fin n → M} : (φ.castLE h').realize v xs ↔ φ.realize v (xs ∘ Fin.castIso h) :=
   by
   subst h
   simp only [cast_le_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]

6cf59007

bump to nightly-2023-06-13-21

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

829520ac

Diff

@@ -93,17 +93,21 @@ theorem realize_relabel {t : L.term α} {g : α → β} {v : β → M} :
 #align first_order.language.term.realize_relabel FirstOrder.Language.Term.realize_relabel
 -/
 
+#print FirstOrder.Language.Term.realize_liftAt /-
 @[simp]
 theorem realize_liftAt {n n' m : ℕ} {t : L.term (Sum α (Fin n))} {v : Sum α (Fin (n + n')) → M} :
     (t.liftAt n' m).realize v =
       t.realize (v ∘ Sum.map id fun i => if ↑i < m then Fin.castAdd n' i else Fin.addNat n' i) :=
   realize_relabel
 #align first_order.language.term.realize_lift_at FirstOrder.Language.Term.realize_liftAt
+-/
 
+#print FirstOrder.Language.Term.realize_constants /-
 @[simp]
 theorem realize_constants {c : L.Constants} {v : α → M} : c.term.realize v = c :=
   funMap_eq_coe_constants
 #align first_order.language.term.realize_constants FirstOrder.Language.Term.realize_constants
+-/
 
 #print FirstOrder.Language.Term.realize_functions_apply₁ /-
 @[simp]
@@ -157,6 +161,7 @@ theorem realize_restrictVar [DecidableEq α] {t : L.term α} {s : Set α} (h : 
 #align first_order.language.term.realize_restrict_var FirstOrder.Language.Term.realize_restrictVar
 -/
 
+#print FirstOrder.Language.Term.realize_restrictVarLeft /-
 @[simp]
 theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.term (Sum α γ)} {s : Set α}
     (h : ↑t.varFinsetLeft ⊆ s) {v : α → M} {xs : γ → M} :
@@ -168,7 +173,9 @@ theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.term (Sum
   · simp_rw [var_finset_left, Finset.coe_biUnion, Set.iUnion_subset_iff] at h 
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var_left FirstOrder.Language.Term.realize_restrictVarLeft
+-/
 
+#print FirstOrder.Language.Term.realize_constantsToVars /-
 @[simp]
 theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
     {t : L[[α]].term β} {v : β → M} :
@@ -185,7 +192,9 @@ theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).
       · simp [ih]
       · exact isEmptyElim f
 #align first_order.language.term.realize_constants_to_vars FirstOrder.Language.Term.realize_constantsToVars
+-/
 
+#print FirstOrder.Language.Term.realize_varsToConstants /-
 @[simp]
 theorem realize_varsToConstants [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
     {t : L.term (Sum α β)} {v : β → M} :
@@ -195,6 +204,7 @@ theorem realize_varsToConstants [L[[α]].Structure M] [(lhomWithConstants L α).
   · cases ab <;> simp [language.con]
   · simp [ih]
 #align first_order.language.term.realize_vars_to_constants FirstOrder.Language.Term.realize_varsToConstants
+-/
 
 #print FirstOrder.Language.Term.realize_constantsVarsEquivLeft /-
 theorem realize_constantsVarsEquivLeft [L[[α]].Structure M]
@@ -215,6 +225,7 @@ end Term
 
 namespace Lhom
 
+#print FirstOrder.Language.LHom.realize_onTerm /-
 @[simp]
 theorem realize_onTerm [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (t : L.term α)
     (v : α → M) : (φ.onTerm t).realize v = t.realize v :=
@@ -223,9 +234,11 @@ theorem realize_onTerm [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
   · rfl
   · simp only [term.realize, Lhom.on_term, Lhom.map_on_function, ih]
 #align first_order.language.Lhom.realize_on_term FirstOrder.Language.LHom.realize_onTerm
+-/
 
 end Lhom
 
+#print FirstOrder.Language.Hom.realize_term /-
 @[simp]
 theorem Hom.realize_term (g : M →[L] N) {t : L.term α} {v : α → M} :
     t.realize (g ∘ v) = g (t.realize v) := by
@@ -236,18 +249,23 @@ theorem Hom.realize_term (g : M →[L] N) {t : L.term α} {v : α → M} :
     ext x
     simp [t_ih x]
 #align first_order.language.hom.realize_term FirstOrder.Language.Hom.realize_term
+-/
 
+#print FirstOrder.Language.Embedding.realize_term /-
 @[simp]
 theorem Embedding.realize_term {v : α → M} (t : L.term α) (g : M ↪[L] N) :
     t.realize (g ∘ v) = g (t.realize v) :=
   g.toHom.realize_term
 #align first_order.language.embedding.realize_term FirstOrder.Language.Embedding.realize_term
+-/
 
+#print FirstOrder.Language.Equiv.realize_term /-
 @[simp]
 theorem Equiv.realize_term {v : α → M} (t : L.term α) (g : M ≃[L] N) :
     t.realize (g ∘ v) = g (t.realize v) :=
   g.toHom.realize_term
 #align first_order.language.equiv.realize_term FirstOrder.Language.Equiv.realize_term
+-/
 
 variable {L} {α} {n : ℕ}
 
@@ -364,6 +382,7 @@ theorem realize_sup : (φ ⊔ ψ).realize v xs ↔ φ.realize v xs ∨ ψ.realiz
 #align first_order.language.bounded_formula.realize_sup FirstOrder.Language.BoundedFormula.realize_sup
 -/
 
+#print FirstOrder.Language.BoundedFormula.realize_foldr_sup /-
 @[simp]
 theorem realize_foldr_sup (l : List (L.BoundedFormula α n)) (v : α → M) (xs : Fin n → M) :
     (l.foldr (· ⊔ ·) ⊥).realize v xs ↔ ∃ φ ∈ l, BoundedFormula.Realize φ v xs :=
@@ -374,6 +393,7 @@ theorem realize_foldr_sup (l : List (L.BoundedFormula α n)) (v : α → M) (xs
     simp_rw [List.foldr_cons, realize_sup, ih, exists_prop, List.mem_cons, or_and_right, exists_or,
       exists_eq_left]
 #align first_order.language.bounded_formula.realize_foldr_sup FirstOrder.Language.BoundedFormula.realize_foldr_sup
+-/
 
 #print FirstOrder.Language.BoundedFormula.realize_all /-
 @[simp]
@@ -398,13 +418,16 @@ theorem realize_iff : (φ.Iff ψ).realize v xs ↔ (φ.realize v xs ↔ ψ.reali
 #align first_order.language.bounded_formula.realize_iff FirstOrder.Language.BoundedFormula.realize_iff
 -/
 
+#print FirstOrder.Language.BoundedFormula.realize_castLe_of_eq /-
 theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
     {v : α → M} {xs : Fin n → M} : (φ.castLE h').realize v xs ↔ φ.realize v (xs ∘ Fin.cast h) :=
   by
   subst h
   simp only [cast_le_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]
 #align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eq
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_mapTermRel_id /-
 theorem realize_mapTermRel_id [L'.Structure M]
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin n))}
     {fr : ∀ n, L.Relations n → L'.Relations n} {n} {φ : L.BoundedFormula α n} {v : α → M}
@@ -422,7 +445,9 @@ theorem realize_mapTermRel_id [L'.Structure M]
   · simp [map_term_rel, realize, ih1, ih2]
   · simp only [map_term_rel, realize, ih, id.def]
 #align first_order.language.bounded_formula.realize_map_term_rel_id FirstOrder.Language.BoundedFormula.realize_mapTermRel_id
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe /-
 theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin (k + n)))}
     {fr : ∀ n, L.Relations n → L'.Relations n} {n} {φ : L.BoundedFormula α n}
@@ -442,14 +467,18 @@ theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
   · simp [map_term_rel, realize, ih1, ih2]
   · simp [map_term_rel, realize, ih, hv]
 #align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_relabel /-
 theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum β (Fin m)} {v : β → M}
     {xs : Fin (m + n) → M} :
     (φ.relabel g).realize v xs ↔
       φ.realize (Sum.elim v (xs ∘ Fin.castAdd n) ∘ g) (xs ∘ Fin.natAdd m) :=
   by rw [relabel, realize_map_term_rel_add_cast_le] <;> intros <;> simp
 #align first_order.language.bounded_formula.realize_relabel FirstOrder.Language.BoundedFormula.realize_relabel
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_liftAt /-
 theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + n') → M}
     (hmn : m + n' ≤ n + 1) :
     (φ.liftAt n' m).realize v xs ↔
@@ -480,14 +509,18 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
       simp only [cast_refl, coe_cast_succ, OrderIso.coe_refl, id.def]
       split_ifs <;> simp
 #align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAt
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_liftAt_one /-
 theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + 1) → M}
     (hmn : m ≤ n) :
     (φ.liftAt 1 m).realize v xs ↔
       φ.realize v (xs ∘ fun i => if ↑i < m then castSucc i else i.succ) :=
   by simp_rw [realize_lift_at (add_le_add_right hmn 1), cast_succ, add_nat_one]
 #align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_one
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_liftAt_one_self /-
 @[simp]
 theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α → M}
     {xs : Fin (n + 1) → M} : (φ.liftAt 1 n).realize v xs ↔ φ.realize v (xs ∘ castSucc) :=
@@ -496,6 +529,7 @@ theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α 
   refine' congr rfl (congr rfl (funext fun i => _))
   rw [if_pos i.is_lt]
 #align first_order.language.bounded_formula.realize_lift_at_one_self FirstOrder.Language.BoundedFormula.realize_liftAt_one_self
+-/
 
 #print FirstOrder.Language.BoundedFormula.realize_subst /-
 theorem realize_subst {φ : L.BoundedFormula α n} {tf : α → L.term β} {v : β → M} {xs : Fin n → M} :
@@ -645,6 +679,7 @@ namespace Lhom
 
 open BoundedFormula
 
+#print FirstOrder.Language.LHom.realize_onBoundedFormula /-
 @[simp]
 theorem realize_onBoundedFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] {n : ℕ}
     (ψ : L.BoundedFormula α n) {v : α → M} {xs : Fin n → M} :
@@ -659,6 +694,7 @@ theorem realize_onBoundedFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpan
   · simp only [on_bounded_formula, ih1, ih2, realize_imp]
   · simp only [on_bounded_formula, ih3, realize_all]
 #align first_order.language.Lhom.realize_on_bounded_formula FirstOrder.Language.LHom.realize_onBoundedFormula
+-/
 
 end Lhom
 
@@ -795,17 +831,21 @@ theorem realize_graph {f : L.Functions n} {x : Fin n → M} {y : M} :
 
 end Formula
 
+#print FirstOrder.Language.LHom.realize_onFormula /-
 @[simp]
 theorem LHom.realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (ψ : L.Formula α)
     {v : α → M} : (φ.onFormula ψ).realize v ↔ ψ.realize v :=
   φ.realize_onBoundedFormula ψ
 #align first_order.language.Lhom.realize_on_formula FirstOrder.Language.LHom.realize_onFormula
+-/
 
+#print FirstOrder.Language.LHom.setOf_realize_onFormula /-
 @[simp]
 theorem LHom.setOf_realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
     (ψ : L.Formula α) : (setOf (φ.onFormula ψ).realize : Set (α → M)) = setOf ψ.realize := by ext;
   simp
 #align first_order.language.Lhom.set_of_realize_on_formula FirstOrder.Language.LHom.setOf_realize_onFormula
+-/
 
 variable (M)
 
@@ -816,7 +856,6 @@ def Sentence.Realize (φ : L.Sentence) : Prop :=
 #align first_order.language.sentence.realize FirstOrder.Language.Sentence.Realize
 -/
 
--- mathport name: sentence.realize
 infixl:51
   " ⊨ " =>-- input using \|= or \vDash, but not using \models
   Sentence.Realize
@@ -865,11 +904,13 @@ theorem realize_equivSentence_symm (φ : L[[α]].Sentence) (v : α → M) :
 
 end Formula
 
+#print FirstOrder.Language.LHom.realize_onSentence /-
 @[simp]
 theorem LHom.realize_onSentence [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
     (ψ : L.Sentence) : M ⊨ φ.onSentence ψ ↔ M ⊨ ψ :=
   φ.realize_onFormula ψ
 #align first_order.language.Lhom.realize_on_sentence FirstOrder.Language.LHom.realize_onSentence
+-/
 
 variable (L)
 
@@ -889,7 +930,6 @@ def ElementarilyEquivalent : Prop :=
 #align first_order.language.elementarily_equivalent FirstOrder.Language.ElementarilyEquivalent
 -/
 
--- mathport name: elementarily_equivalent
 scoped[FirstOrder]
   notation:25 A " ≅[" L "] " B:50 => FirstOrder.Language.ElementarilyEquivalent L A B
 
@@ -902,9 +942,11 @@ theorem mem_completeTheory {φ : Sentence L} : φ ∈ L.completeTheory M ↔ M 
 #align first_order.language.mem_complete_theory FirstOrder.Language.mem_completeTheory
 -/
 
+#print FirstOrder.Language.elementarilyEquivalent_iff /-
 theorem elementarilyEquivalent_iff : M ≅[L] N ↔ ∀ φ : L.Sentence, M ⊨ φ ↔ N ⊨ φ := by
   simp only [elementarily_equivalent, Set.ext_iff, complete_theory, Set.mem_setOf_eq]
 #align first_order.language.elementarily_equivalent_iff FirstOrder.Language.elementarilyEquivalent_iff
+-/
 
 variable (M)
 
@@ -915,7 +957,6 @@ class Theory.Model (T : L.Theory) : Prop where
 #align first_order.language.Theory.model FirstOrder.Language.Theory.Model
 -/
 
--- mathport name: Theory.model
 infixl:51
   " ⊨ " =>-- input using \|= or \vDash, but not using \models
   Theory.Model
@@ -941,10 +982,12 @@ theorem Theory.realize_sentence_of_mem [M ⊨ T] {φ : L.Sentence} (h : φ ∈ T
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.LHom.onTheory_model /-
 @[simp]
 theorem LHom.onTheory_model [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (T : L.Theory) :
     M ⊨ φ.onTheory T ↔ M ⊨ T := by simp [Theory.model_iff, Lhom.on_Theory]
 #align first_order.language.Lhom.on_Theory_model FirstOrder.Language.LHom.onTheory_model
+-/
 
 variable {M} {T}
 
@@ -968,20 +1011,24 @@ theorem Model.mono {T' : L.Theory} (h : M ⊨ T') (hs : T ⊆ T') : M ⊨ T :=
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Theory.Model.union /-
 theorem Model.union {T' : L.Theory} (h : M ⊨ T) (h' : M ⊨ T') : M ⊨ T ∪ T' :=
   by
   simp only [model_iff, Set.mem_union] at *
   exact fun φ hφ => hφ.elim (h _) (h' _)
 #align first_order.language.Theory.model.union FirstOrder.Language.Theory.Model.union
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Theory.model_union_iff /-
 @[simp]
 theorem model_union_iff {T' : L.Theory} : M ⊨ T ∪ T' ↔ M ⊨ T ∧ M ⊨ T' :=
   ⟨fun h => ⟨h.mono (T.subset_union_left T'), h.mono (T.subset_union_right T')⟩, fun h =>
     h.1.union h.2⟩
 #align first_order.language.Theory.model_union_iff FirstOrder.Language.Theory.model_union_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1018,6 +1065,7 @@ variable (M N)
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.realize_iff_of_model_completeTheory /-
 theorem realize_iff_of_model_completeTheory [N ⊨ L.completeTheory M] (φ : L.Sentence) :
     N ⊨ φ ↔ M ⊨ φ :=
   by
@@ -1026,6 +1074,7 @@ theorem realize_iff_of_model_completeTheory [N ⊨ L.completeTheory M] (φ : L.S
   rw [← sentence.realize_not] at *
   exact (L.complete_theory M).realize_sentence_of_mem (mem_complete_theory.2 h)
 #align first_order.language.realize_iff_of_model_complete_theory FirstOrder.Language.realize_iff_of_model_completeTheory
+-/
 
 variable {M N}
 
@@ -1095,6 +1144,7 @@ end BoundedFormula
 
 namespace Equiv
 
+#print FirstOrder.Language.Equiv.realize_boundedFormula /-
 @[simp]
 theorem realize_boundedFormula (g : M ≃[L] N) (φ : L.BoundedFormula α n) {v : α → M}
     {xs : Fin n → M} : φ.realize (g ∘ v) (g ∘ xs) ↔ φ.realize v xs :=
@@ -1115,29 +1165,38 @@ theorem realize_boundedFormula (g : M ≃[L] N) (φ : L.BoundedFormula α n) {v
       rw [← ih3, Fin.comp_snoc, g.apply_symm_apply] at h' 
       exact h'
 #align first_order.language.equiv.realize_bounded_formula FirstOrder.Language.Equiv.realize_boundedFormula
+-/
 
+#print FirstOrder.Language.Equiv.realize_formula /-
 @[simp]
 theorem realize_formula (g : M ≃[L] N) (φ : L.Formula α) {v : α → M} :
     φ.realize (g ∘ v) ↔ φ.realize v := by
   rw [formula.realize, formula.realize, ← g.realize_bounded_formula φ, iff_eq_eq,
     Unique.eq_default (g ∘ default)]
 #align first_order.language.equiv.realize_formula FirstOrder.Language.Equiv.realize_formula
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Equiv.realize_sentence /-
 theorem realize_sentence (g : M ≃[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ := by
   rw [sentence.realize, sentence.realize, ← g.realize_formula, Unique.eq_default (g ∘ default)]
 #align first_order.language.equiv.realize_sentence FirstOrder.Language.Equiv.realize_sentence
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Equiv.theory_model /-
 theorem theory_model (g : M ≃[L] N) [M ⊨ T] : N ⊨ T :=
   ⟨fun φ hφ => (g.realize_sentence φ).1 (Theory.realize_sentence_of_mem T hφ)⟩
 #align first_order.language.equiv.Theory_model FirstOrder.Language.Equiv.theory_model
+-/
 
+#print FirstOrder.Language.Equiv.elementarilyEquivalent /-
 theorem elementarilyEquivalent (g : M ≃[L] N) : M ≅[L] N :=
   elementarilyEquivalent_iff.2 g.realize_sentence
 #align first_order.language.equiv.elementarily_equivalent FirstOrder.Language.Equiv.elementarilyEquivalent
+-/
 
 end Equiv
 
@@ -1207,6 +1266,7 @@ section Cardinality
 variable (L)
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Sentence.realize_cardGe /-
 @[simp]
 theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ (#M) :=
   by
@@ -1226,6 +1286,7 @@ theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ (#M
   · rintro _ i j ij rfl
     simp [ij]
 #align first_order.language.sentence.realize_card_ge FirstOrder.Language.Sentence.realize_cardGe
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 #print FirstOrder.Language.model_infiniteTheory_iff /-
@@ -1259,6 +1320,7 @@ instance model_nonempty [h : Nonempty M] : M ⊨ L.nonemptyTheory :=
 -/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.model_distinctConstantsTheory /-
 theorem model_distinctConstantsTheory {M : Type w} [L[[α]].Structure M] (s : Set α) :
     M ⊨ L.distinctConstantsTheory s ↔ Set.InjOn (fun i : α => (L.con i : M)) s :=
   by
@@ -1274,6 +1336,7 @@ theorem model_distinctConstantsTheory {M : Type w} [L[[α]].Structure M] (s : Se
     simp only [sentence.realize, formula.realize_not, formula.realize_equal, term.realize_constants]
     exact fun contra => ab (h as bs contra)
 #align first_order.language.model_distinct_constants_theory FirstOrder.Language.model_distinctConstantsTheory
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 #print FirstOrder.Language.card_le_of_model_distinctConstantsTheory /-
@@ -1287,54 +1350,74 @@ end Cardinality
 
 namespace ElementarilyEquivalent
 
+#print FirstOrder.Language.ElementarilyEquivalent.symm /-
 @[symm]
 theorem symm (h : M ≅[L] N) : N ≅[L] M :=
   h.symm
 #align first_order.language.elementarily_equivalent.symm FirstOrder.Language.ElementarilyEquivalent.symm
+-/
 
+#print FirstOrder.Language.ElementarilyEquivalent.trans /-
 @[trans]
 theorem trans (MN : M ≅[L] N) (NP : N ≅[L] P) : M ≅[L] P :=
   MN.trans NP
 #align first_order.language.elementarily_equivalent.trans FirstOrder.Language.ElementarilyEquivalent.trans
+-/
 
+#print FirstOrder.Language.ElementarilyEquivalent.completeTheory_eq /-
 theorem completeTheory_eq (h : M ≅[L] N) : L.completeTheory M = L.completeTheory N :=
   h
 #align first_order.language.elementarily_equivalent.complete_theory_eq FirstOrder.Language.ElementarilyEquivalent.completeTheory_eq
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.ElementarilyEquivalent.realize_sentence /-
 theorem realize_sentence (h : M ≅[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ :=
   (elementarilyEquivalent_iff.1 h) φ
 #align first_order.language.elementarily_equivalent.realize_sentence FirstOrder.Language.ElementarilyEquivalent.realize_sentence
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.ElementarilyEquivalent.theory_model_iff /-
 theorem theory_model_iff (h : M ≅[L] N) : M ⊨ T ↔ N ⊨ T := by
   rw [Theory.model_iff_subset_complete_theory, Theory.model_iff_subset_complete_theory,
     h.complete_theory_eq]
 #align first_order.language.elementarily_equivalent.Theory_model_iff FirstOrder.Language.ElementarilyEquivalent.theory_model_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.ElementarilyEquivalent.theory_model /-
 theorem theory_model [MT : M ⊨ T] (h : M ≅[L] N) : N ⊨ T :=
   h.theory_model_iff.1 MT
 #align first_order.language.elementarily_equivalent.Theory_model FirstOrder.Language.ElementarilyEquivalent.theory_model
+-/
 
+#print FirstOrder.Language.ElementarilyEquivalent.nonempty_iff /-
 theorem nonempty_iff (h : M ≅[L] N) : Nonempty M ↔ Nonempty N :=
   (model_nonemptyTheory_iff L).symm.trans (h.theory_model_iff.trans (model_nonemptyTheory_iff L))
 #align first_order.language.elementarily_equivalent.nonempty_iff FirstOrder.Language.ElementarilyEquivalent.nonempty_iff
+-/
 
+#print FirstOrder.Language.ElementarilyEquivalent.nonempty /-
 theorem nonempty [Mn : Nonempty M] (h : M ≅[L] N) : Nonempty N :=
   h.nonempty_iff.1 Mn
 #align first_order.language.elementarily_equivalent.nonempty FirstOrder.Language.ElementarilyEquivalent.nonempty
+-/
 
+#print FirstOrder.Language.ElementarilyEquivalent.infinite_iff /-
 theorem infinite_iff (h : M ≅[L] N) : Infinite M ↔ Infinite N :=
   (model_infiniteTheory_iff L).symm.trans (h.theory_model_iff.trans (model_infiniteTheory_iff L))
 #align first_order.language.elementarily_equivalent.infinite_iff FirstOrder.Language.ElementarilyEquivalent.infinite_iff
+-/
 
+#print FirstOrder.Language.ElementarilyEquivalent.infinite /-
 theorem infinite [Mi : Infinite M] (h : M ≅[L] N) : Infinite N :=
   h.infinite_iff.1 Mi
 #align first_order.language.elementarily_equivalent.infinite FirstOrder.Language.ElementarilyEquivalent.infinite
+-/
 
 end ElementarilyEquivalent

6cf59007

bump to nightly-2023-06-04-18

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

3fb4268b

Diff

@@ -876,7 +876,7 @@ variable (L)
 #print FirstOrder.Language.completeTheory /-
 /-- The complete theory of a structure `M` is the set of all sentences `M` satisfies. -/
 def completeTheory : L.Theory :=
-  { φ | M ⊨ φ }
+  {φ | M ⊨ φ}
 #align first_order.language.complete_theory FirstOrder.Language.completeTheory
 -/

6cf59007

bump to nightly-2023-06-01-16

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

b9c87c70

Diff

@@ -152,7 +152,7 @@ theorem realize_restrictVar [DecidableEq α] {t : L.term α} {s : Set α} (h : 
   by
   induction' t with _ _ _ _ ih
   · rfl
-  · simp_rw [var_finset, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
+  · simp_rw [var_finset, Finset.coe_biUnion, Set.iUnion_subset_iff] at h 
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var FirstOrder.Language.Term.realize_restrictVar
 -/
@@ -165,7 +165,7 @@ theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.term (Sum
   by
   induction' t with a _ _ _ ih
   · cases a <;> rfl
-  · simp_rw [var_finset_left, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
+  · simp_rw [var_finset_left, Finset.coe_biUnion, Set.iUnion_subset_iff] at h 
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var_left FirstOrder.Language.Term.realize_restrictVarLeft
 
@@ -1055,7 +1055,7 @@ theorem realize_exs {φ : L.BoundedFormula α n} {v : α → M} :
     · rintro ⟨xs, x, h⟩
       exact ⟨_, h⟩
     · rintro ⟨xs, h⟩
-      rw [← Fin.snoc_init_self xs] at h
+      rw [← Fin.snoc_init_self xs] at h 
       exact ⟨_, _, h⟩
 #align first_order.language.bounded_formula.realize_exs FirstOrder.Language.BoundedFormula.realize_exs
 -/
@@ -1075,7 +1075,7 @@ theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M
   · rw [to_formula, formula.realize, realize_all, realize_all]
     refine' forall_congr' fun a => _
     have h := ih3 (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a))
-    simp only [Sum.elim_comp_inl, Sum.elim_comp_inr] at h
+    simp only [Sum.elim_comp_inl, Sum.elim_comp_inr] at h 
     rw [← h, realize_relabel, formula.realize]
     rcongr
     · cases x
@@ -1108,11 +1108,11 @@ theorem realize_boundedFormula (g : M ≃[L] N) (φ : L.BoundedFormula α n) {v
     constructor
     · intro h a
       have h' := h (g a)
-      rw [← Fin.comp_snoc, ih3] at h'
+      rw [← Fin.comp_snoc, ih3] at h' 
       exact h'
     · intro h a
       have h' := h (g.symm a)
-      rw [← ih3, Fin.comp_snoc, g.apply_symm_apply] at h'
+      rw [← ih3, Fin.comp_snoc, g.apply_symm_apply] at h' 
       exact h'
 #align first_order.language.equiv.realize_bounded_formula FirstOrder.Language.Equiv.realize_boundedFormula
 
@@ -1221,7 +1221,7 @@ theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ (#M
     contrapose! ij
     have hij := h _ i j ij rfl
     simp only [bounded_formula.realize_not, term.realize, bounded_formula.realize_bd_equal,
-      Sum.elim_inr] at hij
+      Sum.elim_inr] at hij 
     exact hij
   · rintro _ i j ij rfl
     simp [ij]
@@ -1268,7 +1268,7 @@ theorem model_distinctConstantsTheory {M : Type w} [L[[α]].Structure M] (s : Se
   · contrapose! ab
     have h' := h _ a b ⟨⟨as, bs⟩, ab⟩ rfl
     simp only [sentence.realize, formula.realize_not, formula.realize_equal,
-      term.realize_constants] at h'
+      term.realize_constants] at h' 
     exact h'
   · rintro h φ a b ⟨⟨as, bs⟩, ab⟩ rfl
     simp only [sentence.realize, formula.realize_not, formula.realize_equal, term.realize_constants]

6cf59007

bump to nightly-2023-05-28-18

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

8d353152

Diff

@@ -67,7 +67,7 @@ variable {M : Type w} {N P : Type _} [L.Structure M] [L.Structure N] [L.Structur
 
 variable {α : Type u'} {β : Type v'}
 
-open FirstOrder Cardinal
+open scoped FirstOrder Cardinal
 
 open Structure Cardinal Fin

6cf59007

bump to nightly-2023-05-28-06

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

ba5d8b97

Diff

@@ -93,9 +93,6 @@ theorem realize_relabel {t : L.term α} {g : α → β} {v : β → M} :
 #align first_order.language.term.realize_relabel FirstOrder.Language.Term.realize_relabel
 -/
 
-/- warning: first_order.language.term.realize_lift_at -> FirstOrder.Language.Term.realize_liftAt is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align first_order.language.term.realize_lift_at FirstOrder.Language.Term.realize_liftAtₓ'. -/
 @[simp]
 theorem realize_liftAt {n n' m : ℕ} {t : L.term (Sum α (Fin n))} {v : Sum α (Fin (n + n')) → M} :
     (t.liftAt n' m).realize v =
@@ -103,12 +100,6 @@ theorem realize_liftAt {n n' m : ℕ} {t : L.term (Sum α (Fin n))} {v : Sum α
   realize_relabel
 #align first_order.language.term.realize_lift_at FirstOrder.Language.Term.realize_liftAt
 
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 @[simp]
 theorem realize_constants {c : L.Constants} {v : α → M} : c.term.realize v = c :=
   funMap_eq_coe_constants
@@ -166,12 +157,6 @@ theorem realize_restrictVar [DecidableEq α] {t : L.term α} {s : Set α} (h : 
 #align first_order.language.term.realize_restrict_var FirstOrder.Language.Term.realize_restrictVar
 -/
 
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 @[simp]
 theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.term (Sum α γ)} {s : Set α}
     (h : ↑t.varFinsetLeft ⊆ s) {v : α → M} {xs : γ → M} :
@@ -184,12 +169,6 @@ theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.term (Sum
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var_left FirstOrder.Language.Term.realize_restrictVarLeft
 
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 @[simp]
 theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
     {t : L[[α]].term β} {v : β → M} :
@@ -207,12 +186,6 @@ theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).
       · exact isEmptyElim f
 #align first_order.language.term.realize_constants_to_vars FirstOrder.Language.Term.realize_constantsToVars
 
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 @[simp]
 theorem realize_varsToConstants [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
     {t : L.term (Sum α β)} {v : β → M} :
@@ -242,12 +215,6 @@ end Term
 
 namespace Lhom
 
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 @[simp]
 theorem realize_onTerm [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (t : L.term α)
     (v : α → M) : (φ.onTerm t).realize v = t.realize v :=
@@ -259,12 +226,6 @@ theorem realize_onTerm [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
 
 end Lhom
 
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 @[simp]
 theorem Hom.realize_term (g : M →[L] N) {t : L.term α} {v : α → M} :
     t.realize (g ∘ v) = g (t.realize v) := by
@@ -276,24 +237,12 @@ theorem Hom.realize_term (g : M →[L] N) {t : L.term α} {v : α → M} :
     simp [t_ih x]
 #align first_order.language.hom.realize_term FirstOrder.Language.Hom.realize_term
 
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 @[simp]
 theorem Embedding.realize_term {v : α → M} (t : L.term α) (g : M ↪[L] N) :
     t.realize (g ∘ v) = g (t.realize v) :=
   g.toHom.realize_term
 #align first_order.language.embedding.realize_term FirstOrder.Language.Embedding.realize_term
 
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 @[simp]
 theorem Equiv.realize_term {v : α → M} (t : L.term α) (g : M ≃[L] N) :
     t.realize (g ∘ v) = g (t.realize v) :=
@@ -415,12 +364,6 @@ theorem realize_sup : (φ ⊔ ψ).realize v xs ↔ φ.realize v xs ∨ ψ.realiz
 #align first_order.language.bounded_formula.realize_sup FirstOrder.Language.BoundedFormula.realize_sup
 -/
 
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 @[simp]
 theorem realize_foldr_sup (l : List (L.BoundedFormula α n)) (v : α → M) (xs : Fin n → M) :
     (l.foldr (· ⊔ ·) ⊥).realize v xs ↔ ∃ φ ∈ l, BoundedFormula.Realize φ v xs :=
@@ -455,9 +398,6 @@ theorem realize_iff : (φ.Iff ψ).realize v xs ↔ (φ.realize v xs ↔ ψ.reali
 #align first_order.language.bounded_formula.realize_iff FirstOrder.Language.BoundedFormula.realize_iff
 -/
 
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 theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
     {v : α → M} {xs : Fin n → M} : (φ.castLE h').realize v xs ↔ φ.realize v (xs ∘ Fin.cast h) :=
   by
@@ -465,9 +405,6 @@ theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.Boun
   simp only [cast_le_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]
 #align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eq
 
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 theorem realize_mapTermRel_id [L'.Structure M]
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin n))}
     {fr : ∀ n, L.Relations n → L'.Relations n} {n} {φ : L.BoundedFormula α n} {v : α → M}
@@ -486,9 +423,6 @@ theorem realize_mapTermRel_id [L'.Structure M]
   · simp only [map_term_rel, realize, ih, id.def]
 #align first_order.language.bounded_formula.realize_map_term_rel_id FirstOrder.Language.BoundedFormula.realize_mapTermRel_id
 
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 theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin (k + n)))}
     {fr : ∀ n, L.Relations n → L'.Relations n} {n} {φ : L.BoundedFormula α n}
@@ -509,9 +443,6 @@ theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
   · simp [map_term_rel, realize, ih, hv]
 #align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe
 
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-<too large>
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 theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum β (Fin m)} {v : β → M}
     {xs : Fin (m + n) → M} :
     (φ.relabel g).realize v xs ↔
@@ -519,9 +450,6 @@ theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum
   by rw [relabel, realize_map_term_rel_add_cast_le] <;> intros <;> simp
 #align first_order.language.bounded_formula.realize_relabel FirstOrder.Language.BoundedFormula.realize_relabel
 
-/- warning: first_order.language.bounded_formula.realize_lift_at -> FirstOrder.Language.BoundedFormula.realize_liftAt is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAtₓ'. -/
 theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + n') → M}
     (hmn : m + n' ≤ n + 1) :
     (φ.liftAt n' m).realize v xs ↔
@@ -553,9 +481,6 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
       split_ifs <;> simp
 #align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAt
 
-/- warning: first_order.language.bounded_formula.realize_lift_at_one -> FirstOrder.Language.BoundedFormula.realize_liftAt_one is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_oneₓ'. -/
 theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + 1) → M}
     (hmn : m ≤ n) :
     (φ.liftAt 1 m).realize v xs ↔
@@ -563,9 +488,6 @@ theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α →
   by simp_rw [realize_lift_at (add_le_add_right hmn 1), cast_succ, add_nat_one]
 #align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_one
 
-/- warning: first_order.language.bounded_formula.realize_lift_at_one_self -> FirstOrder.Language.BoundedFormula.realize_liftAt_one_self is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at_one_self FirstOrder.Language.BoundedFormula.realize_liftAt_one_selfₓ'. -/
 @[simp]
 theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α → M}
     {xs : Fin (n + 1) → M} : (φ.liftAt 1 n).realize v xs ↔ φ.realize v (xs ∘ castSucc) :=
@@ -723,12 +645,6 @@ namespace Lhom
 
 open BoundedFormula
 
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-Case conversion may be inaccurate. Consider using '#align first_order.language.Lhom.realize_on_bounded_formula FirstOrder.Language.LHom.realize_onBoundedFormulaₓ'. -/
 @[simp]
 theorem realize_onBoundedFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] {n : ℕ}
     (ψ : L.BoundedFormula α n) {v : α → M} {xs : Fin n → M} :
@@ -879,24 +795,12 @@ theorem realize_graph {f : L.Functions n} {x : Fin n → M} {y : M} :
 
 end Formula
 
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-Case conversion may be inaccurate. Consider using '#align first_order.language.Lhom.realize_on_formula FirstOrder.Language.LHom.realize_onFormulaₓ'. -/
 @[simp]
 theorem LHom.realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (ψ : L.Formula α)
     {v : α → M} : (φ.onFormula ψ).realize v ↔ ψ.realize v :=
   φ.realize_onBoundedFormula ψ
 #align first_order.language.Lhom.realize_on_formula FirstOrder.Language.LHom.realize_onFormula
 
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 @[simp]
 theorem LHom.setOf_realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
     (ψ : L.Formula α) : (setOf (φ.onFormula ψ).realize : Set (α → M)) = setOf ψ.realize := by ext;
@@ -961,12 +865,6 @@ theorem realize_equivSentence_symm (φ : L[[α]].Sentence) (v : α → M) :
 
 end Formula
 
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-Case conversion may be inaccurate. Consider using '#align first_order.language.Lhom.realize_on_sentence FirstOrder.Language.LHom.realize_onSentenceₓ'. -/
 @[simp]
 theorem LHom.realize_onSentence [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
     (ψ : L.Sentence) : M ⊨ φ.onSentence ψ ↔ M ⊨ ψ :=
@@ -1004,12 +902,6 @@ theorem mem_completeTheory {φ : Sentence L} : φ ∈ L.completeTheory M ↔ M 
 #align first_order.language.mem_complete_theory FirstOrder.Language.mem_completeTheory
 -/
 
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-Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent_iff FirstOrder.Language.elementarilyEquivalent_iffₓ'. -/
 theorem elementarilyEquivalent_iff : M ≅[L] N ↔ ∀ φ : L.Sentence, M ⊨ φ ↔ N ⊨ φ := by
   simp only [elementarily_equivalent, Set.ext_iff, complete_theory, Set.mem_setOf_eq]
 #align first_order.language.elementarily_equivalent_iff FirstOrder.Language.elementarilyEquivalent_iff
@@ -1047,12 +939,6 @@ theorem Theory.realize_sentence_of_mem [M ⊨ T] {φ : L.Sentence} (h : φ ∈ T
 #align first_order.language.Theory.realize_sentence_of_mem FirstOrder.Language.Theory.realize_sentence_of_mem
 -/
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 @[simp]
@@ -1079,12 +965,6 @@ theorem Model.mono {T' : L.Theory} (h : M ⊨ T') (hs : T ⊆ T') : M ⊨ T :=
 #align first_order.language.Theory.model.mono FirstOrder.Language.Theory.Model.mono
 -/
 
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-Case conversion may be inaccurate. Consider using '#align first_order.language.Theory.model.union FirstOrder.Language.Theory.Model.unionₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1094,12 +974,6 @@ theorem Model.union {T' : L.Theory} (h : M ⊨ T) (h' : M ⊨ T') : M ⊨ T ∪
   exact fun φ hφ => hφ.elim (h _) (h' _)
 #align first_order.language.Theory.model.union FirstOrder.Language.Theory.Model.union
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align first_order.language.Theory.model_union_iff FirstOrder.Language.Theory.model_union_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1141,12 +1015,6 @@ instance model_completeTheory : M ⊨ L.completeTheory M :=
 
 variable (M N)
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1227,12 +1095,6 @@ end BoundedFormula
 
 namespace Equiv
 
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 @[simp]
 theorem realize_boundedFormula (g : M ≃[L] N) (φ : L.BoundedFormula α n) {v : α → M}
     {xs : Fin n → M} : φ.realize (g ∘ v) (g ∘ xs) ↔ φ.realize v xs :=
@@ -1254,12 +1116,6 @@ theorem realize_boundedFormula (g : M ≃[L] N) (φ : L.BoundedFormula α n) {v
       exact h'
 #align first_order.language.equiv.realize_bounded_formula FirstOrder.Language.Equiv.realize_boundedFormula
 
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 @[simp]
 theorem realize_formula (g : M ≃[L] N) (φ : L.Formula α) {v : α → M} :
     φ.realize (g ∘ v) ↔ φ.realize v := by
@@ -1267,36 +1123,18 @@ theorem realize_formula (g : M ≃[L] N) (φ : L.Formula α) {v : α → M} :
     Unique.eq_default (g ∘ default)]
 #align first_order.language.equiv.realize_formula FirstOrder.Language.Equiv.realize_formula
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem realize_sentence (g : M ≃[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ := by
   rw [sentence.realize, sentence.realize, ← g.realize_formula, Unique.eq_default (g ∘ default)]
 #align first_order.language.equiv.realize_sentence FirstOrder.Language.Equiv.realize_sentence
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem theory_model (g : M ≃[L] N) [M ⊨ T] : N ⊨ T :=
   ⟨fun φ hφ => (g.realize_sentence φ).1 (Theory.realize_sentence_of_mem T hφ)⟩
 #align first_order.language.equiv.Theory_model FirstOrder.Language.Equiv.theory_model
 
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 theorem elementarilyEquivalent (g : M ≃[L] N) : M ≅[L] N :=
   elementarilyEquivalent_iff.2 g.realize_sentence
 #align first_order.language.equiv.elementarily_equivalent FirstOrder.Language.Equiv.elementarilyEquivalent
@@ -1368,12 +1206,6 @@ section Cardinality
 
 variable (L)
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 @[simp]
 theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ (#M) :=
@@ -1426,12 +1258,6 @@ instance model_nonempty [h : Nonempty M] : M ⊨ L.nonemptyTheory :=
 #align first_order.language.model_nonempty FirstOrder.Language.model_nonempty
 -/
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem model_distinctConstantsTheory {M : Type w} [L[[α]].Structure M] (s : Set α) :
     M ⊨ L.distinctConstantsTheory s ↔ Set.InjOn (fun i : α => (L.con i : M)) s :=
@@ -1461,56 +1287,26 @@ end Cardinality
 
 namespace ElementarilyEquivalent
 
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 @[symm]
 theorem symm (h : M ≅[L] N) : N ≅[L] M :=
   h.symm
 #align first_order.language.elementarily_equivalent.symm FirstOrder.Language.ElementarilyEquivalent.symm
 
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 @[trans]
 theorem trans (MN : M ≅[L] N) (NP : N ≅[L] P) : M ≅[L] P :=
   MN.trans NP
 #align first_order.language.elementarily_equivalent.trans FirstOrder.Language.ElementarilyEquivalent.trans
 
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 theorem completeTheory_eq (h : M ≅[L] N) : L.completeTheory M = L.completeTheory N :=
   h
 #align first_order.language.elementarily_equivalent.complete_theory_eq FirstOrder.Language.ElementarilyEquivalent.completeTheory_eq
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem realize_sentence (h : M ≅[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ :=
   (elementarilyEquivalent_iff.1 h) φ
 #align first_order.language.elementarily_equivalent.realize_sentence FirstOrder.Language.ElementarilyEquivalent.realize_sentence
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem theory_model_iff (h : M ≅[L] N) : M ⊨ T ↔ N ⊨ T := by
@@ -1518,54 +1314,24 @@ theorem theory_model_iff (h : M ≅[L] N) : M ⊨ T ↔ N ⊨ T := by
     h.complete_theory_eq]
 #align first_order.language.elementarily_equivalent.Theory_model_iff FirstOrder.Language.ElementarilyEquivalent.theory_model_iff
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem theory_model [MT : M ⊨ T] (h : M ≅[L] N) : N ⊨ T :=
   h.theory_model_iff.1 MT
 #align first_order.language.elementarily_equivalent.Theory_model FirstOrder.Language.ElementarilyEquivalent.theory_model
 
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 theorem nonempty_iff (h : M ≅[L] N) : Nonempty M ↔ Nonempty N :=
   (model_nonemptyTheory_iff L).symm.trans (h.theory_model_iff.trans (model_nonemptyTheory_iff L))
 #align first_order.language.elementarily_equivalent.nonempty_iff FirstOrder.Language.ElementarilyEquivalent.nonempty_iff
 
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 theorem nonempty [Mn : Nonempty M] (h : M ≅[L] N) : Nonempty N :=
   h.nonempty_iff.1 Mn
 #align first_order.language.elementarily_equivalent.nonempty FirstOrder.Language.ElementarilyEquivalent.nonempty
 
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 theorem infinite_iff (h : M ≅[L] N) : Infinite M ↔ Infinite N :=
   (model_infiniteTheory_iff L).symm.trans (h.theory_model_iff.trans (model_infiniteTheory_iff L))
 #align first_order.language.elementarily_equivalent.infinite_iff FirstOrder.Language.ElementarilyEquivalent.infinite_iff
 
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 theorem infinite [Mi : Infinite M] (h : M ≅[L] N) : Infinite N :=
   h.infinite_iff.1 Mi
 #align first_order.language.elementarily_equivalent.infinite FirstOrder.Language.ElementarilyEquivalent.infinite

6cf59007

bump to nightly-2023-05-28-04

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

6797a637

Diff

@@ -899,9 +899,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align first_order.language.Lhom.set_of_realize_on_formula FirstOrder.Language.LHom.setOf_realize_onFormulaₓ'. -/
 @[simp]
 theorem LHom.setOf_realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
-    (ψ : L.Formula α) : (setOf (φ.onFormula ψ).realize : Set (α → M)) = setOf ψ.realize :=
-  by
-  ext
+    (ψ : L.Formula α) : (setOf (φ.onFormula ψ).realize : Set (α → M)) = setOf ψ.realize := by ext;
   simp
 #align first_order.language.Lhom.set_of_realize_on_formula FirstOrder.Language.LHom.setOf_realize_onFormula

6cf59007

bump to nightly-2023-05-28-03

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

6c6011cf

Diff

@@ -94,10 +94,7 @@ theorem realize_relabel {t : L.term α} {g : α → β} {v : β → M} :
 -/
 
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x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.addNat n n') i)))) t)
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 Case conversion may be inaccurate. Consider using '#align first_order.language.term.realize_lift_at FirstOrder.Language.Term.realize_liftAtₓ'. -/
 @[simp]
 theorem realize_liftAt {n n' m : ℕ} {t : L.term (Sum α (Fin n))} {v : Sum α (Fin (n + n')) → M} :
@@ -459,10 +456,7 @@ theorem realize_iff : (φ.Iff ψ).realize v xs ↔ (φ.realize v xs ↔ ψ.reali
 -/
 
 /- warning: first_order.language.bounded_formula.realize_cast_le_of_eq -> FirstOrder.Language.BoundedFormula.realize_castLe_of_eq is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eqₓ'. -/
 theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
     {v : α → M} {xs : Fin n → M} : (φ.castLE h').realize v xs ↔ φ.realize v (xs ∘ Fin.cast h) :=
@@ -472,10 +466,7 @@ theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.Boun
 #align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eq
 
 /- warning: first_order.language.bounded_formula.realize_map_term_rel_id -> FirstOrder.Language.BoundedFormula.realize_mapTermRel_id is a dubious translation:
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 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_map_term_rel_id FirstOrder.Language.BoundedFormula.realize_mapTermRel_idₓ'. -/
 theorem realize_mapTermRel_id [L'.Structure M]
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin n))}
@@ -496,10 +487,7 @@ theorem realize_mapTermRel_id [L'.Structure M]
 #align first_order.language.bounded_formula.realize_map_term_rel_id FirstOrder.Language.BoundedFormula.realize_mapTermRel_id
 
 /- warning: first_order.language.bounded_formula.realize_map_term_rel_add_cast_le -> FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLeₓ'. -/
 theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin (k + n)))}
@@ -522,10 +510,7 @@ theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
 #align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_relabel FirstOrder.Language.BoundedFormula.realize_relabelₓ'. -/
 theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum β (Fin m)} {v : β → M}
     {xs : Fin (m + n) → M} :
@@ -535,10 +520,7 @@ theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum
 #align first_order.language.bounded_formula.realize_relabel FirstOrder.Language.BoundedFormula.realize_relabel
 
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 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAtₓ'. -/
 theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + n') → M}
     (hmn : m + n' ≤ n + 1) :
@@ -572,10 +554,7 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
 #align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAt
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_oneₓ'. -/
 theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + 1) → M}
     (hmn : m ≤ n) :
@@ -585,10 +564,7 @@ theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α →
 #align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_one
 
 /- warning: first_order.language.bounded_formula.realize_lift_at_one_self -> FirstOrder.Language.BoundedFormula.realize_liftAt_one_self is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at_one_self FirstOrder.Language.BoundedFormula.realize_liftAt_one_selfₓ'. -/
 @[simp]
 theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α → M}

6cf59007

bump to nightly-2023-05-19-16

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

a31df521

Diff

@@ -97,7 +97,7 @@ theorem realize_relabel {t : L.term α} {g : α → β} {v : β → M} :
 lean 3 declaration is
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 but is expected to have type
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(InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin n) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.addNat n n') i)))) t)
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} {n' : Nat} {m : Nat} {t : FirstOrder.Language.Term.{u1, u2, u4} L (Sum.{u4, 0} α (Fin n))} {v : (Sum.{u4, 0} α (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) -> M}, Eq.{succ u3} M (FirstOrder.Language.Term.realize.{u1, u2, u3, u4} L M _inst_1 (Sum.{u4, 0} α (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) v (FirstOrder.Language.Term.liftAt.{u1, u2, u4} L α n n' m t)) (FirstOrder.Language.Term.realize.{u1, u2, u3, u4} L M _inst_1 (Sum.{u4, 0} α (Fin n)) (Function.comp.{succ u4, succ u4, succ u3} (Sum.{u4, 0} α (Fin n)) (Sum.{u4, 0} α (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) M v (Sum.map.{u4, 0, u4, 0} α α (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (id.{succ u4} α) (fun (i : Fin n) => ite.{1} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (LT.lt.{0} Nat instLTNat (Fin.val n i) m) (Nat.decLt (Fin.val n i) m) (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin n) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.castAdd n n') i) (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin n) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.addNat n n') i)))) t)
 Case conversion may be inaccurate. Consider using '#align first_order.language.term.realize_lift_at FirstOrder.Language.Term.realize_liftAtₓ'. -/
 @[simp]
 theorem realize_liftAt {n n' m : ℕ} {t : L.term (Sum α (Fin n))} {v : Sum α (Fin (n + n')) → M} :
@@ -462,7 +462,7 @@ theorem realize_iff : (φ.Iff ψ).realize v xs ↔ (φ.realize v xs ↔ ψ.reali
 lean 3 declaration is
   forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {m : Nat} {n : Nat} (h : Eq.{1} Nat m n) {h' : LE.le.{0} Nat Nat.hasLe m n} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α m} {v : α -> M} {xs : (Fin n) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n (FirstOrder.Language.BoundedFormula.castLE.{u1, u2, u4} L α m n h' φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α m φ v (Function.comp.{1, 1, succ u3} (Fin m) (Fin n) M xs (coeFn.{1, 1} (OrderIso.{0, 0} (Fin m) (Fin n) (Fin.hasLe m) (Fin.hasLe n)) (fun (_x : RelIso.{0, 0} (Fin m) (Fin n) (LE.le.{0} (Fin m) (Fin.hasLe m)) (LE.le.{0} (Fin n) (Fin.hasLe n))) => (Fin m) -> (Fin n)) (RelIso.hasCoeToFun.{0, 0} (Fin m) (Fin n) (LE.le.{0} (Fin m) (Fin.hasLe m)) (LE.le.{0} (Fin n) (Fin.hasLe n))) (Fin.cast m n h))))
 but is expected to have type
-  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {m : Nat} {n : Nat} (h : Eq.{1} Nat m n) {h' : LE.le.{0} Nat instLENat m n} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α m} {v : α -> M} {xs : (Fin n) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n (FirstOrder.Language.BoundedFormula.castLE.{u1, u2, u4} L α m n h' φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α m φ v (Function.comp.{1, 1, succ u3} (Fin m) (Fin n) M xs (FunLike.coe.{1, 1, 1} (RelIso.{0, 0} (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298)) (Fin m) (fun (_x : Fin m) => Fin n) (RelHomClass.toFunLike.{0, 0, 0} (RelIso.{0, 0} (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298)) (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.instRelHomClassRelIso.{0, 0} (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298))) (Fin.cast m n h))))
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {m : Nat} {n : Nat} (h : Eq.{1} Nat m n) {h' : LE.le.{0} Nat instLENat m n} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α m} {v : α -> M} {xs : (Fin n) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n (FirstOrder.Language.BoundedFormula.castLE.{u1, u2, u4} L α m n h' φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α m φ v (Function.comp.{1, 1, succ u3} (Fin m) (Fin n) M xs (FunLike.coe.{1, 1, 1} (RelIso.{0, 0} (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1285 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1287 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1285 x._@.Mathlib.Order.Hom.Basic._hyg.1287) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1300 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1302 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1300 x._@.Mathlib.Order.Hom.Basic._hyg.1302)) (Fin m) (fun (_x : Fin m) => Fin n) (RelHomClass.toFunLike.{0, 0, 0} (RelIso.{0, 0} (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1285 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1287 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1285 x._@.Mathlib.Order.Hom.Basic._hyg.1287) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1300 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1302 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1300 x._@.Mathlib.Order.Hom.Basic._hyg.1302)) (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1285 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1287 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1285 x._@.Mathlib.Order.Hom.Basic._hyg.1287) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1300 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1302 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1300 x._@.Mathlib.Order.Hom.Basic._hyg.1302) (RelIso.instRelHomClassRelIso.{0, 0} (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1285 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1287 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1285 x._@.Mathlib.Order.Hom.Basic._hyg.1287) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1300 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1302 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1300 x._@.Mathlib.Order.Hom.Basic._hyg.1302))) (Fin.cast m n h))))
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eqₓ'. -/
 theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
     {v : α → M} {xs : Fin n → M} : (φ.castLE h').realize v xs ↔ φ.realize v (xs ∘ Fin.cast h) :=
@@ -499,7 +499,7 @@ theorem realize_mapTermRel_id [L'.Structure M]
 lean 3 declaration is
   forall {L : FirstOrder.Language.{u1, u2}} {L' : FirstOrder.Language.{u6, u7}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {β : Type.{u5}} [_inst_4 : FirstOrder.Language.Structure.{u6, u7, u3} L' M] {k : Nat} {ft : forall (n : Nat), (FirstOrder.Language.Term.{u1, u2, u4} L (Sum.{u4, 0} α (Fin n))) -> (FirstOrder.Language.Term.{u6, u7, u5} L' (Sum.{u5, 0} β (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n))))} {fr : forall (n : Nat), (FirstOrder.Language.Relations.{u1, u2} L n) -> (FirstOrder.Language.Relations.{u6, u7} L' n)} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} (v : forall {n : Nat}, ((Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) -> M) -> α -> M) {v' : β -> M} (xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) -> M), (forall (n : Nat) (t : FirstOrder.Language.Term.{u1, u2, u4} L (Sum.{u4, 0} α (Fin n))) (xs' : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) -> M), Eq.{succ u3} M (FirstOrder.Language.Term.realize.{u6, u7, u3, u5} L' M _inst_4 (Sum.{u5, 0} β (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n))) (Sum.elim.{u5, 0, succ u3} β (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) M v' xs') (ft n t)) (FirstOrder.Language.Term.realize.{u1, u2, u3, u4} L M _inst_1 (Sum.{u4, 0} α (Fin n)) (Sum.elim.{u4, 0, succ u3} α (Fin n) M (v n xs') (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) M xs' (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) (Fin.hasLe n) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)))) (Fin.natAdd k n)))) t)) -> (forall (n : Nat) (R : FirstOrder.Language.Relations.{u1, u2} L n) (x : (Fin n) -> M), Eq.{1} Prop (FirstOrder.Language.Structure.RelMap.{u6, u7, u3} L' M _inst_4 n (fr n R) x) (FirstOrder.Language.Structure.RelMap.{u1, u2, u3} L M _inst_1 n R x)) -> (forall (n : Nat) (xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) -> M) (x : M), Eq.{max (succ u4) (succ u3)} (α -> M) (v (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) (Fin.snoc.{u3} (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n) (fun (ᾰ : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => M) xs x)) (v n xs)) -> (Iff (FirstOrder.Language.BoundedFormula.Realize.{u6, u7, u3, u5} L' M _inst_4 β (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n) (FirstOrder.Language.BoundedFormula.mapTermRel.{u1, u2, u4, u5, u6, u7} L L' α β (fun (n : Nat) => HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n) ft fr (fun (n : Nat) => FirstOrder.Language.BoundedFormula.castLE.{u6, u7, u5} L' β (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) (Eq.le.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat (AddSemigroup.toHasAdd.{0} Nat Nat.addSemigroup)) k (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat (AddSemigroup.toHasAdd.{0} Nat Nat.addSemigroup)) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat (AddSemigroup.toHasAdd.{0} Nat Nat.addSemigroup)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat (AddSemigroup.toHasAdd.{0} Nat Nat.addSemigroup)) k n) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) (Eq.symm.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat (AddSemigroup.toHasAdd.{0} Nat Nat.addSemigroup)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat (AddSemigroup.toHasAdd.{0} Nat Nat.addSemigroup)) k n) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat (AddSemigroup.toHasAdd.{0} Nat Nat.addSemigroup)) k (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat (AddSemigroup.toHasAdd.{0} Nat Nat.addSemigroup)) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (add_assoc.{0} Nat Nat.addSemigroup k n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))) n φ) v' xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ (v n xs) (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) M xs (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) (Fin.hasLe n) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k n)))) (Fin.natAdd k n)))))
 but is expected to have type
-  forall {L : FirstOrder.Language.{u3, u4}} {L' : FirstOrder.Language.{u2, u1}} {M : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u3, u4, u5} L M] {α : Type.{u6}} {β : Type.{u7}} [_inst_4 : FirstOrder.Language.Structure.{u2, u1, u5} L' M] {k : Nat} {ft : forall (n : Nat), (FirstOrder.Language.Term.{u3, u4, u6} L (Sum.{u6, 0} α (Fin n))) -> (FirstOrder.Language.Term.{u2, u1, u7} L' (Sum.{u7, 0} β (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n))))} {fr : forall (n : Nat), (FirstOrder.Language.Relations.{u3, u4} L n) -> (FirstOrder.Language.Relations.{u2, u1} L' n)} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u3, u4, u6} L α n} (v : forall {n : Nat}, ((Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) -> M) -> α -> M) {v' : β -> M} (xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) -> M), (forall (n : Nat) (t : FirstOrder.Language.Term.{u3, u4, u6} L (Sum.{u6, 0} α (Fin n))) (xs' : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) -> M), Eq.{succ u5} M (FirstOrder.Language.Term.realize.{u2, u1, u5, u7} L' M _inst_4 (Sum.{u7, 0} β (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n))) (Sum.elim.{u7, 0, succ u5} β (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) M v' xs') (ft n t)) (FirstOrder.Language.Term.realize.{u3, u4, u5, u6} L M _inst_1 (Sum.{u6, 0} α (Fin n)) (Sum.elim.{u6, 0, succ u5} α (Fin n) M (v n xs') (Function.comp.{1, 1, succ u5} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) M xs' (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.natAdd k n)))) t)) -> (forall (n : Nat) (R : FirstOrder.Language.Relations.{u3, u4} L n) (x : (Fin n) -> M), Eq.{1} Prop 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(HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.natAdd k n)))))
+  forall {L : FirstOrder.Language.{u3, u4}} {L' : FirstOrder.Language.{u2, u1}} {M : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u3, u4, u5} L M] {α : Type.{u6}} {β : Type.{u7}} [_inst_4 : FirstOrder.Language.Structure.{u2, u1, u5} L' M] {k : Nat} {ft : forall (n : Nat), (FirstOrder.Language.Term.{u3, u4, u6} L (Sum.{u6, 0} α (Fin n))) -> (FirstOrder.Language.Term.{u2, u1, u7} L' (Sum.{u7, 0} β (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n))))} {fr : forall (n : Nat), (FirstOrder.Language.Relations.{u3, u4} L n) -> (FirstOrder.Language.Relations.{u2, u1} L' n)} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u3, u4, u6} L α n} (v : forall {n : Nat}, ((Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) -> M) -> α -> M) {v' : β -> M} (xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) -> M), (forall (n : Nat) (t : FirstOrder.Language.Term.{u3, u4, u6} L (Sum.{u6, 0} α (Fin n))) (xs' : (Fin 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k n)) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Fin.natAdd k n)))) t)) -> (forall (n : Nat) (R : FirstOrder.Language.Relations.{u3, u4} L n) (x : (Fin n) -> M), Eq.{1} Prop 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(HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Fin.natAdd k n)))))
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLeₓ'. -/
 theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin (k + n)))}
@@ -525,7 +525,7 @@ theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
 lean 3 declaration is
   forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {β : Type.{u5}} {m : Nat} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {g : α -> (Sum.{u5, 0} β (Fin m))} {v : β -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u5} L M _inst_1 β (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n) (FirstOrder.Language.BoundedFormula.relabel.{u1, u2, u4, u5} L α β m g n φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ (Function.comp.{succ u4, succ u5, succ u3} α (Sum.{u5, 0} β (Fin m)) M (Sum.elim.{u5, 0, succ u3} β (Fin m) M v (Function.comp.{1, 1, succ u3} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) M xs (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) (Fin.hasLe m) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n))) (fun (_x : RelEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) (LE.le.{0} (Fin m) (Fin.hasLe m)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)))) => (Fin m) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) (LE.le.{0} (Fin m) (Fin.hasLe m)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)))) (Fin.castAdd m n)))) g) (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) M xs (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) (Fin.hasLe n) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n)))) (Fin.natAdd m n))))
 but is expected to have type
-  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {β : Type.{u5}} {m : Nat} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {g : α -> (Sum.{u5, 0} β (Fin m))} {v : β -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u5} L M _inst_1 β (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n) (FirstOrder.Language.BoundedFormula.relabel.{u1, u2, u4, u5} L α β m g n φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ (Function.comp.{succ u4, succ u5, succ u3} α (Sum.{u5, 0} β (Fin m)) M (Sum.elim.{u5, 0, succ u3} β (Fin m) M v (Function.comp.{1, 1, succ u3} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) M xs (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin m) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin m) (fun (_x : Fin m) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin m) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin m) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (Lattice.toInf.{0} (Fin m) (DistribLattice.toLattice.{0} (Fin m) (instDistribLattice.{0} (Fin m) (Fin.instLinearOrderFin m)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin m) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (DistribLattice.toLattice.{0} (Fin m) (instDistribLattice.{0} (Fin m) (Fin.instLinearOrderFin m))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin m) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (Fin.instLinearOrderFin m) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castAdd m n)))) g) (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) M xs (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.natAdd m n))))
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {β : Type.{u5}} {m : Nat} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {g : α -> (Sum.{u5, 0} β (Fin m))} {v : β -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u5} L M _inst_1 β (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n) (FirstOrder.Language.BoundedFormula.relabel.{u1, u2, u4, u5} L α β m g n φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ (Function.comp.{succ u4, succ u5, succ u3} α (Sum.{u5, 0} β (Fin m)) M (Sum.elim.{u5, 0, succ u3} β (Fin m) M v (Function.comp.{1, 1, succ u3} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) M xs (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin m) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin m) (fun (_x : Fin m) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin m) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin m) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (Lattice.toInf.{0} (Fin m) (DistribLattice.toLattice.{0} (Fin m) (instDistribLattice.{0} (Fin m) (Fin.instLinearOrderFin m)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin m) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (DistribLattice.toLattice.{0} (Fin m) (instDistribLattice.{0} (Fin m) (Fin.instLinearOrderFin m))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin m) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (Fin.instLinearOrderFin m) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin m) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.castAdd m n)))) g) (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) M xs (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n)) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Fin.natAdd m n))))
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_relabel FirstOrder.Language.BoundedFormula.realize_relabelₓ'. -/
 theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum β (Fin m)} {v : β → M}
     {xs : Fin (m + n) → M} :
@@ -538,7 +538,7 @@ theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum
 lean 3 declaration is
   forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} {n' : Nat} {m : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {v : α -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) -> M}, (LE.le.{0} Nat Nat.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m n') (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> (Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n') (FirstOrder.Language.BoundedFormula.liftAt.{u1, u2, u4} L α n n' m φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) M xs (fun (i : Fin n) => ite.{1} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (LT.lt.{0} Nat Nat.hasLt ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (Fin n) Nat (HasLiftT.mk.{1, 1} (Fin n) Nat (CoeTCₓ.coe.{1, 1} (Fin n) Nat (coeBase.{1, 1} (Fin n) Nat (Fin.coeToNat n)))) i) m) (Nat.decidableLt ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (Fin n) Nat (HasLiftT.mk.{1, 1} (Fin n) Nat (CoeTCₓ.coe.{1, 1} (Fin n) Nat (coeBase.{1, 1} (Fin n) Nat (Fin.coeToNat n)))) i) m) (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (Fin.hasLe n) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n'))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n'))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')))) (Fin.castAdd n n') i) (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (Fin.hasLe n) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n'))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n'))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n n')))) (Fin.addNat n n') i)))))
 but is expected to have type
-  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} {n' : Nat} {m : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {v : α -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) -> M}, (LE.le.{0} Nat instLENat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n') (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> (Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n') (FirstOrder.Language.BoundedFormula.liftAt.{u1, u2, u4} L α n n' m φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) M xs (fun (i : Fin n) => ite.{1} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (LT.lt.{0} Nat instLTNat (Fin.val n i) m) (Nat.decLt (Fin.val n i) m) (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin n) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castAdd n n') i) (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin n) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.addNat n n') i)))))
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} {n' : Nat} {m : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {v : α -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) -> M}, (LE.le.{0} Nat instLENat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m n') (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> (Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n') (FirstOrder.Language.BoundedFormula.liftAt.{u1, u2, u4} L α n n' m φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) M xs (fun (i : Fin n) => ite.{1} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (LT.lt.{0} Nat instLTNat (Fin.val n i) m) (Nat.decLt (Fin.val n i) m) (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin n) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.castAdd n n') i) (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n'))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin n) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n n')) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.addNat n n') i)))))
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAtₓ'. -/
 theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + n') → M}
     (hmn : m + n' ≤ n + 1) :
@@ -575,7 +575,7 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
 lean 3 declaration is
   forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} {m : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {v : α -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> M}, (LE.le.{0} Nat Nat.hasLe m n) -> (Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) (FirstOrder.Language.BoundedFormula.liftAt.{u1, u2, u4} L α n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) m φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) M xs (fun (i : Fin n) => ite.{1} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LT.lt.{0} Nat Nat.hasLt ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (Fin n) Nat (HasLiftT.mk.{1, 1} (Fin n) Nat (CoeTCₓ.coe.{1, 1} (Fin n) Nat (coeBase.{1, 1} (Fin n) Nat (Fin.coeToNat n)))) i) m) (Nat.decidableLt ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (Fin n) Nat (HasLiftT.mk.{1, 1} (Fin n) Nat (CoeTCₓ.coe.{1, 1} (Fin n) Nat (coeBase.{1, 1} (Fin n) Nat (Fin.coeToNat n)))) i) m) (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))) (Fin.castSucc n) i) (Fin.succ n i)))))
 but is expected to have type
-  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} {m : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {v : α -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> M}, (LE.le.{0} Nat instLENat m n) -> (Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (FirstOrder.Language.BoundedFormula.liftAt.{u1, u2, u4} L α n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) m φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) M xs (fun (i : Fin n) => ite.{1} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (LT.lt.{0} Nat instLTNat (Fin.val n i) m) (Nat.decLt (Fin.val n i) m) (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castSucc n) i) (Fin.succ n i)))))
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} {m : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {v : α -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> M}, (LE.le.{0} Nat instLENat m n) -> (Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (FirstOrder.Language.BoundedFormula.liftAt.{u1, u2, u4} L α n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) m φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) M xs (fun (i : Fin n) => ite.{1} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (LT.lt.{0} Nat instLTNat (Fin.val n i) m) (Nat.decLt (Fin.val n i) m) (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.castSucc n) i) (Fin.succ n i)))))
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_oneₓ'. -/
 theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + 1) → M}
     (hmn : m ≤ n) :
@@ -588,7 +588,7 @@ theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α →
 lean 3 declaration is
   forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {v : α -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) (FirstOrder.Language.BoundedFormula.liftAt.{u1, u2, u4} L α n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) n φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) M xs (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))) (Fin.castSucc n))))
 but is expected to have type
-  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {v : α -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (FirstOrder.Language.BoundedFormula.liftAt.{u1, u2, u4} L α n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) n φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) M xs (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castSucc n))))
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {v : α -> M} {xs : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (FirstOrder.Language.BoundedFormula.liftAt.{u1, u2, u4} L α n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) n φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v (Function.comp.{1, 1, succ u3} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) M xs (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.castSucc n))))
 Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at_one_self FirstOrder.Language.BoundedFormula.realize_liftAt_one_selfₓ'. -/
 @[simp]
 theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α → M}

6cf59007

bump to nightly-2023-05-15-12

mathlib commit https://github.com/leanprover-community/mathlib/commit/7ad820c4997738e2f542f8a20f32911f52020e26

f3ad2973

Diff

@@ -1150,10 +1150,10 @@ theorem model_iff_subset_completeTheory : M ⊨ T ↔ T ⊆ L.completeTheory M :
 -/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-#print FirstOrder.Language.Theory.CompleteTheory.subset /-
-theorem CompleteTheory.subset [MT : M ⊨ T] : T ⊆ L.completeTheory M :=
+#print FirstOrder.Language.Theory.completeTheory.subset /-
+theorem completeTheory.subset [MT : M ⊨ T] : T ⊆ L.completeTheory M :=
   model_iff_subset_completeTheory.1 MT
-#align first_order.language.Theory.complete_theory.subset FirstOrder.Language.Theory.CompleteTheory.subset
+#align first_order.language.Theory.complete_theory.subset FirstOrder.Language.Theory.completeTheory.subset
 -/
 
 end Theory

6cf59007

bump to nightly-2023-05-14-08

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

d31da313

Diff

@@ -164,7 +164,7 @@ theorem realize_restrictVar [DecidableEq α] {t : L.term α} {s : Set α} (h : 
   by
   induction' t with _ _ _ _ ih
   · rfl
-  · simp_rw [var_finset, Finset.coe_bunionᵢ, Set.unionᵢ_subset_iff] at h
+  · simp_rw [var_finset, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var FirstOrder.Language.Term.realize_restrictVar
 -/
@@ -183,7 +183,7 @@ theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.term (Sum
   by
   induction' t with a _ _ _ ih
   · cases a <;> rfl
-  · simp_rw [var_finset_left, Finset.coe_bunionᵢ, Set.unionᵢ_subset_iff] at h
+  · simp_rw [var_finset_left, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var_left FirstOrder.Language.Term.realize_restrictVarLeft

6cf59007

bump to nightly-2023-05-14-02

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

fbb256bf

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
 
 ! This file was ported from Lean 3 source module model_theory.semantics
-! leanprover-community/mathlib commit d565b3df44619c1498326936be16f1a935df0728
+! leanprover-community/mathlib commit 6cf5900728239efa287df7761ec2a1ac9cf39b29
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.ModelTheory.Syntax
 
 /-!
 # Basics on First-Order Semantics
+
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
 This file defines the interpretations of first-order terms, formulas, sentences, and theories
 in a style inspired by the [Flypitch project](https://flypitch.github.io/).

d565b3df

bump to nightly-2023-05-12-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

1d98000d

Diff

@@ -70,19 +70,16 @@ open Structure Cardinal Fin
 
 namespace Term
 
-/- warning: first_order.language.term.realize -> FirstOrder.Language.Term.realize is a dubious translation:
-lean 3 declaration is
-  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}}, (α -> M) -> (FirstOrder.Language.Term.{u1, u2, u4} L α) -> M
-but is expected to have type
-  forall {L : FirstOrder.Language.{u1, u3}} {M : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u3, u4} L M] {α : Type.{u2}}, (α -> M) -> (FirstOrder.Language.Term.{u1, u3, u2} L α) -> M
-Case conversion may be inaccurate. Consider using '#align first_order.language.term.realize FirstOrder.Language.Term.realizeₓ'. -/
+#print FirstOrder.Language.Term.realize /-
 /-- A term `t` with variables indexed by `α` can be evaluated by giving a value to each variable. -/
 @[simp]
 def realize (v : α → M) : ∀ t : L.term α, M
   | var k => v k
   | func f ts => funMap f fun i => (ts i).realize
 #align first_order.language.term.realize FirstOrder.Language.Term.realize
+-/
 
+#print FirstOrder.Language.Term.realize_relabel /-
 @[simp]
 theorem realize_relabel {t : L.term α} {g : α → β} {v : β → M} :
     (t.relabel g).realize v = t.realize (v ∘ g) :=
@@ -91,7 +88,14 @@ theorem realize_relabel {t : L.term α} {g : α → β} {v : β → M} :
   · rfl
   · simp [ih]
 #align first_order.language.term.realize_relabel FirstOrder.Language.Term.realize_relabel
+-/
 
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x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.addNat n n') i)))) t)
+Case conversion may be inaccurate. Consider using '#align first_order.language.term.realize_lift_at FirstOrder.Language.Term.realize_liftAtₓ'. -/
 @[simp]
 theorem realize_liftAt {n n' m : ℕ} {t : L.term (Sum α (Fin n))} {v : Sum α (Fin (n + n')) → M} :
     (t.liftAt n' m).realize v =
@@ -99,11 +103,18 @@ theorem realize_liftAt {n n' m : ℕ} {t : L.term (Sum α (Fin n))} {v : Sum α
   realize_relabel
 #align first_order.language.term.realize_lift_at FirstOrder.Language.Term.realize_liftAt
 
+/- warning: first_order.language.term.realize_constants -> FirstOrder.Language.Term.realize_constants is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {c : FirstOrder.Language.Constants.{u1, u2} L} {v : α -> M}, Eq.{succ u3} M (FirstOrder.Language.Term.realize.{u1, u2, u3, u4} L M _inst_1 α v (FirstOrder.Language.Constants.term.{u1, u2, u4} L α c)) ((fun (a : Type.{u1}) (b : Type.{u3}) [self : HasLiftT.{succ u1, succ u3} a b] => self.0) (FirstOrder.Language.Constants.{u1, u2} L) M (HasLiftT.mk.{succ u1, succ u3} (FirstOrder.Language.Constants.{u1, u2} L) M (CoeTCₓ.coe.{succ u1, succ u3} (FirstOrder.Language.Constants.{u1, u2} L) M (FirstOrder.Language.hasCoeT.{u1, u2, u3} L M _inst_1))) c)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {c : FirstOrder.Language.Constants.{u1, u2} L} {v : α -> M}, Eq.{succ u3} M (FirstOrder.Language.Term.realize.{u1, u2, u3, u4} L M _inst_1 α v (FirstOrder.Language.Constants.term.{u1, u2, u4} L α c)) (FirstOrder.Language.constantMap.{u1, u2, u3} L M _inst_1 c)
+Case conversion may be inaccurate. Consider using '#align first_order.language.term.realize_constants FirstOrder.Language.Term.realize_constantsₓ'. -/
 @[simp]
 theorem realize_constants {c : L.Constants} {v : α → M} : c.term.realize v = c :=
   funMap_eq_coe_constants
 #align first_order.language.term.realize_constants FirstOrder.Language.Term.realize_constants
 
+#print FirstOrder.Language.Term.realize_functions_apply₁ /-
 @[simp]
 theorem realize_functions_apply₁ {f : L.Functions 1} {t : L.term α} {v : α → M} :
     (f.apply₁ t).realize v = funMap f ![t.realize v] :=
@@ -112,7 +123,9 @@ theorem realize_functions_apply₁ {f : L.Functions 1} {t : L.term α} {v : α 
   refine' congr rfl (funext fun i => _)
   simp only [Matrix.cons_val_fin_one]
 #align first_order.language.term.realize_functions_apply₁ FirstOrder.Language.Term.realize_functions_apply₁
+-/
 
+#print FirstOrder.Language.Term.realize_functions_apply₂ /-
 @[simp]
 theorem realize_functions_apply₂ {f : L.Functions 2} {t₁ t₂ : L.term α} {v : α → M} :
     (f.apply₂ t₁ t₂).realize v = funMap f ![t₁.realize v, t₂.realize v] :=
@@ -122,11 +135,15 @@ theorem realize_functions_apply₂ {f : L.Functions 2} {t₁ t₂ : L.term α} {
   · simp only [Matrix.cons_val_zero]
   · simp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]
 #align first_order.language.term.realize_functions_apply₂ FirstOrder.Language.Term.realize_functions_apply₂
+-/
 
+#print FirstOrder.Language.Term.realize_con /-
 theorem realize_con {A : Set M} {a : A} {v : α → M} : (L.con a).term.realize v = a :=
   rfl
 #align first_order.language.term.realize_con FirstOrder.Language.Term.realize_con
+-/
 
+#print FirstOrder.Language.Term.realize_subst /-
 @[simp]
 theorem realize_subst {t : L.term α} {tf : α → L.term β} {v : β → M} :
     (t.subst tf).realize v = t.realize fun a => (tf a).realize v :=
@@ -135,7 +152,9 @@ theorem realize_subst {t : L.term α} {tf : α → L.term β} {v : β → M} :
   · rfl
   · simp [ih]
 #align first_order.language.term.realize_subst FirstOrder.Language.Term.realize_subst
+-/
 
+#print FirstOrder.Language.Term.realize_restrictVar /-
 @[simp]
 theorem realize_restrictVar [DecidableEq α] {t : L.term α} {s : Set α} (h : ↑t.varFinset ⊆ s)
     {v : α → M} : (t.restrictVar (Set.inclusion h)).realize (v ∘ coe) = t.realize v :=
@@ -145,7 +164,14 @@ theorem realize_restrictVar [DecidableEq α] {t : L.term α} {s : Set α} (h : 
   · simp_rw [var_finset, Finset.coe_bunionᵢ, Set.unionᵢ_subset_iff] at h
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var FirstOrder.Language.Term.realize_restrictVar
+-/
 
+/- warning: first_order.language.term.realize_restrict_var_left -> FirstOrder.Language.Term.realize_restrictVarLeft is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align first_order.language.term.realize_restrict_var_left FirstOrder.Language.Term.realize_restrictVarLeftₓ'. -/
 @[simp]
 theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.term (Sum α γ)} {s : Set α}
     (h : ↑t.varFinsetLeft ⊆ s) {v : α → M} {xs : γ → M} :
@@ -158,6 +184,12 @@ theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.term (Sum
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var_left FirstOrder.Language.Term.realize_restrictVarLeft
 
+/- warning: first_order.language.term.realize_constants_to_vars -> FirstOrder.Language.Term.realize_constantsToVars is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align first_order.language.term.realize_constants_to_vars FirstOrder.Language.Term.realize_constantsToVarsₓ'. -/
 @[simp]
 theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
     {t : L[[α]].term β} {v : β → M} :
@@ -175,6 +207,12 @@ theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).
       · exact isEmptyElim f
 #align first_order.language.term.realize_constants_to_vars FirstOrder.Language.Term.realize_constantsToVars
 
+/- warning: first_order.language.term.realize_vars_to_constants -> FirstOrder.Language.Term.realize_varsToConstants is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align first_order.language.term.realize_vars_to_constants FirstOrder.Language.Term.realize_varsToConstantsₓ'. -/
 @[simp]
 theorem realize_varsToConstants [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
     {t : L.term (Sum α β)} {v : β → M} :
@@ -185,6 +223,7 @@ theorem realize_varsToConstants [L[[α]].Structure M] [(lhomWithConstants L α).
   · simp [ih]
 #align first_order.language.term.realize_vars_to_constants FirstOrder.Language.Term.realize_varsToConstants
 
+#print FirstOrder.Language.Term.realize_constantsVarsEquivLeft /-
 theorem realize_constantsVarsEquivLeft [L[[α]].Structure M]
     [(lhomWithConstants L α).IsExpansionOn M] {n} {t : L[[α]].term (Sum β (Fin n))} {v : β → M}
     {xs : Fin n → M} :
@@ -197,11 +236,18 @@ theorem realize_constantsVarsEquivLeft [L[[α]].Structure M]
   rcongr
   rcases x with (a | (b | i)) <;> simp
 #align first_order.language.term.realize_constants_vars_equiv_left FirstOrder.Language.Term.realize_constantsVarsEquivLeft
+-/
 
 end Term
 
 namespace Lhom
 
+/- warning: first_order.language.Lhom.realize_on_term -> FirstOrder.Language.LHom.realize_onTerm is a dubious translation:
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+  forall {L : FirstOrder.Language.{u1, u2}} {L' : FirstOrder.Language.{u5, u6}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} [_inst_4 : FirstOrder.Language.Structure.{u5, u6, u3} L' M] (φ : FirstOrder.Language.LHom.{u1, u2, u5, u6} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u1, u2, u5, u6, u3} L L' φ M _inst_1 _inst_4] (t : FirstOrder.Language.Term.{u1, u2, u4} L α) (v : α -> M), Eq.{succ u3} M (FirstOrder.Language.Term.realize.{u5, u6, u3, u4} L' M _inst_4 α v (FirstOrder.Language.LHom.onTerm.{u1, u2, u4, u5, u6} L L' α φ t)) (FirstOrder.Language.Term.realize.{u1, u2, u3, u4} L M _inst_1 α v t)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u3, u4}} {L' : FirstOrder.Language.{u2, u1}} {M : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u3, u4, u5} L M] {α : Type.{u6}} [_inst_4 : FirstOrder.Language.Structure.{u2, u1, u5} L' M] (φ : FirstOrder.Language.LHom.{u3, u4, u2, u1} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u3, u4, u2, u1, u5} L L' φ M _inst_1 _inst_4] (t : FirstOrder.Language.Term.{u3, u4, u6} L α) (v : α -> M), Eq.{succ u5} M (FirstOrder.Language.Term.realize.{u2, u1, u5, u6} L' M _inst_4 α v (FirstOrder.Language.LHom.onTerm.{u3, u4, u6, u2, u1} L L' α φ t)) (FirstOrder.Language.Term.realize.{u3, u4, u5, u6} L M _inst_1 α v t)
+Case conversion may be inaccurate. Consider using '#align first_order.language.Lhom.realize_on_term FirstOrder.Language.LHom.realize_onTermₓ'. -/
 @[simp]
 theorem realize_onTerm [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (t : L.term α)
     (v : α → M) : (φ.onTerm t).realize v = t.realize v :=
@@ -213,6 +259,12 @@ theorem realize_onTerm [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
 
 end Lhom
 
+/- warning: first_order.language.hom.realize_term -> FirstOrder.Language.Hom.realize_term is a dubious translation:
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 @[simp]
 theorem Hom.realize_term (g : M →[L] N) {t : L.term α} {v : α → M} :
     t.realize (g ∘ v) = g (t.realize v) := by
@@ -224,12 +276,24 @@ theorem Hom.realize_term (g : M →[L] N) {t : L.term α} {v : α → M} :
     simp [t_ih x]
 #align first_order.language.hom.realize_term FirstOrder.Language.Hom.realize_term
 
+/- warning: first_order.language.embedding.realize_term -> FirstOrder.Language.Embedding.realize_term is a dubious translation:
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 @[simp]
 theorem Embedding.realize_term {v : α → M} (t : L.term α) (g : M ↪[L] N) :
     t.realize (g ∘ v) = g (t.realize v) :=
   g.toHom.realize_term
 #align first_order.language.embedding.realize_term FirstOrder.Language.Embedding.realize_term
 
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+Case conversion may be inaccurate. Consider using '#align first_order.language.equiv.realize_term FirstOrder.Language.Equiv.realize_termₓ'. -/
 @[simp]
 theorem Equiv.realize_term {v : α → M} (t : L.term α) (g : M ≃[L] N) :
     t.realize (g ∘ v) = g (t.realize v) :=
@@ -242,12 +306,7 @@ namespace BoundedFormula
 
 open Term
 
-/- warning: first_order.language.bounded_formula.realize -> FirstOrder.Language.BoundedFormula.Realize is a dubious translation:
-lean 3 declaration is
-  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {l : Nat}, (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α l) -> (α -> M) -> ((Fin l) -> M) -> Prop
-but is expected to have type
-  forall {L : FirstOrder.Language.{u1, u3}} {M : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u3, u4} L M] {α : Type.{u2}} {l : Nat}, (FirstOrder.Language.BoundedFormula.{u1, u3, u2} L α l) -> (α -> M) -> ((Fin l) -> M) -> Prop
-Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize FirstOrder.Language.BoundedFormula.Realizeₓ'. -/
+#print FirstOrder.Language.BoundedFormula.Realize /-
 /-- A bounded formula can be evaluated as true or false by giving values to each free variable. -/
 def Realize : ∀ {l} (f : L.BoundedFormula α l) (v : α → M) (xs : Fin l → M), Prop
   | _, falsum, v, xs => False
@@ -256,36 +315,48 @@ def Realize : ∀ {l} (f : L.BoundedFormula α l) (v : α → M) (xs : Fin l →
   | _, bounded_formula.imp f₁ f₂, v, xs => realize f₁ v xs → realize f₂ v xs
   | _, bounded_formula.all f, v, xs => ∀ x : M, realize f v (snoc xs x)
 #align first_order.language.bounded_formula.realize FirstOrder.Language.BoundedFormula.Realize
+-/
 
 variable {l : ℕ} {φ ψ : L.BoundedFormula α l} {θ : L.BoundedFormula α l.succ}
 
 variable {v : α → M} {xs : Fin l → M}
 
+#print FirstOrder.Language.BoundedFormula.realize_bot /-
 @[simp]
 theorem realize_bot : (⊥ : L.BoundedFormula α l).realize v xs ↔ False :=
   Iff.rfl
 #align first_order.language.bounded_formula.realize_bot FirstOrder.Language.BoundedFormula.realize_bot
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_not /-
 @[simp]
 theorem realize_not : φ.Not.realize v xs ↔ ¬φ.realize v xs :=
   Iff.rfl
 #align first_order.language.bounded_formula.realize_not FirstOrder.Language.BoundedFormula.realize_not
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_bdEqual /-
 @[simp]
 theorem realize_bdEqual (t₁ t₂ : L.term (Sum α (Fin l))) :
     (t₁.bdEqual t₂).realize v xs ↔ t₁.realize (Sum.elim v xs) = t₂.realize (Sum.elim v xs) :=
   Iff.rfl
 #align first_order.language.bounded_formula.realize_bd_equal FirstOrder.Language.BoundedFormula.realize_bdEqual
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_top /-
 @[simp]
 theorem realize_top : (⊤ : L.BoundedFormula α l).realize v xs ↔ True := by simp [Top.top]
 #align first_order.language.bounded_formula.realize_top FirstOrder.Language.BoundedFormula.realize_top
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_inf /-
 @[simp]
 theorem realize_inf : (φ ⊓ ψ).realize v xs ↔ φ.realize v xs ∧ ψ.realize v xs := by
   simp [Inf.inf, realize]
 #align first_order.language.bounded_formula.realize_inf FirstOrder.Language.BoundedFormula.realize_inf
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_foldr_inf /-
 @[simp]
 theorem realize_foldr_inf (l : List (L.BoundedFormula α n)) (v : α → M) (xs : Fin n → M) :
     (l.foldr (· ⊓ ·) ⊤).realize v xs ↔ ∀ φ ∈ l, BoundedFormula.Realize φ v xs :=
@@ -294,18 +365,24 @@ theorem realize_foldr_inf (l : List (L.BoundedFormula α n)) (v : α → M) (xs
   · simp
   · simp [ih]
 #align first_order.language.bounded_formula.realize_foldr_inf FirstOrder.Language.BoundedFormula.realize_foldr_inf
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_imp /-
 @[simp]
 theorem realize_imp : (φ.imp ψ).realize v xs ↔ φ.realize v xs → ψ.realize v xs := by
   simp only [realize]
 #align first_order.language.bounded_formula.realize_imp FirstOrder.Language.BoundedFormula.realize_imp
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_rel /-
 @[simp]
 theorem realize_rel {k : ℕ} {R : L.Relations k} {ts : Fin k → L.term _} :
     (R.BoundedFormula ts).realize v xs ↔ RelMap R fun i => (ts i).realize (Sum.elim v xs) :=
   Iff.rfl
 #align first_order.language.bounded_formula.realize_rel FirstOrder.Language.BoundedFormula.realize_rel
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_rel₁ /-
 @[simp]
 theorem realize_rel₁ {R : L.Relations 1} {t : L.term _} :
     (R.boundedFormula₁ t).realize v xs ↔ RelMap R ![t.realize (Sum.elim v xs)] :=
@@ -314,7 +391,9 @@ theorem realize_rel₁ {R : L.Relations 1} {t : L.term _} :
   refine' congr rfl (funext fun _ => _)
   simp only [Matrix.cons_val_fin_one]
 #align first_order.language.bounded_formula.realize_rel₁ FirstOrder.Language.BoundedFormula.realize_rel₁
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_rel₂ /-
 @[simp]
 theorem realize_rel₂ {R : L.Relations 2} {t₁ t₂ : L.term _} :
     (R.boundedFormula₂ t₁ t₂).realize v xs ↔
@@ -325,14 +404,23 @@ theorem realize_rel₂ {R : L.Relations 2} {t₁ t₂ : L.term _} :
   · simp only [Matrix.cons_val_zero]
   · simp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]
 #align first_order.language.bounded_formula.realize_rel₂ FirstOrder.Language.BoundedFormula.realize_rel₂
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_sup /-
 @[simp]
 theorem realize_sup : (φ ⊔ ψ).realize v xs ↔ φ.realize v xs ∨ ψ.realize v xs :=
   by
   simp only [realize, Sup.sup, realize_not, eq_iff_iff]
   tauto
 #align first_order.language.bounded_formula.realize_sup FirstOrder.Language.BoundedFormula.realize_sup
+-/
 
+/- warning: first_order.language.bounded_formula.realize_foldr_sup -> FirstOrder.Language.BoundedFormula.realize_foldr_sup is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} (l : List.{max u1 u2 u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n)) (v : α -> M) (xs : (Fin n) -> M), Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n (List.foldr.{max u1 u2 u4, max u1 u2 u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (Sup.sup.{max u1 u2 u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (FirstOrder.Language.BoundedFormula.hasSup.{u1, u2, u4} L α n)) (Bot.bot.{max u1 u2 u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (FirstOrder.Language.BoundedFormula.hasBot.{u1, u2, u4} L α n)) l) v xs) (Exists.{succ (max u1 u2 u4)} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (fun (φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) => Exists.{0} (Membership.Mem.{max u1 u2 u4, max u1 u2 u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (List.{max u1 u2 u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n)) (List.hasMem.{max u1 u2 u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n)) φ l) (fun (H : Membership.Mem.{max u1 u2 u4, max u1 u2 u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (List.{max u1 u2 u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n)) (List.hasMem.{max u1 u2 u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n)) φ l) => FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v xs)))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {n : Nat} (l : List.{max (max u1 u2) u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n)) (v : α -> M) (xs : (Fin n) -> M), Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n (List.foldr.{max (max u1 u4) u2, max (max u1 u4) u2} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (fun (x._@.Mathlib.ModelTheory.Semantics._hyg.4548 : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (x._@.Mathlib.ModelTheory.Semantics._hyg.4550 : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) => Sup.sup.{max (max u1 u4) u2} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (FirstOrder.Language.BoundedFormula.instSupBoundedFormula.{u1, u2, u4} L α n) x._@.Mathlib.ModelTheory.Semantics._hyg.4548 x._@.Mathlib.ModelTheory.Semantics._hyg.4550) (Bot.bot.{max (max u1 u4) u2} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (FirstOrder.Language.BoundedFormula.instBotBoundedFormula.{u1, u2, u4} L α n)) l) v xs) (Exists.{succ (max (max u1 u4) u2)} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (fun (φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) => And (Membership.mem.{max (max u1 u4) u2, max (max u1 u4) u2} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) (List.{max (max u1 u2) u4} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n)) (List.instMembershipList.{max (max u1 u4) u2} (FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n)) φ l) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v xs)))
+Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_foldr_sup FirstOrder.Language.BoundedFormula.realize_foldr_supₓ'. -/
 @[simp]
 theorem realize_foldr_sup (l : List (L.BoundedFormula α n)) (v : α → M) (xs : Fin n → M) :
     (l.foldr (· ⊔ ·) ⊥).realize v xs ↔ ∃ φ ∈ l, BoundedFormula.Realize φ v xs :=
@@ -344,30 +432,48 @@ theorem realize_foldr_sup (l : List (L.BoundedFormula α n)) (v : α → M) (xs
       exists_eq_left]
 #align first_order.language.bounded_formula.realize_foldr_sup FirstOrder.Language.BoundedFormula.realize_foldr_sup
 
+#print FirstOrder.Language.BoundedFormula.realize_all /-
 @[simp]
 theorem realize_all : (all θ).realize v xs ↔ ∀ a : M, θ.realize v (Fin.snoc xs a) :=
   Iff.rfl
 #align first_order.language.bounded_formula.realize_all FirstOrder.Language.BoundedFormula.realize_all
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_ex /-
 @[simp]
 theorem realize_ex : θ.ex.realize v xs ↔ ∃ a : M, θ.realize v (Fin.snoc xs a) :=
   by
   rw [bounded_formula.ex, realize_not, realize_all, not_forall]
   simp_rw [realize_not, Classical.not_not]
 #align first_order.language.bounded_formula.realize_ex FirstOrder.Language.BoundedFormula.realize_ex
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_iff /-
 @[simp]
 theorem realize_iff : (φ.Iff ψ).realize v xs ↔ (φ.realize v xs ↔ ψ.realize v xs) := by
   simp only [bounded_formula.iff, realize_inf, realize_imp, and_imp, ← iff_def]
 #align first_order.language.bounded_formula.realize_iff FirstOrder.Language.BoundedFormula.realize_iff
+-/
 
-theorem realize_castLE_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
+/- warning: first_order.language.bounded_formula.realize_cast_le_of_eq -> FirstOrder.Language.BoundedFormula.realize_castLe_of_eq is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {m : Nat} {n : Nat} (h : Eq.{1} Nat m n) {h' : LE.le.{0} Nat instLENat m n} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α m} {v : α -> M} {xs : (Fin n) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n (FirstOrder.Language.BoundedFormula.castLE.{u1, u2, u4} L α m n h' φ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α m φ v (Function.comp.{1, 1, succ u3} (Fin m) (Fin n) M xs (FunLike.coe.{1, 1, 1} (RelIso.{0, 0} (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298)) (Fin m) (fun (_x : Fin m) => Fin n) (RelHomClass.toFunLike.{0, 0, 0} (RelIso.{0, 0} (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298)) (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.instRelHomClassRelIso.{0, 0} (Fin m) (Fin n) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : Fin m) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : Fin m) => LE.le.{0} (Fin m) (instLEFin m) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298))) (Fin.cast m n h))))
+Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eqₓ'. -/
+theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
     {v : α → M} {xs : Fin n → M} : (φ.castLE h').realize v xs ↔ φ.realize v (xs ∘ Fin.cast h) :=
   by
   subst h
   simp only [cast_le_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]
-#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLE_of_eq
+#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eq
 
+/- warning: first_order.language.bounded_formula.realize_map_term_rel_id -> FirstOrder.Language.BoundedFormula.realize_mapTermRel_id is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {L' : FirstOrder.Language.{u6, u7}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} {β : Type.{u5}} [_inst_4 : FirstOrder.Language.Structure.{u6, u7, u3} L' M] {ft : forall (n : Nat), (FirstOrder.Language.Term.{u1, u2, u4} L (Sum.{u4, 0} α (Fin n))) -> (FirstOrder.Language.Term.{u6, u7, u5} L' (Sum.{u5, 0} β (Fin n)))} {fr : forall (n : Nat), (FirstOrder.Language.Relations.{u1, u2} L n) -> (FirstOrder.Language.Relations.{u6, u7} L' n)} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n} {v : α -> M} {v' : β -> M} {xs : (Fin n) -> M}, (forall (n : Nat) (t : FirstOrder.Language.Term.{u1, u2, u4} L (Sum.{u4, 0} α (Fin n))) (xs : (Fin n) -> M), Eq.{succ u3} M (FirstOrder.Language.Term.realize.{u6, u7, u3, u5} L' M _inst_4 (Sum.{u5, 0} β (Fin n)) (Sum.elim.{u5, 0, succ u3} β (Fin n) M v' xs) (ft n t)) (FirstOrder.Language.Term.realize.{u1, u2, u3, u4} L M _inst_1 (Sum.{u4, 0} α (Fin n)) (Sum.elim.{u4, 0, succ u3} α (Fin n) M v xs) t)) -> (forall (n : Nat) (R : FirstOrder.Language.Relations.{u1, u2} L n) (x : (Fin n) -> M), Eq.{1} Prop (FirstOrder.Language.Structure.RelMap.{u6, u7, u3} L' M _inst_4 n (fr n R) x) (FirstOrder.Language.Structure.RelMap.{u1, u2, u3} L M _inst_1 n R x)) -> (Iff (FirstOrder.Language.BoundedFormula.Realize.{u6, u7, u3, u5} L' M _inst_4 β n (FirstOrder.Language.BoundedFormula.mapTermRel.{u1, u2, u4, u5, u6, u7} L L' α β (fun (n : Nat) => n) ft fr (fun (_x : Nat) => id.{succ (max u6 u7 u5)} (FirstOrder.Language.BoundedFormula.{u6, u7, u5} L' β (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) _x (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) n φ) v' xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v xs))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u3, u4}} {L' : FirstOrder.Language.{u2, u1}} {M : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u3, u4, u5} L M] {α : Type.{u6}} {β : Type.{u7}} [_inst_4 : FirstOrder.Language.Structure.{u2, u1, u5} L' M] {ft : forall (n : Nat), (FirstOrder.Language.Term.{u3, u4, u6} L (Sum.{u6, 0} α (Fin n))) -> (FirstOrder.Language.Term.{u2, u1, u7} L' (Sum.{u7, 0} β (Fin n)))} {fr : forall (n : Nat), (FirstOrder.Language.Relations.{u3, u4} L n) -> (FirstOrder.Language.Relations.{u2, u1} L' n)} {n : Nat} {φ : FirstOrder.Language.BoundedFormula.{u3, u4, u6} L α n} {v : α -> M} {v' : β -> M} {xs : (Fin n) -> M}, (forall (n : Nat) (t : FirstOrder.Language.Term.{u3, u4, u6} L (Sum.{u6, 0} α (Fin n))) (xs : (Fin n) -> M), Eq.{succ u5} M (FirstOrder.Language.Term.realize.{u2, u1, u5, u7} L' M _inst_4 (Sum.{u7, 0} β (Fin n)) (Sum.elim.{u7, 0, succ u5} β (Fin n) M v' xs) (ft n t)) (FirstOrder.Language.Term.realize.{u3, u4, u5, u6} L M _inst_1 (Sum.{u6, 0} α (Fin n)) (Sum.elim.{u6, 0, succ u5} α (Fin n) M v xs) t)) -> (forall (n : Nat) (R : FirstOrder.Language.Relations.{u3, u4} L n) (x : (Fin n) -> M), Eq.{1} Prop (FirstOrder.Language.Structure.RelMap.{u2, u1, u5} L' M _inst_4 n (fr n R) x) (FirstOrder.Language.Structure.RelMap.{u3, u4, u5} L M _inst_1 n R x)) -> (Iff (FirstOrder.Language.BoundedFormula.Realize.{u2, u1, u5, u7} L' M _inst_4 β n (FirstOrder.Language.BoundedFormula.mapTermRel.{u3, u4, u6, u7, u2, u1} L L' α β (fun (n : Nat) => n) ft fr (fun (_x : Nat) => id.{max (max (succ u7) (succ u1)) (succ u2)} (FirstOrder.Language.BoundedFormula.{u2, u1, u7} L' β (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) _x (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) n φ) v' xs) (FirstOrder.Language.BoundedFormula.Realize.{u3, u4, u5, u6} L M _inst_1 α n φ v xs))
+Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_map_term_rel_id FirstOrder.Language.BoundedFormula.realize_mapTermRel_idₓ'. -/
 theorem realize_mapTermRel_id [L'.Structure M]
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin n))}
     {fr : ∀ n, L.Relations n → L'.Relations n} {n} {φ : L.BoundedFormula α n} {v : α → M}
@@ -386,7 +492,13 @@ theorem realize_mapTermRel_id [L'.Structure M]
   · simp only [map_term_rel, realize, ih, id.def]
 #align first_order.language.bounded_formula.realize_map_term_rel_id FirstOrder.Language.BoundedFormula.realize_mapTermRel_id
 
-theorem realize_mapTermRel_add_castLE [L'.Structure M] {k : ℕ}
+/- warning: first_order.language.bounded_formula.realize_map_term_rel_add_cast_le -> FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe is a dubious translation:
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(HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k n)) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.natAdd k n)))))
+Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLeₓ'. -/
+theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin (k + n)))}
     {fr : ∀ n, L.Relations n → L'.Relations n} {n} {φ : L.BoundedFormula α n}
     (v : ∀ {n}, (Fin (k + n) → M) → α → M) {v' : β → M} (xs : Fin (k + n) → M)
@@ -404,8 +516,14 @@ theorem realize_mapTermRel_add_castLE [L'.Structure M] {k : ℕ}
   · simp [map_term_rel, realize, h1, h2]
   · simp [map_term_rel, realize, ih1, ih2]
   · simp [map_term_rel, realize, ih, hv]
-#align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLE
+#align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe
 
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+Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_relabel FirstOrder.Language.BoundedFormula.realize_relabelₓ'. -/
 theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum β (Fin m)} {v : β → M}
     {xs : Fin (m + n) → M} :
     (φ.relabel g).realize v xs ↔
@@ -413,6 +531,12 @@ theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum
   by rw [relabel, realize_map_term_rel_add_cast_le] <;> intros <;> simp
 #align first_order.language.bounded_formula.realize_relabel FirstOrder.Language.BoundedFormula.realize_relabel
 
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+Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAtₓ'. -/
 theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + n') → M}
     (hmn : m + n' ≤ n + 1) :
     (φ.liftAt n' m).realize v xs ↔
@@ -444,6 +568,12 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
       split_ifs <;> simp
 #align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAt
 
+/- warning: first_order.language.bounded_formula.realize_lift_at_one -> FirstOrder.Language.BoundedFormula.realize_liftAt_one is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_oneₓ'. -/
 theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + 1) → M}
     (hmn : m ≤ n) :
     (φ.liftAt 1 m).realize v xs ↔
@@ -451,6 +581,12 @@ theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α →
   by simp_rw [realize_lift_at (add_le_add_right hmn 1), cast_succ, add_nat_one]
 #align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_one
 
+/- warning: first_order.language.bounded_formula.realize_lift_at_one_self -> FirstOrder.Language.BoundedFormula.realize_liftAt_one_self is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align first_order.language.bounded_formula.realize_lift_at_one_self FirstOrder.Language.BoundedFormula.realize_liftAt_one_selfₓ'. -/
 @[simp]
 theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α → M}
     {xs : Fin (n + 1) → M} : (φ.liftAt 1 n).realize v xs ↔ φ.realize v (xs ∘ castSucc) :=
@@ -460,6 +596,7 @@ theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α 
   rw [if_pos i.is_lt]
 #align first_order.language.bounded_formula.realize_lift_at_one_self FirstOrder.Language.BoundedFormula.realize_liftAt_one_self
 
+#print FirstOrder.Language.BoundedFormula.realize_subst /-
 theorem realize_subst {φ : L.BoundedFormula α n} {tf : α → L.term β} {v : β → M} {xs : Fin n → M} :
     (φ.subst tf).realize v xs ↔ φ.realize (fun a => (tf a).realize v) xs :=
   realize_mapTermRel_id
@@ -471,7 +608,9 @@ theorem realize_subst {φ : L.BoundedFormula α n} {tf : α → L.term β} {v :
         · rfl)
     (by simp)
 #align first_order.language.bounded_formula.realize_subst FirstOrder.Language.BoundedFormula.realize_subst
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_restrictFreeVar /-
 @[simp]
 theorem realize_restrictFreeVar [DecidableEq α] {n : ℕ} {φ : L.BoundedFormula α n} {s : Set α}
     (h : ↑φ.freeVarFinset ⊆ s) {v : α → M} {xs : Fin n → M} :
@@ -484,7 +623,9 @@ theorem realize_restrictFreeVar [DecidableEq α] {n : ℕ} {φ : L.BoundedFormul
   · simp [restrict_free_var, realize, ih1, ih2]
   · simp [restrict_free_var, realize, ih3]
 #align first_order.language.bounded_formula.realize_restrict_free_var FirstOrder.Language.BoundedFormula.realize_restrictFreeVar
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_constantsVarsEquiv /-
 theorem realize_constantsVarsEquiv [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
     {n} {φ : L[[α]].BoundedFormula β n} {v : β → M} {xs : Fin n → M} :
     (constantsVarsEquiv φ).realize (Sum.elim (fun a => ↑(L.con a)) v) xs ↔ φ.realize v xs :=
@@ -498,7 +639,9 @@ theorem realize_constantsVarsEquiv [L[[α]].Structure M] [(lhomWithConstants L 
   · simp
   · exact isEmptyElim R
 #align first_order.language.bounded_formula.realize_constants_vars_equiv FirstOrder.Language.BoundedFormula.realize_constantsVarsEquiv
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_relabelEquiv /-
 @[simp]
 theorem realize_relabelEquiv {g : α ≃ β} {k} {φ : L.BoundedFormula α k} {v : β → M}
     {xs : Fin k → M} : (relabelEquiv g φ).realize v xs ↔ φ.realize (v ∘ g) xs :=
@@ -509,9 +652,11 @@ theorem realize_relabelEquiv {g : α ≃ β} {k} {φ : L.BoundedFormula α k} {v
   refine' congr (congr rfl _) rfl
   ext (i | i) <;> rfl
 #align first_order.language.bounded_formula.realize_relabel_equiv FirstOrder.Language.BoundedFormula.realize_relabelEquiv
+-/
 
 variable [Nonempty M]
 
+#print FirstOrder.Language.BoundedFormula.realize_all_liftAt_one_self /-
 theorem realize_all_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α → M}
     {xs : Fin n → M} : (φ.liftAt 1 n).all.realize v xs ↔ φ.realize v xs :=
   by
@@ -523,7 +668,9 @@ theorem realize_all_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v :
   · refine' (congr rfl (funext fun i => _)).mp h
     simp
 #align first_order.language.bounded_formula.realize_all_lift_at_one_self FirstOrder.Language.BoundedFormula.realize_all_liftAt_one_self
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_toPrenexImpRight /-
 theorem realize_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQF φ) (hψ : IsPrenex ψ)
     {v : α → M} {xs : Fin n → M} : (φ.toPrenexImpRight ψ).realize v xs ↔ (φ.imp ψ).realize v xs :=
   by
@@ -545,7 +692,9 @@ theorem realize_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQF φ)
       · inhabit M
         exact ⟨default, fun h'' => (h h'').elim⟩
 #align first_order.language.bounded_formula.realize_to_prenex_imp_right FirstOrder.Language.BoundedFormula.realize_toPrenexImpRight
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_toPrenexImp /-
 theorem realize_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ) (hψ : IsPrenex ψ)
     {v : α → M} {xs : Fin n → M} : (φ.toPrenexImp ψ).realize v xs ↔ (φ.imp ψ).realize v xs :=
   by
@@ -567,7 +716,9 @@ theorem realize_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ)
   · refine' trans (forall_congr' fun _ => ih hψ.lift_at) _
     simp
 #align first_order.language.bounded_formula.realize_to_prenex_imp FirstOrder.Language.BoundedFormula.realize_toPrenexImp
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_toPrenex /-
 @[simp]
 theorem realize_toPrenex (φ : L.BoundedFormula α n) {v : α → M} :
     ∀ {xs : Fin n → M}, φ.toPrenex.realize v xs ↔ φ.realize v xs :=
@@ -581,6 +732,7 @@ theorem realize_toPrenex (φ : L.BoundedFormula α n) {v : α → M} :
   · rw [realize_all, to_prenex, realize_all]
     exact forall_congr' fun a => h
 #align first_order.language.bounded_formula.realize_to_prenex FirstOrder.Language.BoundedFormula.realize_toPrenex
+-/
 
 end BoundedFormula
 
@@ -592,6 +744,12 @@ namespace Lhom
 
 open BoundedFormula
 
+/- warning: first_order.language.Lhom.realize_on_bounded_formula -> FirstOrder.Language.LHom.realize_onBoundedFormula is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {L' : FirstOrder.Language.{u5, u6}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} [_inst_4 : FirstOrder.Language.Structure.{u5, u6, u3} L' M] (φ : FirstOrder.Language.LHom.{u1, u2, u5, u6} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u1, u2, u5, u6, u3} L L' φ M _inst_1 _inst_4] {n : Nat} (ψ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) {v : α -> M} {xs : (Fin n) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u5, u6, u3, u4} L' M _inst_4 α n (FirstOrder.Language.LHom.onBoundedFormula.{u1, u2, u4, u5, u6} L L' α φ n ψ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n ψ v xs)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u3, u4}} {L' : FirstOrder.Language.{u2, u1}} {M : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u3, u4, u5} L M] {α : Type.{u6}} [_inst_4 : FirstOrder.Language.Structure.{u2, u1, u5} L' M] (φ : FirstOrder.Language.LHom.{u3, u4, u2, u1} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u3, u4, u2, u1, u5} L L' φ M _inst_1 _inst_4] {n : Nat} (ψ : FirstOrder.Language.BoundedFormula.{u3, u4, u6} L α n) {v : α -> M} {xs : (Fin n) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u2, u1, u5, u6} L' M _inst_4 α n (FirstOrder.Language.LHom.onBoundedFormula.{u3, u4, u6, u2, u1} L L' α φ n ψ) v xs) (FirstOrder.Language.BoundedFormula.Realize.{u3, u4, u5, u6} L M _inst_1 α n ψ v xs)
+Case conversion may be inaccurate. Consider using '#align first_order.language.Lhom.realize_on_bounded_formula FirstOrder.Language.LHom.realize_onBoundedFormulaₓ'. -/
 @[simp]
 theorem realize_onBoundedFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] {n : ℕ}
     (ψ : L.BoundedFormula α n) {v : α → M} {xs : Fin n → M} :
@@ -615,44 +773,59 @@ attribute [protected] bounded_formula.imp bounded_formula.all
 
 namespace Formula
 
+#print FirstOrder.Language.Formula.Realize /-
 /-- A formula can be evaluated as true or false by giving values to each free variable. -/
 def Realize (φ : L.Formula α) (v : α → M) : Prop :=
   φ.realize v default
 #align first_order.language.formula.realize FirstOrder.Language.Formula.Realize
+-/
 
 variable {M} {φ ψ : L.Formula α} {v : α → M}
 
+#print FirstOrder.Language.Formula.realize_not /-
 @[simp]
 theorem realize_not : φ.Not.realize v ↔ ¬φ.realize v :=
   Iff.rfl
 #align first_order.language.formula.realize_not FirstOrder.Language.Formula.realize_not
+-/
 
+#print FirstOrder.Language.Formula.realize_bot /-
 @[simp]
 theorem realize_bot : (⊥ : L.Formula α).realize v ↔ False :=
   Iff.rfl
 #align first_order.language.formula.realize_bot FirstOrder.Language.Formula.realize_bot
+-/
 
+#print FirstOrder.Language.Formula.realize_top /-
 @[simp]
 theorem realize_top : (⊤ : L.Formula α).realize v ↔ True :=
   BoundedFormula.realize_top
 #align first_order.language.formula.realize_top FirstOrder.Language.Formula.realize_top
+-/
 
+#print FirstOrder.Language.Formula.realize_inf /-
 @[simp]
 theorem realize_inf : (φ ⊓ ψ).realize v ↔ φ.realize v ∧ ψ.realize v :=
   BoundedFormula.realize_inf
 #align first_order.language.formula.realize_inf FirstOrder.Language.Formula.realize_inf
+-/
 
+#print FirstOrder.Language.Formula.realize_imp /-
 @[simp]
 theorem realize_imp : (φ.imp ψ).realize v ↔ φ.realize v → ψ.realize v :=
   BoundedFormula.realize_imp
 #align first_order.language.formula.realize_imp FirstOrder.Language.Formula.realize_imp
+-/
 
+#print FirstOrder.Language.Formula.realize_rel /-
 @[simp]
 theorem realize_rel {k : ℕ} {R : L.Relations k} {ts : Fin k → L.term α} :
     (R.Formula ts).realize v ↔ RelMap R fun i => (ts i).realize v :=
   BoundedFormula.realize_rel.trans (by simp)
 #align first_order.language.formula.realize_rel FirstOrder.Language.Formula.realize_rel
+-/
 
+#print FirstOrder.Language.Formula.realize_rel₁ /-
 @[simp]
 theorem realize_rel₁ {R : L.Relations 1} {t : L.term _} :
     (R.formula₁ t).realize v ↔ RelMap R ![t.realize v] :=
@@ -661,7 +834,9 @@ theorem realize_rel₁ {R : L.Relations 1} {t : L.term _} :
   refine' congr rfl (funext fun _ => _)
   simp only [Matrix.cons_val_fin_one]
 #align first_order.language.formula.realize_rel₁ FirstOrder.Language.Formula.realize_rel₁
+-/
 
+#print FirstOrder.Language.Formula.realize_rel₂ /-
 @[simp]
 theorem realize_rel₂ {R : L.Relations 2} {t₁ t₂ : L.term _} :
     (R.formula₂ t₁ t₂).realize v ↔ RelMap R ![t₁.realize v, t₂.realize v] :=
@@ -671,17 +846,23 @@ theorem realize_rel₂ {R : L.Relations 2} {t₁ t₂ : L.term _} :
   · simp only [Matrix.cons_val_zero]
   · simp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]
 #align first_order.language.formula.realize_rel₂ FirstOrder.Language.Formula.realize_rel₂
+-/
 
+#print FirstOrder.Language.Formula.realize_sup /-
 @[simp]
 theorem realize_sup : (φ ⊔ ψ).realize v ↔ φ.realize v ∨ ψ.realize v :=
   BoundedFormula.realize_sup
 #align first_order.language.formula.realize_sup FirstOrder.Language.Formula.realize_sup
+-/
 
+#print FirstOrder.Language.Formula.realize_iff /-
 @[simp]
 theorem realize_iff : (φ.Iff ψ).realize v ↔ (φ.realize v ↔ ψ.realize v) :=
   BoundedFormula.realize_iff
 #align first_order.language.formula.realize_iff FirstOrder.Language.Formula.realize_iff
+-/
 
+#print FirstOrder.Language.Formula.realize_relabel /-
 @[simp]
 theorem realize_relabel {φ : L.Formula α} {g : α → β} {v : β → M} :
     (φ.relabel g).realize v ↔ φ.realize (v ∘ g) :=
@@ -689,19 +870,25 @@ theorem realize_relabel {φ : L.Formula α} {g : α → β} {v : β → M} :
   rw [realize, realize, relabel, bounded_formula.realize_relabel, iff_eq_eq, Fin.castAdd_zero]
   exact congr rfl (funext finZeroElim)
 #align first_order.language.formula.realize_relabel FirstOrder.Language.Formula.realize_relabel
+-/
 
+#print FirstOrder.Language.Formula.realize_relabel_sum_inr /-
 theorem realize_relabel_sum_inr (φ : L.Formula (Fin n)) {v : Empty → M} {x : Fin n → M} :
     (BoundedFormula.relabel Sum.inr φ).realize v x ↔ φ.realize x := by
   rw [bounded_formula.realize_relabel, formula.realize, Sum.elim_comp_inr, Fin.castAdd_zero,
     cast_refl, OrderIso.coe_refl, Function.comp.right_id,
     Subsingleton.elim (x ∘ (nat_add n : Fin 0 → Fin n)) default]
 #align first_order.language.formula.realize_relabel_sum_inr FirstOrder.Language.Formula.realize_relabel_sum_inr
+-/
 
+#print FirstOrder.Language.Formula.realize_equal /-
 @[simp]
 theorem realize_equal {t₁ t₂ : L.term α} {x : α → M} :
     (t₁.equal t₂).realize x ↔ t₁.realize x = t₂.realize x := by simp [term.equal, realize]
 #align first_order.language.formula.realize_equal FirstOrder.Language.Formula.realize_equal
+-/
 
+#print FirstOrder.Language.Formula.realize_graph /-
 @[simp]
 theorem realize_graph {f : L.Functions n} {x : Fin n → M} {y : M} :
     (Formula.graph f).realize (Fin.cons y x : _ → M) ↔ funMap f x = y :=
@@ -709,15 +896,28 @@ theorem realize_graph {f : L.Functions n} {x : Fin n → M} {y : M} :
   simp only [formula.graph, term.realize, realize_equal, Fin.cons_zero, Fin.cons_succ]
   rw [eq_comm]
 #align first_order.language.formula.realize_graph FirstOrder.Language.Formula.realize_graph
+-/
 
 end Formula
 
+/- warning: first_order.language.Lhom.realize_on_formula -> FirstOrder.Language.LHom.realize_onFormula is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {L' : FirstOrder.Language.{u5, u6}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} [_inst_4 : FirstOrder.Language.Structure.{u5, u6, u3} L' M] (φ : FirstOrder.Language.LHom.{u1, u2, u5, u6} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u1, u2, u5, u6, u3} L L' φ M _inst_1 _inst_4] (ψ : FirstOrder.Language.Formula.{u1, u2, u4} L α) {v : α -> M}, Iff (FirstOrder.Language.Formula.Realize.{u5, u6, u3, u4} L' M _inst_4 α (FirstOrder.Language.LHom.onFormula.{u1, u2, u4, u5, u6} L L' α φ ψ) v) (FirstOrder.Language.Formula.Realize.{u1, u2, u3, u4} L M _inst_1 α ψ v)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u3, u4}} {L' : FirstOrder.Language.{u2, u1}} {M : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u3, u4, u5} L M] {α : Type.{u6}} [_inst_4 : FirstOrder.Language.Structure.{u2, u1, u5} L' M] (φ : FirstOrder.Language.LHom.{u3, u4, u2, u1} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u3, u4, u2, u1, u5} L L' φ M _inst_1 _inst_4] (ψ : FirstOrder.Language.Formula.{u3, u4, u6} L α) {v : α -> M}, Iff (FirstOrder.Language.Formula.Realize.{u2, u1, u5, u6} L' M _inst_4 α (FirstOrder.Language.LHom.onFormula.{u3, u4, u6, u2, u1} L L' α φ ψ) v) (FirstOrder.Language.Formula.Realize.{u3, u4, u5, u6} L M _inst_1 α ψ v)
+Case conversion may be inaccurate. Consider using '#align first_order.language.Lhom.realize_on_formula FirstOrder.Language.LHom.realize_onFormulaₓ'. -/
 @[simp]
 theorem LHom.realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (ψ : L.Formula α)
     {v : α → M} : (φ.onFormula ψ).realize v ↔ ψ.realize v :=
   φ.realize_onBoundedFormula ψ
 #align first_order.language.Lhom.realize_on_formula FirstOrder.Language.LHom.realize_onFormula
 
+/- warning: first_order.language.Lhom.set_of_realize_on_formula -> FirstOrder.Language.LHom.setOf_realize_onFormula is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {L' : FirstOrder.Language.{u5, u6}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {α : Type.{u4}} [_inst_4 : FirstOrder.Language.Structure.{u5, u6, u3} L' M] (φ : FirstOrder.Language.LHom.{u1, u2, u5, u6} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u1, u2, u5, u6, u3} L L' φ M _inst_1 _inst_4] (ψ : FirstOrder.Language.Formula.{u1, u2, u4} L α), Eq.{succ (max u4 u3)} (Set.{max u4 u3} (α -> M)) (setOf.{max u4 u3} (α -> M) (FirstOrder.Language.Formula.Realize.{u5, u6, u3, u4} L' M _inst_4 α (FirstOrder.Language.LHom.onFormula.{u1, u2, u4, u5, u6} L L' α φ ψ))) (setOf.{max u4 u3} (α -> M) (FirstOrder.Language.Formula.Realize.{u1, u2, u3, u4} L M _inst_1 α ψ))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u3, u4}} {L' : FirstOrder.Language.{u2, u1}} {M : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u3, u4, u5} L M] {α : Type.{u6}} [_inst_4 : FirstOrder.Language.Structure.{u2, u1, u5} L' M] (φ : FirstOrder.Language.LHom.{u3, u4, u2, u1} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u3, u4, u2, u1, u5} L L' φ M _inst_1 _inst_4] (ψ : FirstOrder.Language.Formula.{u3, u4, u6} L α), Eq.{max (succ u6) (succ u5)} (Set.{max u6 u5} (α -> M)) (setOf.{max u6 u5} (α -> M) (FirstOrder.Language.Formula.Realize.{u2, u1, u5, u6} L' M _inst_4 α (FirstOrder.Language.LHom.onFormula.{u3, u4, u6, u2, u1} L L' α φ ψ))) (setOf.{max u6 u5} (α -> M) (FirstOrder.Language.Formula.Realize.{u3, u4, u5, u6} L M _inst_1 α ψ))
+Case conversion may be inaccurate. Consider using '#align first_order.language.Lhom.set_of_realize_on_formula FirstOrder.Language.LHom.setOf_realize_onFormulaₓ'. -/
 @[simp]
 theorem LHom.setOf_realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
     (ψ : L.Formula α) : (setOf (φ.onFormula ψ).realize : Set (α → M)) = setOf ψ.realize :=
@@ -728,23 +928,28 @@ theorem LHom.setOf_realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsE
 
 variable (M)
 
+#print FirstOrder.Language.Sentence.Realize /-
 /-- A sentence can be evaluated as true or false in a structure. -/
 def Sentence.Realize (φ : L.Sentence) : Prop :=
   φ.realize (default : _ → M)
 #align first_order.language.sentence.realize FirstOrder.Language.Sentence.Realize
+-/
 
 -- mathport name: sentence.realize
 infixl:51
   " ⊨ " =>-- input using \|= or \vDash, but not using \models
   Sentence.Realize
 
+#print FirstOrder.Language.Sentence.realize_not /-
 @[simp]
 theorem Sentence.realize_not {φ : L.Sentence} : M ⊨ φ.Not ↔ ¬M ⊨ φ :=
   Iff.rfl
 #align first_order.language.sentence.realize_not FirstOrder.Language.Sentence.realize_not
+-/
 
 namespace Formula
 
+#print FirstOrder.Language.Formula.realize_equivSentence_symm_con /-
 @[simp]
 theorem realize_equivSentence_symm_con [L[[α]].Structure M]
     [(L.lhomWithConstants α).IsExpansionOn M] (φ : L[[α]].Sentence) :
@@ -757,13 +962,17 @@ theorem realize_equivSentence_symm_con [L[[α]].Structure M]
   · rfl
   · exact i.elim
 #align first_order.language.formula.realize_equiv_sentence_symm_con FirstOrder.Language.Formula.realize_equivSentence_symm_con
+-/
 
+#print FirstOrder.Language.Formula.realize_equivSentence /-
 @[simp]
 theorem realize_equivSentence [L[[α]].Structure M] [(L.lhomWithConstants α).IsExpansionOn M]
     (φ : L.Formula α) : (equivSentence φ).realize M ↔ φ.realize fun a => (L.con a : M) := by
   rw [← realize_equiv_sentence_symm_con M (equiv_sentence φ), _root_.equiv.symm_apply_apply]
 #align first_order.language.formula.realize_equiv_sentence FirstOrder.Language.Formula.realize_equivSentence
+-/
 
+#print FirstOrder.Language.Formula.realize_equivSentence_symm /-
 theorem realize_equivSentence_symm (φ : L[[α]].Sentence) (v : α → M) :
     (equivSentence.symm φ).realize v ↔
       @Sentence.Realize _ M (@Language.withConstantsStructure L M _ α (constantsOn.structure v))
@@ -771,9 +980,16 @@ theorem realize_equivSentence_symm (φ : L[[α]].Sentence) (v : α → M) :
   letI := constants_on.Structure v
   realize_equiv_sentence_symm_con M φ
 #align first_order.language.formula.realize_equiv_sentence_symm FirstOrder.Language.Formula.realize_equivSentence_symm
+-/
 
 end Formula
 
+/- warning: first_order.language.Lhom.realize_on_sentence -> FirstOrder.Language.LHom.realize_onSentence is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {L' : FirstOrder.Language.{u4, u5}} (M : Type.{u3}) [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_4 : FirstOrder.Language.Structure.{u4, u5, u3} L' M] (φ : FirstOrder.Language.LHom.{u1, u2, u4, u5} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u1, u2, u4, u5, u3} L L' φ M _inst_1 _inst_4] (ψ : FirstOrder.Language.Sentence.{u1, u2} L), Iff (FirstOrder.Language.Sentence.Realize.{u4, u5, u3} L' M _inst_4 (FirstOrder.Language.LHom.onSentence.{u1, u2, u4, u5} L L' φ ψ)) (FirstOrder.Language.Sentence.Realize.{u1, u2, u3} L M _inst_1 ψ)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u3, u4}} {L' : FirstOrder.Language.{u2, u1}} (M : Type.{u5}) [_inst_1 : FirstOrder.Language.Structure.{u3, u4, u5} L M] [_inst_4 : FirstOrder.Language.Structure.{u2, u1, u5} L' M] (φ : FirstOrder.Language.LHom.{u3, u4, u2, u1} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u3, u4, u2, u1, u5} L L' φ M _inst_1 _inst_4] (ψ : FirstOrder.Language.Sentence.{u3, u4} L), Iff (FirstOrder.Language.Sentence.Realize.{u2, u1, u5} L' M _inst_4 (FirstOrder.Language.LHom.onSentence.{u3, u4, u2, u1} L L' φ ψ)) (FirstOrder.Language.Sentence.Realize.{u3, u4, u5} L M _inst_1 ψ)
+Case conversion may be inaccurate. Consider using '#align first_order.language.Lhom.realize_on_sentence FirstOrder.Language.LHom.realize_onSentenceₓ'. -/
 @[simp]
 theorem LHom.realize_onSentence [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
     (ψ : L.Sentence) : M ⊨ φ.onSentence ψ ↔ M ⊨ ψ :=
@@ -782,17 +998,21 @@ theorem LHom.realize_onSentence [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpans
 
 variable (L)
 
+#print FirstOrder.Language.completeTheory /-
 /-- The complete theory of a structure `M` is the set of all sentences `M` satisfies. -/
 def completeTheory : L.Theory :=
   { φ | M ⊨ φ }
 #align first_order.language.complete_theory FirstOrder.Language.completeTheory
+-/
 
 variable (N)
 
+#print FirstOrder.Language.ElementarilyEquivalent /-
 /-- Two structures are elementarily equivalent when they satisfy the same sentences. -/
 def ElementarilyEquivalent : Prop :=
   L.completeTheory M = L.completeTheory N
 #align first_order.language.elementarily_equivalent FirstOrder.Language.ElementarilyEquivalent
+-/
 
 -- mathport name: elementarily_equivalent
 scoped[FirstOrder]
@@ -800,21 +1020,31 @@ scoped[FirstOrder]
 
 variable {L} {M} {N}
 
+#print FirstOrder.Language.mem_completeTheory /-
 @[simp]
 theorem mem_completeTheory {φ : Sentence L} : φ ∈ L.completeTheory M ↔ M ⊨ φ :=
   Iff.rfl
 #align first_order.language.mem_complete_theory FirstOrder.Language.mem_completeTheory
+-/
 
+/- warning: first_order.language.elementarily_equivalent_iff -> FirstOrder.Language.elementarilyEquivalent_iff is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N], Iff (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) (forall (φ : FirstOrder.Language.Sentence.{u1, u2} L), Iff (FirstOrder.Language.Sentence.Realize.{u1, u2, u3} L M _inst_1 φ) (FirstOrder.Language.Sentence.Realize.{u1, u2, u4} L N _inst_2 φ))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N], Iff (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2) (forall (φ : FirstOrder.Language.Sentence.{u2, u3} L), Iff (FirstOrder.Language.Sentence.Realize.{u2, u3, u4} L M _inst_1 φ) (FirstOrder.Language.Sentence.Realize.{u2, u3, u1} L N _inst_2 φ))
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent_iff FirstOrder.Language.elementarilyEquivalent_iffₓ'. -/
 theorem elementarilyEquivalent_iff : M ≅[L] N ↔ ∀ φ : L.Sentence, M ⊨ φ ↔ N ⊨ φ := by
   simp only [elementarily_equivalent, Set.ext_iff, complete_theory, Set.mem_setOf_eq]
 #align first_order.language.elementarily_equivalent_iff FirstOrder.Language.elementarilyEquivalent_iff
 
 variable (M)
 
+#print FirstOrder.Language.Theory.Model /-
 /-- A model of a theory is a structure in which every sentence is realized as true. -/
 class Theory.Model (T : L.Theory) : Prop where
   realize_of_mem : ∀ φ ∈ T, M ⊨ φ
 #align first_order.language.Theory.model FirstOrder.Language.Theory.Model
+-/
 
 -- mathport name: Theory.model
 infixl:51
@@ -825,17 +1055,27 @@ variable {M} (T : L.Theory)
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Theory.model_iff /-
 @[simp]
 theorem Theory.model_iff : M ⊨ T ↔ ∀ φ ∈ T, M ⊨ φ :=
   ⟨fun h => h.realize_of_mem, fun h => ⟨h⟩⟩
 #align first_order.language.Theory.model_iff FirstOrder.Language.Theory.model_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Theory.realize_sentence_of_mem /-
 theorem Theory.realize_sentence_of_mem [M ⊨ T] {φ : L.Sentence} (h : φ ∈ T) : M ⊨ φ :=
   Theory.Model.realize_of_mem φ h
 #align first_order.language.Theory.realize_sentence_of_mem FirstOrder.Language.Theory.realize_sentence_of_mem
+-/
 
+/- warning: first_order.language.Lhom.on_Theory_model -> FirstOrder.Language.LHom.onTheory_model is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {L' : FirstOrder.Language.{u4, u5}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_4 : FirstOrder.Language.Structure.{u4, u5, u3} L' M] (φ : FirstOrder.Language.LHom.{u1, u2, u4, u5} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u1, u2, u4, u5, u3} L L' φ M _inst_1 _inst_4] (T : FirstOrder.Language.Theory.{u1, u2} L), Iff (FirstOrder.Language.Theory.Model.{u4, u5, u3} L' M _inst_4 (FirstOrder.Language.LHom.onTheory.{u1, u2, u4, u5} L L' φ T)) (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u3, u4}} {L' : FirstOrder.Language.{u2, u1}} {M : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u3, u4, u5} L M] [_inst_4 : FirstOrder.Language.Structure.{u2, u1, u5} L' M] (φ : FirstOrder.Language.LHom.{u3, u4, u2, u1} L L') [_inst_5 : FirstOrder.Language.LHom.IsExpansionOn.{u3, u4, u2, u1, u5} L L' φ M _inst_1 _inst_4] (T : FirstOrder.Language.Theory.{u3, u4} L), Iff (FirstOrder.Language.Theory.Model.{u2, u1, u5} L' M _inst_4 (FirstOrder.Language.LHom.onTheory.{u3, u4, u2, u1} L L' φ T)) (FirstOrder.Language.Theory.Model.{u3, u4, u5} L M _inst_1 T)
+Case conversion may be inaccurate. Consider using '#align first_order.language.Lhom.on_Theory_model FirstOrder.Language.LHom.onTheory_modelₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 @[simp]
@@ -846,18 +1086,28 @@ theorem LHom.onTheory_model [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionO
 variable {M} {T}
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.model_empty /-
 instance model_empty : M ⊨ (∅ : L.Theory) :=
   ⟨fun φ hφ => (Set.not_mem_empty φ hφ).elim⟩
 #align first_order.language.model_empty FirstOrder.Language.model_empty
+-/
 
 namespace Theory
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Theory.Model.mono /-
 theorem Model.mono {T' : L.Theory} (h : M ⊨ T') (hs : T ⊆ T') : M ⊨ T :=
   ⟨fun φ hφ => T'.realize_sentence_of_mem (hs hφ)⟩
 #align first_order.language.Theory.model.mono FirstOrder.Language.Theory.Model.mono
+-/
 
+/- warning: first_order.language.Theory.model.union -> FirstOrder.Language.Theory.Model.union is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {T : FirstOrder.Language.Theory.{u1, u2} L} {T' : FirstOrder.Language.Theory.{u1, u2} L}, (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T) -> (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T') -> (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 (Union.union.{max u1 u2} (FirstOrder.Language.Theory.{u1, u2} L) (Set.hasUnion.{max u1 u2} (FirstOrder.Language.Sentence.{u1, u2} L)) T T'))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {T : FirstOrder.Language.Theory.{u1, u2} L} {T' : FirstOrder.Language.Theory.{u1, u2} L}, (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T) -> (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T') -> (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 (Union.union.{max u2 u1} (FirstOrder.Language.Theory.{u1, u2} L) (Set.instUnionSet.{max u1 u2} (FirstOrder.Language.Sentence.{u1, u2} L)) T T'))
+Case conversion may be inaccurate. Consider using '#align first_order.language.Theory.model.union FirstOrder.Language.Theory.Model.unionₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -867,6 +1117,12 @@ theorem Model.union {T' : L.Theory} (h : M ⊨ T) (h' : M ⊨ T') : M ⊨ T ∪
   exact fun φ hφ => hφ.elim (h _) (h' _)
 #align first_order.language.Theory.model.union FirstOrder.Language.Theory.Model.union
 
+/- warning: first_order.language.Theory.model_union_iff -> FirstOrder.Language.Theory.model_union_iff is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {T : FirstOrder.Language.Theory.{u1, u2} L} {T' : FirstOrder.Language.Theory.{u1, u2} L}, Iff (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 (Union.union.{max u1 u2} (FirstOrder.Language.Theory.{u1, u2} L) (Set.hasUnion.{max u1 u2} (FirstOrder.Language.Sentence.{u1, u2} L)) T T')) (And (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T) (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T'))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] {T : FirstOrder.Language.Theory.{u1, u2} L} {T' : FirstOrder.Language.Theory.{u1, u2} L}, Iff (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 (Union.union.{max u2 u1} (FirstOrder.Language.Theory.{u1, u2} L) (Set.instUnionSet.{max u1 u2} (FirstOrder.Language.Sentence.{u1, u2} L)) T T')) (And (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T) (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T'))
+Case conversion may be inaccurate. Consider using '#align first_order.language.Theory.model_union_iff FirstOrder.Language.Theory.model_union_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -878,28 +1134,42 @@ theorem model_union_iff {T' : L.Theory} : M ⊨ T ∪ T' ↔ M ⊨ T ∧ M ⊨ T
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Theory.model_singleton_iff /-
 theorem model_singleton_iff {φ : L.Sentence} : M ⊨ ({φ} : L.Theory) ↔ M ⊨ φ := by simp
 #align first_order.language.Theory.model_singleton_iff FirstOrder.Language.Theory.model_singleton_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Theory.model_iff_subset_completeTheory /-
 theorem model_iff_subset_completeTheory : M ⊨ T ↔ T ⊆ L.completeTheory M :=
   T.model_iff
 #align first_order.language.Theory.model_iff_subset_complete_theory FirstOrder.Language.Theory.model_iff_subset_completeTheory
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Theory.CompleteTheory.subset /-
 theorem CompleteTheory.subset [MT : M ⊨ T] : T ⊆ L.completeTheory M :=
   model_iff_subset_completeTheory.1 MT
 #align first_order.language.Theory.complete_theory.subset FirstOrder.Language.Theory.CompleteTheory.subset
+-/
 
 end Theory
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.model_completeTheory /-
 instance model_completeTheory : M ⊨ L.completeTheory M :=
   Theory.model_iff_subset_completeTheory.2 (subset_refl _)
 #align first_order.language.model_complete_theory FirstOrder.Language.model_completeTheory
+-/
 
 variable (M N)
 
+/- warning: first_order.language.realize_iff_of_model_complete_theory -> FirstOrder.Language.realize_iff_of_model_completeTheory is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} (M : Type.{u3}) (N : Type.{u4}) [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N] [_inst_4 : FirstOrder.Language.Theory.Model.{u1, u2, u4} L N _inst_2 (FirstOrder.Language.completeTheory.{u1, u2, u3} L M _inst_1)] (φ : FirstOrder.Language.Sentence.{u1, u2} L), Iff (FirstOrder.Language.Sentence.Realize.{u1, u2, u4} L N _inst_2 φ) (FirstOrder.Language.Sentence.Realize.{u1, u2, u3} L M _inst_1 φ)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} (M : Type.{u4}) (N : Type.{u1}) [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N] [_inst_4 : FirstOrder.Language.Theory.Model.{u2, u3, u1} L N _inst_2 (FirstOrder.Language.completeTheory.{u2, u3, u4} L M _inst_1)] (φ : FirstOrder.Language.Sentence.{u2, u3} L), Iff (FirstOrder.Language.Sentence.Realize.{u2, u3, u1} L N _inst_2 φ) (FirstOrder.Language.Sentence.Realize.{u2, u3, u4} L M _inst_1 φ)
+Case conversion may be inaccurate. Consider using '#align first_order.language.realize_iff_of_model_complete_theory FirstOrder.Language.realize_iff_of_model_completeTheoryₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -916,6 +1186,7 @@ variable {M N}
 
 namespace BoundedFormula
 
+#print FirstOrder.Language.BoundedFormula.realize_alls /-
 @[simp]
 theorem realize_alls {φ : L.BoundedFormula α n} {v : α → M} :
     φ.alls.realize v ↔ ∀ xs : Fin n → M, φ.realize v xs :=
@@ -925,7 +1196,9 @@ theorem realize_alls {φ : L.BoundedFormula α n} {v : α → M} :
   · simp only [alls, ih, realize]
     exact ⟨fun h xs => Fin.snoc_init_self xs ▸ h _ _, fun h xs x => h (Fin.snoc xs x)⟩
 #align first_order.language.bounded_formula.realize_alls FirstOrder.Language.BoundedFormula.realize_alls
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_exs /-
 @[simp]
 theorem realize_exs {φ : L.BoundedFormula α n} {v : α → M} :
     φ.exs.realize v ↔ ∃ xs : Fin n → M, φ.realize v xs :=
@@ -940,7 +1213,9 @@ theorem realize_exs {φ : L.BoundedFormula α n} {v : α → M} :
       rw [← Fin.snoc_init_self xs] at h
       exact ⟨_, _, h⟩
 #align first_order.language.bounded_formula.realize_exs FirstOrder.Language.BoundedFormula.realize_exs
+-/
 
+#print FirstOrder.Language.BoundedFormula.realize_toFormula /-
 @[simp]
 theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M) :
     φ.toFormula.realize v ↔ φ.realize (v ∘ Sum.inl) (v ∘ Sum.inr) :=
@@ -969,11 +1244,18 @@ theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M
           rw [← cast_succ, snoc_cast_succ]
     · exact Subsingleton.elim _ _
 #align first_order.language.bounded_formula.realize_to_formula FirstOrder.Language.BoundedFormula.realize_toFormula
+-/
 
 end BoundedFormula
 
 namespace Equiv
 
+/- warning: first_order.language.equiv.realize_bounded_formula -> FirstOrder.Language.Equiv.realize_boundedFormula is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u5} L N] {α : Type.{u4}} {n : Nat} (g : FirstOrder.Language.Equiv.{u1, u2, u3, u5} L M N _inst_1 _inst_2) (φ : FirstOrder.Language.BoundedFormula.{u1, u2, u4} L α n) {v : α -> M} {xs : (Fin n) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u5, u4} L N _inst_2 α n φ (Function.comp.{succ u4, succ u3, succ u5} α M N (coeFn.{max (succ u3) (succ u5), max (succ u3) (succ u5)} (FirstOrder.Language.Equiv.{u1, u2, u3, u5} L M N _inst_1 _inst_2) (fun (_x : FirstOrder.Language.Equiv.{u1, u2, u3, u5} L M N _inst_1 _inst_2) => M -> N) (FirstOrder.Language.Equiv.hasCoeToFun.{u1, u2, u3, u5} L M N _inst_1 _inst_2) g) v) (Function.comp.{1, succ u3, succ u5} (Fin n) M N (coeFn.{max (succ u3) (succ u5), max (succ u3) (succ u5)} (FirstOrder.Language.Equiv.{u1, u2, u3, u5} L M N _inst_1 _inst_2) (fun (_x : FirstOrder.Language.Equiv.{u1, u2, u3, u5} L M N _inst_1 _inst_2) => M -> N) (FirstOrder.Language.Equiv.hasCoeToFun.{u1, u2, u3, u5} L M N _inst_1 _inst_2) g) xs)) (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u3, u4} L M _inst_1 α n φ v xs)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N] {α : Type.{u5}} {n : Nat} (g : FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) (φ : FirstOrder.Language.BoundedFormula.{u2, u3, u5} L α n) {v : α -> M} {xs : (Fin n) -> M}, Iff (FirstOrder.Language.BoundedFormula.Realize.{u2, u3, u1, u5} L N _inst_2 α n φ (Function.comp.{succ u5, succ u4, succ u1} α M N (FunLike.coe.{max (succ u4) (succ u1), succ u4, succ u1} (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) M (fun (_x : M) => (fun (a._@.Mathlib.ModelTheory.Basic._hyg.8209 : M) => N) _x) (EmbeddingLike.toFunLike.{max (succ u4) (succ u1), succ u4, succ u1} (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) M N (EquivLike.toEmbeddingLike.{max (succ u4) (succ u1), succ u4, succ u1} (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) M N (FirstOrder.Language.Equiv.instEquivLikeEquiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2))) g) v) (Function.comp.{1, succ u4, succ u1} (Fin n) M N (FunLike.coe.{max (succ u4) (succ u1), succ u4, succ u1} (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) M (fun (_x : M) => (fun (a._@.Mathlib.ModelTheory.Basic._hyg.8209 : M) => N) _x) (EmbeddingLike.toFunLike.{max (succ u4) (succ u1), succ u4, succ u1} (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) M N (EquivLike.toEmbeddingLike.{max (succ u4) (succ u1), succ u4, succ u1} (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) M N (FirstOrder.Language.Equiv.instEquivLikeEquiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2))) g) xs)) (FirstOrder.Language.BoundedFormula.Realize.{u2, u3, u4, u5} L M _inst_1 α n φ v xs)
+Case conversion may be inaccurate. Consider using '#align first_order.language.equiv.realize_bounded_formula FirstOrder.Language.Equiv.realize_boundedFormulaₓ'. -/
 @[simp]
 theorem realize_boundedFormula (g : M ≃[L] N) (φ : L.BoundedFormula α n) {v : α → M}
     {xs : Fin n → M} : φ.realize (g ∘ v) (g ∘ xs) ↔ φ.realize v xs :=
@@ -995,6 +1277,12 @@ theorem realize_boundedFormula (g : M ≃[L] N) (φ : L.BoundedFormula α n) {v
       exact h'
 #align first_order.language.equiv.realize_bounded_formula FirstOrder.Language.Equiv.realize_boundedFormula
 
+/- warning: first_order.language.equiv.realize_formula -> FirstOrder.Language.Equiv.realize_formula is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u5} L N] {α : Type.{u4}} (g : FirstOrder.Language.Equiv.{u1, u2, u3, u5} L M N _inst_1 _inst_2) (φ : FirstOrder.Language.Formula.{u1, u2, u4} L α) {v : α -> M}, Iff (FirstOrder.Language.Formula.Realize.{u1, u2, u5, u4} L N _inst_2 α φ (Function.comp.{succ u4, succ u3, succ u5} α M N (coeFn.{max (succ u3) (succ u5), max (succ u3) (succ u5)} (FirstOrder.Language.Equiv.{u1, u2, u3, u5} L M N _inst_1 _inst_2) (fun (_x : FirstOrder.Language.Equiv.{u1, u2, u3, u5} L M N _inst_1 _inst_2) => M -> N) (FirstOrder.Language.Equiv.hasCoeToFun.{u1, u2, u3, u5} L M N _inst_1 _inst_2) g) v)) (FirstOrder.Language.Formula.Realize.{u1, u2, u3, u4} L M _inst_1 α φ v)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N] {α : Type.{u5}} (g : FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) (φ : FirstOrder.Language.Formula.{u2, u3, u5} L α) {v : α -> M}, Iff (FirstOrder.Language.Formula.Realize.{u2, u3, u1, u5} L N _inst_2 α φ (Function.comp.{succ u5, succ u4, succ u1} α M N (FunLike.coe.{max (succ u4) (succ u1), succ u4, succ u1} (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) M (fun (_x : M) => (fun (a._@.Mathlib.ModelTheory.Basic._hyg.8209 : M) => N) _x) (EmbeddingLike.toFunLike.{max (succ u4) (succ u1), succ u4, succ u1} (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) M N (EquivLike.toEmbeddingLike.{max (succ u4) (succ u1), succ u4, succ u1} (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) M N (FirstOrder.Language.Equiv.instEquivLikeEquiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2))) g) v)) (FirstOrder.Language.Formula.Realize.{u2, u3, u4, u5} L M _inst_1 α φ v)
+Case conversion may be inaccurate. Consider using '#align first_order.language.equiv.realize_formula FirstOrder.Language.Equiv.realize_formulaₓ'. -/
 @[simp]
 theorem realize_formula (g : M ≃[L] N) (φ : L.Formula α) {v : α → M} :
     φ.realize (g ∘ v) ↔ φ.realize v := by
@@ -1002,18 +1290,36 @@ theorem realize_formula (g : M ≃[L] N) (φ : L.Formula α) {v : α → M} :
     Unique.eq_default (g ∘ default)]
 #align first_order.language.equiv.realize_formula FirstOrder.Language.Equiv.realize_formula
 
+/- warning: first_order.language.equiv.realize_sentence -> FirstOrder.Language.Equiv.realize_sentence is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N], (FirstOrder.Language.Equiv.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (forall (φ : FirstOrder.Language.Sentence.{u1, u2} L), Iff (FirstOrder.Language.Sentence.Realize.{u1, u2, u3} L M _inst_1 φ) (FirstOrder.Language.Sentence.Realize.{u1, u2, u4} L N _inst_2 φ))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N], (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (forall (φ : FirstOrder.Language.Sentence.{u2, u3} L), Iff (FirstOrder.Language.Sentence.Realize.{u2, u3, u4} L M _inst_1 φ) (FirstOrder.Language.Sentence.Realize.{u2, u3, u1} L N _inst_2 φ))
+Case conversion may be inaccurate. Consider using '#align first_order.language.equiv.realize_sentence FirstOrder.Language.Equiv.realize_sentenceₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem realize_sentence (g : M ≃[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ := by
   rw [sentence.realize, sentence.realize, ← g.realize_formula, Unique.eq_default (g ∘ default)]
 #align first_order.language.equiv.realize_sentence FirstOrder.Language.Equiv.realize_sentence
 
+/- warning: first_order.language.equiv.Theory_model -> FirstOrder.Language.Equiv.theory_model is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N] {T : FirstOrder.Language.Theory.{u1, u2} L}, (FirstOrder.Language.Equiv.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (forall [_inst_4 : FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T], FirstOrder.Language.Theory.Model.{u1, u2, u4} L N _inst_2 T)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N] {T : FirstOrder.Language.Theory.{u2, u3} L}, (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (forall [_inst_4 : FirstOrder.Language.Theory.Model.{u2, u3, u4} L M _inst_1 T], FirstOrder.Language.Theory.Model.{u2, u3, u1} L N _inst_2 T)
+Case conversion may be inaccurate. Consider using '#align first_order.language.equiv.Theory_model FirstOrder.Language.Equiv.theory_modelₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem theory_model (g : M ≃[L] N) [M ⊨ T] : N ⊨ T :=
   ⟨fun φ hφ => (g.realize_sentence φ).1 (Theory.realize_sentence_of_mem T hφ)⟩
 #align first_order.language.equiv.Theory_model FirstOrder.Language.Equiv.theory_model
 
+/- warning: first_order.language.equiv.elementarily_equivalent -> FirstOrder.Language.Equiv.elementarilyEquivalent is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N], (FirstOrder.Language.Equiv.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N], (FirstOrder.Language.Equiv.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2)
+Case conversion may be inaccurate. Consider using '#align first_order.language.equiv.elementarily_equivalent FirstOrder.Language.Equiv.elementarilyEquivalentₓ'. -/
 theorem elementarilyEquivalent (g : M ≃[L] N) : M ≅[L] N :=
   elementarilyEquivalent_iff.2 g.realize_sentence
 #align first_order.language.equiv.elementarily_equivalent FirstOrder.Language.Equiv.elementarilyEquivalent
@@ -1027,45 +1333,57 @@ open BoundedFormula
 variable {r : L.Relations 2}
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Relations.realize_reflexive /-
 @[simp]
 theorem realize_reflexive : M ⊨ r.Reflexive ↔ Reflexive fun x y : M => RelMap r ![x, y] :=
   forall_congr' fun _ => realize_rel₂
 #align first_order.language.relations.realize_reflexive FirstOrder.Language.Relations.realize_reflexive
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Relations.realize_irreflexive /-
 @[simp]
 theorem realize_irreflexive : M ⊨ r.Irreflexive ↔ Irreflexive fun x y : M => RelMap r ![x, y] :=
   forall_congr' fun _ => not_congr realize_rel₂
 #align first_order.language.relations.realize_irreflexive FirstOrder.Language.Relations.realize_irreflexive
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Relations.realize_symmetric /-
 @[simp]
 theorem realize_symmetric : M ⊨ r.Symmetric ↔ Symmetric fun x y : M => RelMap r ![x, y] :=
   forall_congr' fun _ => forall_congr' fun _ => imp_congr realize_rel₂ realize_rel₂
 #align first_order.language.relations.realize_symmetric FirstOrder.Language.Relations.realize_symmetric
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Relations.realize_antisymmetric /-
 @[simp]
 theorem realize_antisymmetric :
     M ⊨ r.antisymmetric ↔ AntiSymmetric fun x y : M => RelMap r ![x, y] :=
   forall_congr' fun _ =>
     forall_congr' fun _ => imp_congr realize_rel₂ (imp_congr realize_rel₂ Iff.rfl)
 #align first_order.language.relations.realize_antisymmetric FirstOrder.Language.Relations.realize_antisymmetric
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Relations.realize_transitive /-
 @[simp]
 theorem realize_transitive : M ⊨ r.Transitive ↔ Transitive fun x y : M => RelMap r ![x, y] :=
   forall_congr' fun _ =>
     forall_congr' fun _ =>
       forall_congr' fun _ => imp_congr realize_rel₂ (imp_congr realize_rel₂ realize_rel₂)
 #align first_order.language.relations.realize_transitive FirstOrder.Language.Relations.realize_transitive
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Relations.realize_total /-
 @[simp]
 theorem realize_total : M ⊨ r.Total ↔ Total fun x y : M => RelMap r ![x, y] :=
   forall_congr' fun _ =>
     forall_congr' fun _ => realize_sup.trans (or_congr realize_rel₂ realize_rel₂)
 #align first_order.language.relations.realize_total FirstOrder.Language.Relations.realize_total
+-/
 
 end Relations
 
@@ -1073,6 +1391,12 @@ section Cardinality
 
 variable (L)
 
+/- warning: first_order.language.sentence.realize_card_ge -> FirstOrder.Language.Sentence.realize_cardGe is a dubious translation:
+lean 3 declaration is
+  forall (L : FirstOrder.Language.{u1, u2}) {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] (n : Nat), Iff (FirstOrder.Language.Sentence.Realize.{u1, u2, u3} L M _inst_1 (FirstOrder.Language.Sentence.cardGe.{u1, u2} L n)) (LE.le.{succ u3} Cardinal.{u3} Cardinal.hasLe.{u3} ((fun (a : Type) (b : Type.{succ u3}) [self : HasLiftT.{1, succ (succ u3)} a b] => self.0) Nat Cardinal.{u3} (HasLiftT.mk.{1, succ (succ u3)} Nat Cardinal.{u3} (CoeTCₓ.coe.{1, succ (succ u3)} Nat Cardinal.{u3} (Nat.castCoe.{succ u3} Cardinal.{u3} Cardinal.hasNatCast.{u3}))) n) (Cardinal.mk.{u3} M))
+but is expected to have type
+  forall (L : FirstOrder.Language.{u1, u2}) {M : Type.{u3}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] (n : Nat), Iff (FirstOrder.Language.Sentence.Realize.{u1, u2, u3} L M _inst_1 (FirstOrder.Language.Sentence.cardGe.{u1, u2} L n)) (LE.le.{succ u3} Cardinal.{u3} Cardinal.instLECardinal.{u3} (Nat.cast.{succ u3} Cardinal.{u3} Cardinal.instNatCastCardinal.{u3} n) (Cardinal.mk.{u3} M))
+Case conversion may be inaccurate. Consider using '#align first_order.language.sentence.realize_card_ge FirstOrder.Language.Sentence.realize_cardGeₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 @[simp]
 theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ (#M) :=
@@ -1095,28 +1419,42 @@ theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ (#M
 #align first_order.language.sentence.realize_card_ge FirstOrder.Language.Sentence.realize_cardGe
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.model_infiniteTheory_iff /-
 @[simp]
 theorem model_infiniteTheory_iff : M ⊨ L.infiniteTheory ↔ Infinite M := by
   simp [infinite_theory, infinite_iff, aleph_0_le]
 #align first_order.language.model_infinite_theory_iff FirstOrder.Language.model_infiniteTheory_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.model_infiniteTheory /-
 instance model_infiniteTheory [h : Infinite M] : M ⊨ L.infiniteTheory :=
   L.model_infiniteTheory_iff.2 h
 #align first_order.language.model_infinite_theory FirstOrder.Language.model_infiniteTheory
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.model_nonemptyTheory_iff /-
 @[simp]
 theorem model_nonemptyTheory_iff : M ⊨ L.nonemptyTheory ↔ Nonempty M := by
   simp only [nonempty_theory, Theory.model_iff, Set.mem_singleton_iff, forall_eq,
     sentence.realize_card_ge, Nat.cast_one, one_le_iff_ne_zero, mk_ne_zero_iff]
 #align first_order.language.model_nonempty_theory_iff FirstOrder.Language.model_nonemptyTheory_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.model_nonempty /-
 instance model_nonempty [h : Nonempty M] : M ⊨ L.nonemptyTheory :=
   L.model_nonemptyTheory_iff.2 h
 #align first_order.language.model_nonempty FirstOrder.Language.model_nonempty
+-/
 
+/- warning: first_order.language.model_distinct_constants_theory -> FirstOrder.Language.model_distinctConstantsTheory is a dubious translation:
+lean 3 declaration is
+  forall (L : FirstOrder.Language.{u1, u2}) {α : Type.{u4}} {M : Type.{u3}} [_inst_4 : FirstOrder.Language.Structure.{max u1 u4, u2, u3} (FirstOrder.Language.withConstants.{u1, u2, u4} L α) M] (s : Set.{u4} α), Iff (FirstOrder.Language.Theory.Model.{max u1 u4, u2, u3} (FirstOrder.Language.withConstants.{u1, u2, u4} L α) M _inst_4 (FirstOrder.Language.distinctConstantsTheory.{u1, u2, u4} L α s)) (Set.InjOn.{u4, u3} α M (fun (i : α) => (fun (a : Type.{max u1 u4}) (b : Type.{u3}) [self : HasLiftT.{succ (max u1 u4), succ u3} a b] => self.0) (FirstOrder.Language.Constants.{max u1 u4, u2} (FirstOrder.Language.withConstants.{u1, u2, u4} L α)) M (HasLiftT.mk.{succ (max u1 u4), succ u3} (FirstOrder.Language.Constants.{max u1 u4, u2} (FirstOrder.Language.withConstants.{u1, u2, u4} L α)) M (CoeTCₓ.coe.{succ (max u1 u4), succ u3} (FirstOrder.Language.Constants.{max u1 u4, u2} (FirstOrder.Language.withConstants.{u1, u2, u4} L α)) M (FirstOrder.Language.hasCoeT.{max u1 u4, u2, u3} (FirstOrder.Language.withConstants.{u1, u2, u4} L α) M _inst_4))) (FirstOrder.Language.con.{u1, u2, u4} L α i)) s)
+but is expected to have type
+  forall (L : FirstOrder.Language.{u1, u2}) {α : Type.{u4}} {M : Type.{u3}} [_inst_4 : FirstOrder.Language.Structure.{max u1 u4, u2, u3} (FirstOrder.Language.withConstants.{u1, u2, u4} L α) M] (s : Set.{u4} α), Iff (FirstOrder.Language.Theory.Model.{max u1 u4, u2, u3} (FirstOrder.Language.withConstants.{u1, u2, u4} L α) M _inst_4 (FirstOrder.Language.distinctConstantsTheory.{u1, u2, u4} L α s)) (Set.InjOn.{u4, u3} α M (fun (i : α) => FirstOrder.Language.constantMap.{max u1 u4, u2, u3} (FirstOrder.Language.withConstants.{u1, u2, u4} L α) M _inst_4 (FirstOrder.Language.con.{u1, u2, u4} L α i)) s)
+Case conversion may be inaccurate. Consider using '#align first_order.language.model_distinct_constants_theory FirstOrder.Language.model_distinctConstantsTheoryₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem model_distinctConstantsTheory {M : Type w} [L[[α]].Structure M] (s : Set α) :
     M ⊨ L.distinctConstantsTheory s ↔ Set.InjOn (fun i : α => (L.con i : M)) s :=
@@ -1135,35 +1473,67 @@ theorem model_distinctConstantsTheory {M : Type w} [L[[α]].Structure M] (s : Se
 #align first_order.language.model_distinct_constants_theory FirstOrder.Language.model_distinctConstantsTheory
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.card_le_of_model_distinctConstantsTheory /-
 theorem card_le_of_model_distinctConstantsTheory (s : Set α) (M : Type w) [L[[α]].Structure M]
     [h : M ⊨ L.distinctConstantsTheory s] : Cardinal.lift.{w} (#s) ≤ Cardinal.lift.{u'} (#M) :=
   lift_mk_le'.2 ⟨⟨_, Set.injOn_iff_injective.1 ((L.model_distinctConstantsTheory s).1 h)⟩⟩
 #align first_order.language.card_le_of_model_distinct_constants_theory FirstOrder.Language.card_le_of_model_distinctConstantsTheory
+-/
 
 end Cardinality
 
 namespace ElementarilyEquivalent
 
+/- warning: first_order.language.elementarily_equivalent.symm -> FirstOrder.Language.ElementarilyEquivalent.symm is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N], (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u4, u3} L N M _inst_2 _inst_1)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N], (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u1, u4} L N M _inst_2 _inst_1)
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent.symm FirstOrder.Language.ElementarilyEquivalent.symmₓ'. -/
 @[symm]
 theorem symm (h : M ≅[L] N) : N ≅[L] M :=
   h.symm
 #align first_order.language.elementarily_equivalent.symm FirstOrder.Language.ElementarilyEquivalent.symm
 
+/- warning: first_order.language.elementarily_equivalent.trans -> FirstOrder.Language.ElementarilyEquivalent.trans is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} {P : Type.{u5}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N] [_inst_3 : FirstOrder.Language.Structure.{u1, u2, u5} L P], (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u4, u5} L N P _inst_2 _inst_3) -> (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u5} L M P _inst_1 _inst_3)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u3, u4}} {M : Type.{u5}} {N : Type.{u2}} {P : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u3, u4, u5} L M] [_inst_2 : FirstOrder.Language.Structure.{u3, u4, u2} L N] [_inst_3 : FirstOrder.Language.Structure.{u3, u4, u1} L P], (FirstOrder.Language.ElementarilyEquivalent.{u3, u4, u5, u2} L M N _inst_1 _inst_2) -> (FirstOrder.Language.ElementarilyEquivalent.{u3, u4, u2, u1} L N P _inst_2 _inst_3) -> (FirstOrder.Language.ElementarilyEquivalent.{u3, u4, u5, u1} L M P _inst_1 _inst_3)
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent.trans FirstOrder.Language.ElementarilyEquivalent.transₓ'. -/
 @[trans]
 theorem trans (MN : M ≅[L] N) (NP : N ≅[L] P) : M ≅[L] P :=
   MN.trans NP
 #align first_order.language.elementarily_equivalent.trans FirstOrder.Language.ElementarilyEquivalent.trans
 
+/- warning: first_order.language.elementarily_equivalent.complete_theory_eq -> FirstOrder.Language.ElementarilyEquivalent.completeTheory_eq is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N], (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (Eq.{succ (max u1 u2)} (FirstOrder.Language.Theory.{u1, u2} L) (FirstOrder.Language.completeTheory.{u1, u2, u3} L M _inst_1) (FirstOrder.Language.completeTheory.{u1, u2, u4} L N _inst_2))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N], (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (Eq.{max (succ u2) (succ u3)} (FirstOrder.Language.Theory.{u2, u3} L) (FirstOrder.Language.completeTheory.{u2, u3, u4} L M _inst_1) (FirstOrder.Language.completeTheory.{u2, u3, u1} L N _inst_2))
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent.complete_theory_eq FirstOrder.Language.ElementarilyEquivalent.completeTheory_eqₓ'. -/
 theorem completeTheory_eq (h : M ≅[L] N) : L.completeTheory M = L.completeTheory N :=
   h
 #align first_order.language.elementarily_equivalent.complete_theory_eq FirstOrder.Language.ElementarilyEquivalent.completeTheory_eq
 
+/- warning: first_order.language.elementarily_equivalent.realize_sentence -> FirstOrder.Language.ElementarilyEquivalent.realize_sentence is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N], (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (forall (φ : FirstOrder.Language.Sentence.{u1, u2} L), Iff (FirstOrder.Language.Sentence.Realize.{u1, u2, u3} L M _inst_1 φ) (FirstOrder.Language.Sentence.Realize.{u1, u2, u4} L N _inst_2 φ))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N], (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (forall (φ : FirstOrder.Language.Sentence.{u2, u3} L), Iff (FirstOrder.Language.Sentence.Realize.{u2, u3, u4} L M _inst_1 φ) (FirstOrder.Language.Sentence.Realize.{u2, u3, u1} L N _inst_2 φ))
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent.realize_sentence FirstOrder.Language.ElementarilyEquivalent.realize_sentenceₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem realize_sentence (h : M ≅[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ :=
   (elementarilyEquivalent_iff.1 h) φ
 #align first_order.language.elementarily_equivalent.realize_sentence FirstOrder.Language.ElementarilyEquivalent.realize_sentence
 
+/- warning: first_order.language.elementarily_equivalent.Theory_model_iff -> FirstOrder.Language.ElementarilyEquivalent.theory_model_iff is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N] {T : FirstOrder.Language.Theory.{u1, u2} L}, (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (Iff (FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T) (FirstOrder.Language.Theory.Model.{u1, u2, u4} L N _inst_2 T))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N] {T : FirstOrder.Language.Theory.{u2, u3} L}, (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (Iff (FirstOrder.Language.Theory.Model.{u2, u3, u4} L M _inst_1 T) (FirstOrder.Language.Theory.Model.{u2, u3, u1} L N _inst_2 T))
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent.Theory_model_iff FirstOrder.Language.ElementarilyEquivalent.theory_model_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem theory_model_iff (h : M ≅[L] N) : M ⊨ T ↔ N ⊨ T := by
@@ -1171,24 +1541,54 @@ theorem theory_model_iff (h : M ≅[L] N) : M ⊨ T ↔ N ⊨ T := by
     h.complete_theory_eq]
 #align first_order.language.elementarily_equivalent.Theory_model_iff FirstOrder.Language.ElementarilyEquivalent.theory_model_iff
 
+/- warning: first_order.language.elementarily_equivalent.Theory_model -> FirstOrder.Language.ElementarilyEquivalent.theory_model is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N] {T : FirstOrder.Language.Theory.{u1, u2} L} [MT : FirstOrder.Language.Theory.Model.{u1, u2, u3} L M _inst_1 T], (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (FirstOrder.Language.Theory.Model.{u1, u2, u4} L N _inst_2 T)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N] {T : FirstOrder.Language.Theory.{u2, u3} L} [MT : FirstOrder.Language.Theory.Model.{u2, u3, u4} L M _inst_1 T], (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (FirstOrder.Language.Theory.Model.{u2, u3, u1} L N _inst_2 T)
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent.Theory_model FirstOrder.Language.ElementarilyEquivalent.theory_modelₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem theory_model [MT : M ⊨ T] (h : M ≅[L] N) : N ⊨ T :=
   h.theory_model_iff.1 MT
 #align first_order.language.elementarily_equivalent.Theory_model FirstOrder.Language.ElementarilyEquivalent.theory_model
 
+/- warning: first_order.language.elementarily_equivalent.nonempty_iff -> FirstOrder.Language.ElementarilyEquivalent.nonempty_iff is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N], (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (Iff (Nonempty.{succ u3} M) (Nonempty.{succ u4} N))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N], (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (Iff (Nonempty.{succ u4} M) (Nonempty.{succ u1} N))
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent.nonempty_iff FirstOrder.Language.ElementarilyEquivalent.nonempty_iffₓ'. -/
 theorem nonempty_iff (h : M ≅[L] N) : Nonempty M ↔ Nonempty N :=
   (model_nonemptyTheory_iff L).symm.trans (h.theory_model_iff.trans (model_nonemptyTheory_iff L))
 #align first_order.language.elementarily_equivalent.nonempty_iff FirstOrder.Language.ElementarilyEquivalent.nonempty_iff
 
+/- warning: first_order.language.elementarily_equivalent.nonempty -> FirstOrder.Language.ElementarilyEquivalent.nonempty is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N] [Mn : Nonempty.{succ u3} M], (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (Nonempty.{succ u4} N)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N] [Mn : Nonempty.{succ u4} M], (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (Nonempty.{succ u1} N)
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent.nonempty FirstOrder.Language.ElementarilyEquivalent.nonemptyₓ'. -/
 theorem nonempty [Mn : Nonempty M] (h : M ≅[L] N) : Nonempty N :=
   h.nonempty_iff.1 Mn
 #align first_order.language.elementarily_equivalent.nonempty FirstOrder.Language.ElementarilyEquivalent.nonempty
 
+/- warning: first_order.language.elementarily_equivalent.infinite_iff -> FirstOrder.Language.ElementarilyEquivalent.infinite_iff is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N], (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (Iff (Infinite.{succ u3} M) (Infinite.{succ u4} N))
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N], (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (Iff (Infinite.{succ u4} M) (Infinite.{succ u1} N))
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent.infinite_iff FirstOrder.Language.ElementarilyEquivalent.infinite_iffₓ'. -/
 theorem infinite_iff (h : M ≅[L] N) : Infinite M ↔ Infinite N :=
   (model_infiniteTheory_iff L).symm.trans (h.theory_model_iff.trans (model_infiniteTheory_iff L))
 #align first_order.language.elementarily_equivalent.infinite_iff FirstOrder.Language.ElementarilyEquivalent.infinite_iff
 
+/- warning: first_order.language.elementarily_equivalent.infinite -> FirstOrder.Language.ElementarilyEquivalent.infinite is a dubious translation:
+lean 3 declaration is
+  forall {L : FirstOrder.Language.{u1, u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : FirstOrder.Language.Structure.{u1, u2, u3} L M] [_inst_2 : FirstOrder.Language.Structure.{u1, u2, u4} L N] [Mi : Infinite.{succ u3} M], (FirstOrder.Language.ElementarilyEquivalent.{u1, u2, u3, u4} L M N _inst_1 _inst_2) -> (Infinite.{succ u4} N)
+but is expected to have type
+  forall {L : FirstOrder.Language.{u2, u3}} {M : Type.{u4}} {N : Type.{u1}} [_inst_1 : FirstOrder.Language.Structure.{u2, u3, u4} L M] [_inst_2 : FirstOrder.Language.Structure.{u2, u3, u1} L N] [Mi : Infinite.{succ u4} M], (FirstOrder.Language.ElementarilyEquivalent.{u2, u3, u4, u1} L M N _inst_1 _inst_2) -> (Infinite.{succ u1} N)
+Case conversion may be inaccurate. Consider using '#align first_order.language.elementarily_equivalent.infinite FirstOrder.Language.ElementarilyEquivalent.infiniteₓ'. -/
 theorem infinite [Mi : Infinite M] (h : M ≅[L] N) : Infinite N :=
   h.infinite_iff.1 Mi
 #align first_order.language.elementarily_equivalent.infinite FirstOrder.Language.ElementarilyEquivalent.infinite

d565b3df

bump to nightly-2023-05-09-13

mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7

6ff25244

Diff

@@ -438,7 +438,7 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
       · rw [if_neg h]
         refine' (congr rfl (ext _)).trans (snoc_last _ _)
         simp
-    · simp only [Function.comp_apply, Fin.snoc_cast_succ]
+    · simp only [Function.comp_apply, Fin.snoc_castSucc]
       refine' (congr rfl (ext _)).trans (snoc_cast_succ _ _ _)
       simp only [cast_refl, coe_cast_succ, OrderIso.coe_refl, id.def]
       split_ifs <;> simp

d565b3df

bump to nightly-2023-04-17-14

mathlib commit https://github.com/leanprover-community/mathlib/commit/17ad94b4953419f3e3ce3e77da3239c62d1d09f0

ce473067

Diff

@@ -361,12 +361,12 @@ theorem realize_iff : (φ.Iff ψ).realize v xs ↔ (φ.realize v xs ↔ ψ.reali
   simp only [bounded_formula.iff, realize_inf, realize_imp, and_imp, ← iff_def]
 #align first_order.language.bounded_formula.realize_iff FirstOrder.Language.BoundedFormula.realize_iff
 
-theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
+theorem realize_castLE_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
     {v : α → M} {xs : Fin n → M} : (φ.castLE h').realize v xs ↔ φ.realize v (xs ∘ Fin.cast h) :=
   by
   subst h
   simp only [cast_le_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]
-#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eq
+#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLE_of_eq
 
 theorem realize_mapTermRel_id [L'.Structure M]
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin n))}
@@ -386,7 +386,7 @@ theorem realize_mapTermRel_id [L'.Structure M]
   · simp only [map_term_rel, realize, ih, id.def]
 #align first_order.language.bounded_formula.realize_map_term_rel_id FirstOrder.Language.BoundedFormula.realize_mapTermRel_id
 
-theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
+theorem realize_mapTermRel_add_castLE [L'.Structure M] {k : ℕ}
     {ft : ∀ n, L.term (Sum α (Fin n)) → L'.term (Sum β (Fin (k + n)))}
     {fr : ∀ n, L.Relations n → L'.Relations n} {n} {φ : L.BoundedFormula α n}
     (v : ∀ {n}, (Fin (k + n) → M) → α → M) {v' : β → M} (xs : Fin (k + n) → M)
@@ -395,7 +395,7 @@ theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
         (ft n t).realize (Sum.elim v' xs') = t.realize (Sum.elim (v xs') (xs' ∘ Fin.natAdd _)))
     (h2 : ∀ (n) (R : L.Relations n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x)
     (hv : ∀ (n) (xs : Fin (k + n) → M) (x : M), @v (n + 1) (snoc xs x : Fin _ → M) = v xs) :
-    (φ.mapTermRel ft fr fun n => castLe (add_assoc _ _ _).symm.le).realize v' xs ↔
+    (φ.mapTermRel ft fr fun n => castLE (add_assoc _ _ _).symm.le).realize v' xs ↔
       φ.realize (v xs) (xs ∘ Fin.natAdd _) :=
   by
   induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih
@@ -404,7 +404,7 @@ theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
   · simp [map_term_rel, realize, h1, h2]
   · simp [map_term_rel, realize, ih1, ih2]
   · simp [map_term_rel, realize, ih, hv]
-#align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe
+#align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLE
 
 theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum β (Fin m)} {v : β → M}
     {xs : Fin (m + n) → M} :
@@ -524,7 +524,7 @@ theorem realize_all_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v :
     simp
 #align first_order.language.bounded_formula.realize_all_lift_at_one_self FirstOrder.Language.BoundedFormula.realize_all_liftAt_one_self
 
-theorem realize_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQf φ) (hψ : IsPrenex ψ)
+theorem realize_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQF φ) (hψ : IsPrenex ψ)
     {v : α → M} {xs : Fin n → M} : (φ.toPrenexImpRight ψ).realize v xs ↔ (φ.imp ψ).realize v xs :=
   by
   revert φ

d565b3df

bump to nightly-2023-04-13-00

mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65

b0c3fa7f

Diff

@@ -209,7 +209,7 @@ theorem realize_onTerm [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
   induction' t with _ n f ts ih
   · rfl
   · simp only [term.realize, Lhom.on_term, Lhom.map_on_function, ih]
-#align first_order.language.Lhom.realize_on_term FirstOrder.Language.Lhom.realize_onTerm
+#align first_order.language.Lhom.realize_on_term FirstOrder.Language.LHom.realize_onTerm
 
 end Lhom
 
@@ -605,7 +605,7 @@ theorem realize_onBoundedFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpan
     rfl
   · simp only [on_bounded_formula, ih1, ih2, realize_imp]
   · simp only [on_bounded_formula, ih3, realize_all]
-#align first_order.language.Lhom.realize_on_bounded_formula FirstOrder.Language.Lhom.realize_onBoundedFormula
+#align first_order.language.Lhom.realize_on_bounded_formula FirstOrder.Language.LHom.realize_onBoundedFormula
 
 end Lhom
 
@@ -713,18 +713,18 @@ theorem realize_graph {f : L.Functions n} {x : Fin n → M} {y : M} :
 end Formula
 
 @[simp]
-theorem Lhom.realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (ψ : L.Formula α)
+theorem LHom.realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (ψ : L.Formula α)
     {v : α → M} : (φ.onFormula ψ).realize v ↔ ψ.realize v :=
   φ.realize_onBoundedFormula ψ
-#align first_order.language.Lhom.realize_on_formula FirstOrder.Language.Lhom.realize_onFormula
+#align first_order.language.Lhom.realize_on_formula FirstOrder.Language.LHom.realize_onFormula
 
 @[simp]
-theorem Lhom.setOf_realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
+theorem LHom.setOf_realize_onFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
     (ψ : L.Formula α) : (setOf (φ.onFormula ψ).realize : Set (α → M)) = setOf ψ.realize :=
   by
   ext
   simp
-#align first_order.language.Lhom.set_of_realize_on_formula FirstOrder.Language.Lhom.setOf_realize_onFormula
+#align first_order.language.Lhom.set_of_realize_on_formula FirstOrder.Language.LHom.setOf_realize_onFormula
 
 variable (M)
 
@@ -775,10 +775,10 @@ theorem realize_equivSentence_symm (φ : L[[α]].Sentence) (v : α → M) :
 end Formula
 
 @[simp]
-theorem Lhom.realize_onSentence [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
+theorem LHom.realize_onSentence [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M]
     (ψ : L.Sentence) : M ⊨ φ.onSentence ψ ↔ M ⊨ ψ :=
   φ.realize_onFormula ψ
-#align first_order.language.Lhom.realize_on_sentence FirstOrder.Language.Lhom.realize_onSentence
+#align first_order.language.Lhom.realize_on_sentence FirstOrder.Language.LHom.realize_onSentence
 
 variable (L)
 
@@ -839,9 +839,9 @@ theorem Theory.realize_sentence_of_mem [M ⊨ T] {φ : L.Sentence} (h : φ ∈ T
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 @[simp]
-theorem Lhom.onTheory_model [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (T : L.Theory) :
+theorem LHom.onTheory_model [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] (T : L.Theory) :
     M ⊨ φ.onTheory T ↔ M ⊨ T := by simp [Theory.model_iff, Lhom.on_Theory]
-#align first_order.language.Lhom.on_Theory_model FirstOrder.Language.Lhom.onTheory_model
+#align first_order.language.Lhom.on_Theory_model FirstOrder.Language.LHom.onTheory_model
 
 variable {M} {T}

d565b3df

bump to nightly-2023-04-12-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65

06c79b19

Diff

@@ -362,7 +362,7 @@ theorem realize_iff : (φ.Iff ψ).realize v xs ↔ (φ.realize v xs ↔ ψ.reali
 #align first_order.language.bounded_formula.realize_iff FirstOrder.Language.BoundedFormula.realize_iff
 
 theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
-    {v : α → M} {xs : Fin n → M} : (φ.castLe h').realize v xs ↔ φ.realize v (xs ∘ Fin.cast h) :=
+    {v : α → M} {xs : Fin n → M} : (φ.castLE h').realize v xs ↔ φ.realize v (xs ∘ Fin.cast h) :=
   by
   subst h
   simp only [cast_le_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]

d565b3df

bump to nightly-2023-03-27-14

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce86f4e05e9a9b8da5e316b22c76ce76440c56a1

b92a7b6a

Diff

@@ -1080,7 +1080,7 @@ theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ (#M
   rw [← lift_mk_fin, ← lift_le, lift_lift, lift_mk_le, sentence.card_ge, sentence.realize,
     bounded_formula.realize_exs]
   simp_rw [bounded_formula.realize_foldr_inf]
-  simp only [Function.comp_apply, List.mem_map', Prod.exists, Ne.def, List.mem_product,
+  simp only [Function.comp_apply, List.mem_map, Prod.exists, Ne.def, List.mem_product,
     List.mem_finRange, forall_exists_index, and_imp, List.mem_filter, true_and_iff]
   refine' ⟨_, fun xs => ⟨xs.some, _⟩⟩
   · rintro ⟨xs, h⟩

d565b3df

bump to nightly-2023-02-28-04

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

2097f8af

Diff

@@ -283,7 +283,7 @@ theorem realize_top : (⊤ : L.BoundedFormula α l).realize v xs ↔ True := by
 
 @[simp]
 theorem realize_inf : (φ ⊓ ψ).realize v xs ↔ φ.realize v xs ∧ ψ.realize v xs := by
-  simp [HasInf.inf, realize]
+  simp [Inf.inf, realize]
 #align first_order.language.bounded_formula.realize_inf FirstOrder.Language.BoundedFormula.realize_inf
 
 @[simp]
@@ -329,7 +329,7 @@ theorem realize_rel₂ {R : L.Relations 2} {t₁ t₂ : L.term _} :
 @[simp]
 theorem realize_sup : (φ ⊔ ψ).realize v xs ↔ φ.realize v xs ∨ ψ.realize v xs :=
   by
-  simp only [realize, HasSup.sup, realize_not, eq_iff_iff]
+  simp only [realize, Sup.sup, realize_not, eq_iff_iff]
   tauto
 #align first_order.language.bounded_formula.realize_sup FirstOrder.Language.BoundedFormula.realize_sup

d565b3df

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

d565b3df

chore: remove unnecessary cdots (#12417)

These · are scoping when there is a single active goal.

These were found using a modification of the linter at #12339.

5450e922

Diff

@@ -452,9 +452,9 @@ theorem realize_subst {φ : L.BoundedFormula α n} {tf : α → L.Term β} {v :
     (fun n t x => by
       rw [Term.realize_subst]
       rcongr a
-      · cases a
-        · simp only [Sum.elim_inl, Function.comp_apply, Term.realize_relabel, Sum.elim_comp_inl]
-        · rfl)
+      cases a
+      · simp only [Sum.elim_inl, Function.comp_apply, Term.realize_relabel, Sum.elim_comp_inl]
+      · rfl)
     (by simp)
 #align first_order.language.bounded_formula.realize_subst FirstOrder.Language.BoundedFormula.realize_subst
 
@@ -924,8 +924,8 @@ theorem _root_.FirstOrder.Language.Formula.realize_iAlls
   · intro x
     rw [Formula.Realize, iff_iff_eq]
     congr
-    · funext i
-      exact i.elim0
+    funext i
+    exact i.elim0
 
 @[simp]
 theorem realize_iAlls [Finite γ] {f : α → β ⊕ γ}
@@ -951,8 +951,8 @@ theorem _root_.FirstOrder.Language.Formula.realize_iExs
   · intro x
     rw [Formula.Realize, iff_iff_eq]
     congr
-    · funext i
-      exact i.elim0
+    funext i
+    exact i.elim0
 
 @[simp]
 theorem realize_iExs [Finite γ] {f : α → β ⊕ γ}

d565b3df

chore: avoid id.def (adaptation for nightly-2024-03-27) (#11829)

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

b8a0f589

Diff

@@ -370,7 +370,7 @@ theorem realize_mapTermRel_id [L'.Structure M]
   · simp [mapTermRel, Realize, h1]
   · simp [mapTermRel, Realize, h1, h2]
   · simp [mapTermRel, Realize, ih1, ih2]
-  · simp only [mapTermRel, Realize, ih, id.def]
+  · simp only [mapTermRel, Realize, ih, id]
 #align first_order.language.bounded_formula.realize_map_term_rel_id FirstOrder.Language.BoundedFormula.realize_mapTermRel_id
 
 theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}

d565b3df

chore: avoid Ne.def (adaptation for nightly-2024-03-27) (#11813)

08686b25

Diff

@@ -1102,7 +1102,7 @@ theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ #M
   rw [← lift_mk_fin, ← lift_le.{0}, lift_lift, lift_mk_le, Sentence.cardGe, Sentence.Realize,
     BoundedFormula.realize_exs]
   simp_rw [BoundedFormula.realize_foldr_inf]
-  simp only [Function.comp_apply, List.mem_map, Prod.exists, Ne.def, List.mem_product,
+  simp only [Function.comp_apply, List.mem_map, Prod.exists, Ne, List.mem_product,
     List.mem_finRange, forall_exists_index, and_imp, List.mem_filter, true_and_iff]
   refine' ⟨_, fun xs => ⟨xs.some, _⟩⟩
   · rintro ⟨xs, h⟩

d565b3df

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

@@ -57,9 +57,7 @@ namespace FirstOrder
 namespace Language
 
 variable {L : Language.{u, v}} {L' : Language}
-
 variable {M : Type w} {N P : Type*} [L.Structure M] [L.Structure N] [L.Structure P]
-
 variable {α : Type u'} {β : Type v'} {γ : Type*}
 
 open FirstOrder Cardinal
@@ -257,7 +255,6 @@ def Realize : ∀ {l} (_f : L.BoundedFormula α l) (_v : α → M) (_xs : Fin l
 #align first_order.language.bounded_formula.realize FirstOrder.Language.BoundedFormula.Realize
 
 variable {l : ℕ} {φ ψ : L.BoundedFormula α l} {θ : L.BoundedFormula α l.succ}
-
 variable {v : α → M} {xs : Fin l → M}
 
 @[simp]

d565b3df

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

@@ -68,7 +68,7 @@ open Structure Cardinal Fin
 
 namespace Term
 
---Porting note: universes in different order
+-- Porting note: universes in different order
 /-- A term `t` with variables indexed by `α` can be evaluated by giving a value to each variable. -/
 def realize (v : α → M) : ∀ _t : L.Term α, M
   | var k => v k
@@ -165,13 +165,13 @@ theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).
   · cases n
     · cases f
       · simp only [realize, ih, Nat.zero_eq, constantsOn, mk₂_Functions]
-        --Porting note: below lemma does not work with simp for some reason
+        -- Porting note: below lemma does not work with simp for some reason
         rw [withConstants_funMap_sum_inl]
       · simp only [realize, constantsToVars, Sum.elim_inl, funMap_eq_coe_constants]
         rfl
     · cases' f with _ f
       · simp only [realize, ih, constantsOn, mk₂_Functions]
-        --Porting note: below lemma does not work with simp for some reason
+        -- Porting note: below lemma does not work with simp for some reason
         rw [withConstants_funMap_sum_inl]
       · exact isEmptyElim f
 #align first_order.language.term.realize_constants_to_vars FirstOrder.Language.Term.realize_constantsToVars
@@ -182,11 +182,11 @@ theorem realize_varsToConstants [L[[α]].Structure M] [(lhomWithConstants L α).
     t.varsToConstants.realize v = t.realize (Sum.elim (fun a => ↑(L.con a)) v) := by
   induction' t with ab n f ts ih
   · cases' ab with a b
-    --Porting note: both cases were `simp [Language.con]`
+    -- Porting note: both cases were `simp [Language.con]`
     · simp [Language.con, realize, funMap_eq_coe_constants]
     · simp [realize, constantMap]
   · simp only [realize, constantsOn, mk₂_Functions, ih]
-    --Porting note: below lemma does not work with simp for some reason
+    -- Porting note: below lemma does not work with simp for some reason
     rw [withConstants_funMap_sum_inl]
 #align first_order.language.term.realize_vars_to_constants FirstOrder.Language.Term.realize_varsToConstants
 
@@ -246,7 +246,7 @@ namespace BoundedFormula
 
 open Term
 
---Porting note: universes in different order
+-- Porting note: universes in different order
 /-- A bounded formula can be evaluated as true or false by giving values to each free variable. -/
 def Realize : ∀ {l} (_f : L.BoundedFormula α l) (_v : α → M) (_xs : Fin l → M), Prop
   | _, falsum, _v, _xs => False
@@ -572,7 +572,7 @@ theorem realize_toPrenex (φ : L.BoundedFormula α n) {v : α → M} :
 end BoundedFormula
 
 
---Porting note: no `protected` attribute in Lean4
+-- Porting note: no `protected` attribute in Lean4
 -- attribute [protected] bounded_formula.falsum bounded_formula.equal bounded_formula.rel
 
 -- attribute [protected] bounded_formula.imp bounded_formula.all
@@ -599,7 +599,7 @@ set_option linter.uppercaseLean3 false in
 
 end LHom
 
---Porting note: no `protected` attribute in Lean4
+-- Porting note: no `protected` attribute in Lean4
 -- attribute [protected] bounded_formula.falsum bounded_formula.equal bounded_formula.rel
 
 -- attribute [protected] bounded_formula.imp bounded_formula.all

d565b3df

feat(ModelTheory/Basic): additional simp lemmas (#9953)

Also results about injectivity.

c9843f72

Diff

@@ -736,7 +736,7 @@ namespace Formula
 theorem realize_equivSentence_symm_con [L[[α]].Structure M]
     [(L.lhomWithConstants α).IsExpansionOn M] (φ : L[[α]].Sentence) :
     ((equivSentence.symm φ).Realize fun a => (L.con a : M)) ↔ φ.Realize M := by
-  simp only [equivSentence, Equiv.symm_symm, Equiv.coe_trans, Realize,
+  simp only [equivSentence, _root_.Equiv.symm_symm, Equiv.coe_trans, Realize,
     BoundedFormula.realize_relabelEquiv, Function.comp]
   refine' _root_.trans _ BoundedFormula.realize_constantsVarsEquiv
   rw [iff_iff_eq]

d565b3df

chore: reduce imports (#9830)

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

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

f4c4ca61

Diff

@@ -5,6 +5,7 @@ Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
 -/
 import Mathlib.Data.Finset.Basic
 import Mathlib.ModelTheory.Syntax
+import Mathlib.Data.List.ProdSigma
 
 #align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"

d565b3df

chore(Function): rename some lemmas (#9738)

Merge Function.left_id and Function.comp.left_id into Function.id_comp.
Merge Function.right_id and Function.comp.right_id into Function.comp_id.
Merge Function.comp_const_right and Function.comp_const into Function.comp_const, use explicit arguments.
Move Function.const_comp to Mathlib.Init.Function, use explicit arguments.

87c10369

Diff

@@ -355,7 +355,7 @@ theorem realize_iff : (φ.iff ψ).Realize v xs ↔ (φ.Realize v xs ↔ ψ.Reali
 theorem realize_castLE_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
     {v : α → M} {xs : Fin n → M} : (φ.castLE h').Realize v xs ↔ φ.Realize v (xs ∘ cast h) := by
   subst h
-  simp only [castLE_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]
+  simp only [castLE_rfl, cast_refl, OrderIso.coe_refl, Function.comp_id]
 #align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLE_of_eq
 
 theorem realize_mapTermRel_id [L'.Structure M]
@@ -680,7 +680,7 @@ theorem realize_relabel {φ : L.Formula α} {g : α → β} {v : β → M} :
 theorem realize_relabel_sum_inr (φ : L.Formula (Fin n)) {v : Empty → M} {x : Fin n → M} :
     (BoundedFormula.relabel Sum.inr φ).Realize v x ↔ φ.Realize x := by
   rw [BoundedFormula.realize_relabel, Formula.Realize, Sum.elim_comp_inr, Fin.castAdd_zero,
-    cast_refl, Function.comp.right_id,
+    cast_refl, Function.comp_id,
     Subsingleton.elim (x ∘ (natAdd n : Fin 0 → Fin n)) default]
 #align first_order.language.formula.realize_relabel_sum_inr FirstOrder.Language.Formula.realize_relabel_sum_inr

d565b3df

chore: cleanup simp calls (#9539)

Remove some unnecessary arguments in simp calls, which will become problematic when the simp algorithm changes in leanprover/lean4#3124.

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

320fe23c

Diff

@@ -182,7 +182,7 @@ theorem realize_varsToConstants [L[[α]].Structure M] [(lhomWithConstants L α).
   induction' t with ab n f ts ih
   · cases' ab with a b
     --Porting note: both cases were `simp [Language.con]`
-    · simp [Language.con, realize, constantMap, funMap_eq_coe_constants]
+    · simp [Language.con, realize, funMap_eq_coe_constants]
     · simp [realize, constantMap]
   · simp only [realize, constantsOn, mk₂_Functions, ih]
     --Porting note: below lemma does not work with simp for some reason

d565b3df

feat: make imp_false a simp-lemma (#8744)

This shortens various proofs
There is already a unification hint in core from ¬ p to p → False, which is probably why we didn't miss this simp-lemma too badly.
There was an inconsistency that simp pull out most connectives into Filter.eventually/Filter.frequently, but pushed negations inside. This loops together with this new simp-lemma, and the fact that there is a unification hint . So I removed some simp attributes in Filter.Basic.

d9ec1450

Diff

@@ -281,7 +281,7 @@ theorem realize_top : (⊤ : L.BoundedFormula α l).Realize v xs ↔ True := by
 
 @[simp]
 theorem realize_inf : (φ ⊓ ψ).Realize v xs ↔ φ.Realize v xs ∧ ψ.Realize v xs := by
-  simp [Inf.inf, Realize]; tauto
+  simp [Inf.inf, Realize]
 #align first_order.language.bounded_formula.realize_inf FirstOrder.Language.BoundedFormula.realize_inf
 
 @[simp]

d565b3df

chore: add missing hypothesis names to by_cases (#8533)

I've also got a change to make this required, but I'd like to land this first.

2009db69

Diff

@@ -524,7 +524,7 @@ theorem realize_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQF φ)
     refine' ⟨_, fun h' => _⟩
     · rintro ⟨a, ha⟩ h
       exact ⟨a, ha h⟩
-    · by_cases φ.Realize v xs
+    · by_cases h : φ.Realize v xs
       · obtain ⟨a, ha⟩ := h' h
         exact ⟨a, fun _ => ha⟩
       · inhabit M
@@ -544,7 +544,7 @@ theorem realize_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ)
     refine' ⟨_, fun h' => _⟩
     · rintro ⟨a, ha⟩ h
       exact ha (h a)
-    · by_cases ψ.Realize v xs
+    · by_cases h : ψ.Realize v xs
       · inhabit M
         exact ⟨default, fun _h'' => h⟩
       · obtain ⟨a, ha⟩ := not_forall.1 (h ∘ h')

d565b3df

fix: attribute [simp] ... in -> attribute [local simp] ... in (#7678)

Mathlib.Logic.Unique contains the line attribute [simp] eq_iff_true_of_subsingleton in ...:

https://github.com/leanprover-community/mathlib4/blob/96a11c7aac574c00370c2b3dab483cb676405c5d/Mathlib/Logic/Unique.lean#L255-L256

Despite what the in part may imply, this adds the lemma to the simp set "globally", including for downstream files; it is likely that attribute [local simp] eq_iff_true_of_subsingleton in ... was meant instead (or maybe scoped simp, but I think "scoped" refers to the current namespace). Indeed, the relevant lemma is not marked with @[simp] for possible slowness: https://github.com/leanprover/std4/blob/846e9e1d6bb534774d1acd2dc430e70987da3c18/Std/Logic.lean#L749. Adding it to the simp set causes the example at https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Regression.20in.20simp to slow down.

This PR changes this and fixes the relevant downstream simps. There was also one ocurrence of attribute [simp] FullSubcategory.comp_def FullSubcategory.id_def in in Mathlib.CategoryTheory.Monoidal.Subcategory but that was much easier to fix.

https://github.com/leanprover-community/mathlib4/blob/bc49eb9ba756a233370b4b68bcdedd60402f71ed/Mathlib/CategoryTheory/Monoidal/Subcategory.lean#L118-L119

1fe87718

Diff

@@ -934,7 +934,7 @@ theorem realize_iAlls [Finite γ] {f : α → β ⊕ γ}
     {φ : L.Formula α} {v : β → M} {v' : Fin 0 → M} :
     BoundedFormula.Realize (φ.iAlls f) v v' ↔
       ∀ (i : γ → M), φ.Realize (fun a => Sum.elim v i (f a)) := by
-  rw [← Formula.realize_iAlls, iff_iff_eq]; congr; simp
+  rw [← Formula.realize_iAlls, iff_iff_eq]; congr; simp [eq_iff_true_of_subsingleton]
 
 @[simp]
 theorem _root_.FirstOrder.Language.Formula.realize_iExs
@@ -961,7 +961,7 @@ theorem realize_iExs [Finite γ] {f : α → β ⊕ γ}
     {φ : L.Formula α} {v : β → M} {v' : Fin 0 → M} :
     BoundedFormula.Realize (φ.iExs f) v v' ↔
       ∃ (i : γ → M), φ.Realize (fun a => Sum.elim v i (f a)) := by
-  rw [← Formula.realize_iExs, iff_iff_eq]; congr; simp
+  rw [← Formula.realize_iExs, iff_iff_eq]; congr; simp [eq_iff_true_of_subsingleton]
 
 @[simp]
 theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M) :

d565b3df

Revert "chore: revert #7703 (#7710)"

This reverts commit f3695eb2.

ab56fa28

Diff

@@ -476,11 +476,13 @@ theorem realize_constantsVarsEquiv [L[[α]].Structure M] [(lhomWithConstants L 
     {n} {φ : L[[α]].BoundedFormula β n} {v : β → M} {xs : Fin n → M} :
     (constantsVarsEquiv φ).Realize (Sum.elim (fun a => ↑(L.con a)) v) xs ↔ φ.Realize v xs := by
   refine' realize_mapTermRel_id (fun n t xs => realize_constantsVarsEquivLeft) fun n R xs => _
-  rw [← (lhomWithConstants L α).map_onRelation
+  -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+  erw [← (lhomWithConstants L α).map_onRelation
       (Equiv.sumEmpty (L.Relations n) ((constantsOn α).Relations n) R) xs]
   rcongr
   cases' R with R R
-  · simp
+  · -- This used to be `simp` before leanprover/lean4#2644
+    simp; erw [Equiv.sumEmpty_apply_inl]
   · exact isEmptyElim R
 #align first_order.language.bounded_formula.realize_constants_vars_equiv FirstOrder.Language.BoundedFormula.realize_constantsVarsEquiv

d565b3df

chore: revert #7703 (#7710)

This reverts commit 26eb2b0a.

f3695eb2

Diff

@@ -476,13 +476,11 @@ theorem realize_constantsVarsEquiv [L[[α]].Structure M] [(lhomWithConstants L 
     {n} {φ : L[[α]].BoundedFormula β n} {v : β → M} {xs : Fin n → M} :
     (constantsVarsEquiv φ).Realize (Sum.elim (fun a => ↑(L.con a)) v) xs ↔ φ.Realize v xs := by
   refine' realize_mapTermRel_id (fun n t xs => realize_constantsVarsEquivLeft) fun n R xs => _
-  -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-  erw [← (lhomWithConstants L α).map_onRelation
+  rw [← (lhomWithConstants L α).map_onRelation
       (Equiv.sumEmpty (L.Relations n) ((constantsOn α).Relations n) R) xs]
   rcongr
   cases' R with R R
-  · -- This used to be `simp` before leanprover/lean4#2644
-    simp; erw [Equiv.sumEmpty_apply_inl]
+  · simp
   · exact isEmptyElim R
 #align first_order.language.bounded_formula.realize_constants_vars_equiv FirstOrder.Language.BoundedFormula.realize_constantsVarsEquiv

d565b3df

chore: bump toolchain to v4.2.0-rc2 (#7703)

This includes all the changes from #7606.

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

26eb2b0a

Diff

@@ -476,11 +476,13 @@ theorem realize_constantsVarsEquiv [L[[α]].Structure M] [(lhomWithConstants L 
     {n} {φ : L[[α]].BoundedFormula β n} {v : β → M} {xs : Fin n → M} :
     (constantsVarsEquiv φ).Realize (Sum.elim (fun a => ↑(L.con a)) v) xs ↔ φ.Realize v xs := by
   refine' realize_mapTermRel_id (fun n t xs => realize_constantsVarsEquivLeft) fun n R xs => _
-  rw [← (lhomWithConstants L α).map_onRelation
+  -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+  erw [← (lhomWithConstants L α).map_onRelation
       (Equiv.sumEmpty (L.Relations n) ((constantsOn α).Relations n) R) xs]
   rcongr
   cases' R with R R
-  · simp
+  · -- This used to be `simp` before leanprover/lean4#2644
+    simp; erw [Equiv.sumEmpty_apply_inl]
   · exact isEmptyElim R
 #align first_order.language.bounded_formula.realize_constants_vars_equiv FirstOrder.Language.BoundedFormula.realize_constantsVarsEquiv

d565b3df

chore: replace Fin.castIso and Fin.revPerm with Fin.cast and Fin.rev for the bump of Std (#5847)

Some theorems in Data.Fin.Basic are copied to Std at the recent commit in Std. These are written using Fin.cast and Fin.rev, so declarations using Fin.castIso and Fin.revPerm in Mathlib should be rewritten.

Co-authored-by: Pol'tta / Miyahara Kō <52843868+Komyyy@users.noreply.github.com> Co-authored-by: Johan Commelin <johan@commelin.net>

45b31dda

Diff

@@ -353,9 +353,9 @@ theorem realize_iff : (φ.iff ψ).Realize v xs ↔ (φ.Realize v xs ↔ ψ.Reali
 #align first_order.language.bounded_formula.realize_iff FirstOrder.Language.BoundedFormula.realize_iff
 
 theorem realize_castLE_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
-    {v : α → M} {xs : Fin n → M} : (φ.castLE h').Realize v xs ↔ φ.Realize v (xs ∘ castIso h) := by
+    {v : α → M} {xs : Fin n → M} : (φ.castLE h').Realize v xs ↔ φ.Realize v (xs ∘ cast h) := by
   subst h
-  simp only [castLE_rfl, castIso_refl, OrderIso.coe_refl, Function.comp.right_id]
+  simp only [castLE_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]
 #align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLE_of_eq
 
 theorem realize_mapTermRel_id [L'.Structure M]
@@ -419,7 +419,7 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
       by_cases h : k < m
       · rw [if_pos h]
         refine' (congr rfl (ext _)).trans (snoc_last _ _)
-        simp only [coe_orderIso_apply, coe_castAdd, val_last, self_eq_add_right]
+        simp only [coe_cast, coe_castAdd, val_last, self_eq_add_right]
         refine'
           le_antisymm (le_of_add_le_add_left ((hmn.trans (Nat.succ_le_of_lt h)).trans _)) n'.zero_le
         rw [add_zero]
@@ -428,7 +428,7 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
         simp
     · simp only [Function.comp_apply, Fin.snoc_castSucc]
       refine' (congr rfl (ext _)).trans (snoc_castSucc _ _ _)
-      simp only [coe_castSucc, coe_orderIso_apply]
+      simp only [coe_castSucc, coe_cast]
       split_ifs <;> simp
 #align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAt

d565b3df

fix: remove simp from BoundedFormula.subst and add to realize_subst (#6635)

mapTermRel doesn't have any nice simp lemmas about it, so it's best not to unfold subst.

e79d2a79

Diff

@@ -447,6 +447,7 @@ theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α 
   rw [if_pos i.is_lt]
 #align first_order.language.bounded_formula.realize_lift_at_one_self FirstOrder.Language.BoundedFormula.realize_liftAt_one_self
 
+@[simp]
 theorem realize_subst {φ : L.BoundedFormula α n} {tf : α → L.Term β} {v : β → M} {xs : Fin n → M} :
     (φ.subst tf).Realize v xs ↔ φ.Realize (fun a => (tf a).realize v) xs :=
   realize_mapTermRel_id

d565b3df

feat(ModelTheory): iAlls and iExs (#6613)

90725365

Diff

@@ -59,7 +59,7 @@ variable {L : Language.{u, v}} {L' : Language}
 
 variable {M : Type w} {N P : Type*} [L.Structure M] [L.Structure N] [L.Structure P]
 
-variable {α : Type u'} {β : Type v'}
+variable {α : Type u'} {β : Type v'} {γ : Type*}
 
 open FirstOrder Cardinal
 
@@ -907,6 +907,59 @@ theorem realize_exs {φ : L.BoundedFormula α n} {v : α → M} :
       exact ⟨_, _, h⟩
 #align first_order.language.bounded_formula.realize_exs FirstOrder.Language.BoundedFormula.realize_exs
 
+@[simp]
+theorem _root_.FirstOrder.Language.Formula.realize_iAlls
+    [Finite γ] {f : α → β ⊕ γ}
+    {φ : L.Formula α} {v : β → M} : (φ.iAlls f).Realize v ↔
+      ∀ (i : γ → M), φ.Realize (fun a => Sum.elim v i (f a)) := by
+  let e := Classical.choice (Classical.choose_spec (Finite.exists_equiv_fin γ))
+  rw [Formula.iAlls]
+  simp only [Nat.add_zero, realize_alls, realize_relabel, Function.comp,
+    castAdd_zero, castIso_refl, OrderIso.refl_apply, Sum.elim_map, id_eq]
+  refine Equiv.forall_congr ?_ ?_
+  · exact ⟨fun v => v ∘ e, fun v => v ∘ e.symm,
+      fun _ => by simp [Function.comp],
+      fun _ => by simp [Function.comp]⟩
+  · intro x
+    rw [Formula.Realize, iff_iff_eq]
+    congr
+    · funext i
+      exact i.elim0
+
+@[simp]
+theorem realize_iAlls [Finite γ] {f : α → β ⊕ γ}
+    {φ : L.Formula α} {v : β → M} {v' : Fin 0 → M} :
+    BoundedFormula.Realize (φ.iAlls f) v v' ↔
+      ∀ (i : γ → M), φ.Realize (fun a => Sum.elim v i (f a)) := by
+  rw [← Formula.realize_iAlls, iff_iff_eq]; congr; simp
+
+@[simp]
+theorem _root_.FirstOrder.Language.Formula.realize_iExs
+    [Finite γ] {f : α → β ⊕ γ}
+    {φ : L.Formula α} {v : β → M} : (φ.iExs f).Realize v ↔
+      ∃ (i : γ → M), φ.Realize (fun a => Sum.elim v i (f a)) := by
+  let e := Classical.choice (Classical.choose_spec (Finite.exists_equiv_fin γ))
+  rw [Formula.iExs]
+  simp only [Nat.add_zero, realize_exs, realize_relabel, Function.comp,
+    castAdd_zero, castIso_refl, OrderIso.refl_apply, Sum.elim_map, id_eq]
+  rw [← not_iff_not, not_exists, not_exists]
+  refine Equiv.forall_congr ?_ ?_
+  · exact ⟨fun v => v ∘ e, fun v => v ∘ e.symm,
+      fun _ => by simp [Function.comp],
+      fun _ => by simp [Function.comp]⟩
+  · intro x
+    rw [Formula.Realize, iff_iff_eq]
+    congr
+    · funext i
+      exact i.elim0
+
+@[simp]
+theorem realize_iExs [Finite γ] {f : α → β ⊕ γ}
+    {φ : L.Formula α} {v : β → M} {v' : Fin 0 → M} :
+    BoundedFormula.Realize (φ.iExs f) v v' ↔
+      ∃ (i : γ → M), φ.Realize (fun a => Sum.elim v i (f a)) := by
+  rw [← Formula.realize_iExs, iff_iff_eq]; congr; simp
+
 @[simp]
 theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M) :
     φ.toFormula.Realize v ↔ φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr) := by

d565b3df

feat(ModelTheory): simp attribute on realize_relabel (#6656)

05b606a9

Diff

@@ -394,6 +394,7 @@ theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
   · simp [mapTermRel, Realize, ih, hv]
 #align first_order.language.bounded_formula.realize_map_term_rel_add_cast_le FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe
 
+@[simp]
 theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum β (Fin m)} {v : β → M}
     {xs : Fin (m + n) → M} :
     (φ.relabel g).Realize v xs ↔

d565b3df

feat(ModelTheory): sups of finite sets of formulas (#6654)

Co-authored-by: Chris Hughes <33847686+ChrisHughes24@users.noreply.github.com>

efa84af4

Diff

@@ -935,6 +935,20 @@ theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M
     · exact Fin.elim0 x
 #align first_order.language.bounded_formula.realize_to_formula FirstOrder.Language.BoundedFormula.realize_toFormula
 
+@[simp]
+theorem realize_iSup (s : Finset β) (f : β → L.BoundedFormula α n)
+    (v : α → M) (v' : Fin n → M) :
+    (iSup s f).Realize v v' ↔ ∃ b ∈ s, (f b).Realize v v' := by
+  simp only [iSup, realize_foldr_sup, List.mem_map, Finset.mem_toList,
+    exists_exists_and_eq_and]
+
+@[simp]
+theorem realize_iInf (s : Finset β) (f : β → L.BoundedFormula α n)
+    (v : α → M) (v' : Fin n → M) :
+    (iInf s f).Realize v v' ↔ ∀ b ∈ s, (f b).Realize v v' := by
+  simp only [iInf, realize_foldr_inf, List.mem_map, Finset.mem_toList,
+    forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
+
 end BoundedFormula
 
 namespace Equiv

d565b3df

chore: remove unused simps (#6632)

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

916c75c1

Diff

@@ -332,7 +332,7 @@ theorem realize_foldr_sup (l : List (L.BoundedFormula α n)) (v : α → M) (xs
     (l.foldr (· ⊔ ·) ⊥).Realize v xs ↔ ∃ φ ∈ l, BoundedFormula.Realize φ v xs := by
   induction' l with φ l ih
   · simp
-  · simp_rw [List.foldr_cons, realize_sup, ih, exists_prop, List.mem_cons, or_and_right, exists_or,
+  · simp_rw [List.foldr_cons, realize_sup, ih, List.mem_cons, or_and_right, exists_or,
       exists_eq_left]
 #align first_order.language.bounded_formula.realize_foldr_sup FirstOrder.Language.BoundedFormula.realize_foldr_sup

d565b3df

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

@@ -57,7 +57,7 @@ namespace Language
 
 variable {L : Language.{u, v}} {L' : Language}
 
-variable {M : Type w} {N P : Type _} [L.Structure M] [L.Structure N] [L.Structure P]
+variable {M : Type w} {N P : Type*} [L.Structure M] [L.Structure N] [L.Structure P]
 
 variable {α : Type u'} {β : Type v'}
 
@@ -145,7 +145,7 @@ theorem realize_restrictVar [DecidableEq α] {t : L.Term α} {s : Set α} (h : 
 #align first_order.language.term.realize_restrict_var FirstOrder.Language.Term.realize_restrictVar
 
 @[simp]
-theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.Term (Sum α γ)} {s : Set α}
+theorem realize_restrictVarLeft [DecidableEq α] {γ : Type*} {t : L.Term (Sum α γ)} {s : Set α}
     (h : ↑t.varFinsetLeft ⊆ s) {v : α → M} {xs : γ → M} :
     (t.restrictVarLeft (Set.inclusion h)).realize (Sum.elim (v ∘ (↑)) xs) =
       t.realize (Sum.elim v xs) := by

d565b3df

chore: tidy various files (#6291)

c4679d7b

Diff

@@ -352,11 +352,11 @@ theorem realize_iff : (φ.iff ψ).Realize v xs ↔ (φ.Realize v xs ↔ ψ.Reali
   simp only [BoundedFormula.iff, realize_inf, realize_imp, and_imp, ← iff_def]
 #align first_order.language.bounded_formula.realize_iff FirstOrder.Language.BoundedFormula.realize_iff
 
-theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
+theorem realize_castLE_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
     {v : α → M} {xs : Fin n → M} : (φ.castLE h').Realize v xs ↔ φ.Realize v (xs ∘ castIso h) := by
   subst h
   simp only [castLE_rfl, castIso_refl, OrderIso.coe_refl, Function.comp.right_id]
-#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eq
+#align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLE_of_eq
 
 theorem realize_mapTermRel_id [L'.Structure M]
     {ft : ∀ n, L.Term (Sum α (Fin n)) → L'.Term (Sum β (Fin n))}
@@ -412,10 +412,10 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
   · simp [mapTermRel, Realize, realize_rel, realize_liftAt, Sum.elim_comp_map]
   · simp only [mapTermRel, Realize, ih1 hmn, ih2 hmn]
   · have h : k + 1 + n' = k + n' + 1 := by rw [add_assoc, add_comm 1 n', ← add_assoc]
-    simp only [mapTermRel, Realize, realize_castLe_of_eq h, ih3 (hmn.trans k.succ.le_succ)]
+    simp only [mapTermRel, Realize, realize_castLE_of_eq h, ih3 (hmn.trans k.succ.le_succ)]
     refine' forall_congr' fun x => iff_eq_eq.mpr (congr rfl (funext (Fin.lastCases _ fun i => _)))
     · simp only [Function.comp_apply, val_last, snoc_last]
-      by_cases k < m
+      by_cases h : k < m
       · rw [if_pos h]
         refine' (congr rfl (ext _)).trans (snoc_last _ _)
         simp only [coe_orderIso_apply, coe_castAdd, val_last, self_eq_add_right]
@@ -716,10 +716,9 @@ nonrec def Sentence.Realize (φ : L.Sentence) : Prop :=
   φ.Realize (default : _ → M)
 #align first_order.language.sentence.realize FirstOrder.Language.Sentence.Realize
 
+-- input using \|= or \vDash, but not using \models
 @[inherit_doc Sentence.Realize]
-infixl:51
-  " ⊨ " =>-- input using \|= or \vDash, but not using \models
-  Sentence.Realize
+infixl:51 " ⊨ " => Sentence.Realize
 
 @[simp]
 theorem Sentence.realize_not {φ : L.Sentence} : M ⊨ φ.not ↔ ¬M ⊨ φ :=
@@ -801,10 +800,9 @@ class Theory.Model (T : L.Theory) : Prop where
 set_option linter.uppercaseLean3 false in
 #align first_order.language.Theory.model FirstOrder.Language.Theory.Model
 
+-- input using \|= or \vDash, but not using \models
 @[inherit_doc Theory.Model]
-infixl:51
-  " ⊨ " =>-- input using \|= or \vDash, but not using \models
-  Theory.Model
+infixl:51 " ⊨ " => Theory.Model
 
 variable {M} (T : L.Theory)

d565b3df

chore: bump to nightly-2023-07-15 (#5992)

Various adaptations to changes when Fin API was moved to Std. One notable change is that many lemmas are now stated in terms of i ≠ 0 (for i : Fin n) rather then i.1 ≠ 0, and as a consequence many Fin.vne_of_ne applications have been added or removed, mostly removed.

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

24924f96

Diff

@@ -676,7 +676,7 @@ theorem realize_relabel {φ : L.Formula α} {g : α → β} {v : β → M} :
 theorem realize_relabel_sum_inr (φ : L.Formula (Fin n)) {v : Empty → M} {x : Fin n → M} :
     (BoundedFormula.relabel Sum.inr φ).Realize v x ↔ φ.Realize x := by
   rw [BoundedFormula.realize_relabel, Formula.Realize, Sum.elim_comp_inr, Fin.castAdd_zero,
-    castIso_refl, OrderIso.coe_refl, Function.comp.right_id,
+    cast_refl, Function.comp.right_id,
     Subsingleton.elim (x ∘ (natAdd n : Fin 0 → Fin n)) default]
 #align first_order.language.formula.realize_relabel_sum_inr FirstOrder.Language.Formula.realize_relabel_sum_inr

d565b3df

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

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-
-! This file was ported from Lean 3 source module model_theory.semantics
-! leanprover-community/mathlib commit d565b3df44619c1498326936be16f1a935df0728
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Finset.Basic
 import Mathlib.ModelTheory.Syntax
 
+#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
+
 /-!
 # Basics on First-Order Semantics
 This file defines the interpretations of first-order terms, formulas, sentences, and theories

d565b3df

chore: bump to nightly-2023-07-01 (#5409)

depends on: https://github.com/leanprover/std4/pull/163
depends on: #5584
depends on: #5715
depends on: #5729
depends on: https://github.com/leanprover/std4/pull/173

Co-authored-by: Komyyy <pol_tta@outlook.jp> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

906f7221

Diff

@@ -100,7 +100,7 @@ theorem realize_relabel {t : L.Term α} {g : α → β} {v : β → M} :
 theorem realize_liftAt {n n' m : ℕ} {t : L.Term (Sum α (Fin n))} {v : Sum α (Fin (n + n')) → M} :
     (t.liftAt n' m).realize v =
       t.realize (v ∘ Sum.map id fun i : Fin _ =>
-        if ↑i < m then Fin.castAdd n' i else Fin.addNat n' i) :=
+        if ↑i < m then Fin.castAdd n' i else Fin.addNat i n') :=
   realize_relabel
 #align first_order.language.term.realize_lift_at FirstOrder.Language.Term.realize_liftAt
 
@@ -407,7 +407,7 @@ theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum
 theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + n') → M}
     (hmn : m + n' ≤ n + 1) :
     (φ.liftAt n' m).Realize v xs ↔
-      φ.Realize v (xs ∘ fun i => if ↑i < m then Fin.castAdd n' i else Fin.addNat n' i) := by
+      φ.Realize v (xs ∘ fun i => if ↑i < m then Fin.castAdd n' i else Fin.addNat i n') := by
   rw [liftAt]
   induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 k _ ih3
   · simp [mapTermRel, Realize]
@@ -428,22 +428,22 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
       · rw [if_neg h]
         refine' (congr rfl (ext _)).trans (snoc_last _ _)
         simp
-    · simp only [Function.comp_apply, Fin.snoc_castSuccEmb]
-      refine' (congr rfl (ext _)).trans (snoc_castSuccEmb _ _ _)
-      simp only [coe_castSuccEmb, coe_orderIso_apply]
+    · simp only [Function.comp_apply, Fin.snoc_castSucc]
+      refine' (congr rfl (ext _)).trans (snoc_castSucc _ _ _)
+      simp only [coe_castSucc, coe_orderIso_apply]
       split_ifs <;> simp
 #align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAt
 
 theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + 1) → M}
     (hmn : m ≤ n) :
     (φ.liftAt 1 m).Realize v xs ↔
-      φ.Realize v (xs ∘ fun i => if ↑i < m then castSuccEmb i else i.succ) := by
-  simp [realize_liftAt (add_le_add_right hmn 1), castSuccEmb]
+      φ.Realize v (xs ∘ fun i => if ↑i < m then castSucc i else i.succ) := by
+  simp [realize_liftAt (add_le_add_right hmn 1), castSucc]
 #align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_one
 
 @[simp]
 theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α → M}
-    {xs : Fin (n + 1) → M} : (φ.liftAt 1 n).Realize v xs ↔ φ.Realize v (xs ∘ castSuccEmb) := by
+    {xs : Fin (n + 1) → M} : (φ.liftAt 1 n).Realize v xs ↔ φ.Realize v (xs ∘ castSucc) := by
   rw [realize_liftAt_one (refl n), iff_eq_eq]
   refine' congr rfl (congr rfl (funext fun i => _))
   rw [if_pos i.is_lt]
@@ -514,12 +514,12 @@ theorem realize_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQF φ)
   induction' hψ with _ _ hψ _ _ _hψ ih _ _ _hψ ih
   · rw [hψ.toPrenexImpRight]
   · refine' _root_.trans (forall_congr' fun _ => ih hφ.liftAt) _
-    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSuccEmb, realize_all]
+    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_all]
     exact ⟨fun h1 a h2 => h1 h2 a, fun h1 h2 a => h1 a h2⟩
   · unfold toPrenexImpRight
     rw [realize_ex]
     refine' _root_.trans (exists_congr fun _ => ih hφ.liftAt) _
-    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSuccEmb, realize_ex]
+    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_ex]
     refine' ⟨_, fun h' => _⟩
     · rintro ⟨a, ha⟩ h
       exact ⟨a, ha h⟩
@@ -539,7 +539,7 @@ theorem realize_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ)
   · unfold toPrenexImp
     rw [realize_ex]
     refine' _root_.trans (exists_congr fun _ => ih hψ.liftAt) _
-    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSuccEmb, realize_all]
+    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_all]
     refine' ⟨_, fun h' => _⟩
     · rintro ⟨a, ha⟩ h
       exact ha (h a)
@@ -933,9 +933,9 @@ theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M
         · rw [Sum.elim_inr, Sum.elim_inr,
             finSumFinEquiv_symm_last, Sum.map_inr, Sum.elim_inr]
           simp [Fin.snoc]
-        · simp only [castSuccEmb, Function.comp_apply, Sum.elim_inr,
+        · simp only [castSucc, Function.comp_apply, Sum.elim_inr,
             finSumFinEquiv_symm_apply_castAdd, Sum.map_inl, Sum.elim_inl]
-          rw [← castSuccEmb]
+          rw [← castSucc]
           simp
     · exact Fin.elim0 x
 #align first_order.language.bounded_formula.realize_to_formula FirstOrder.Language.BoundedFormula.realize_toFormula

d565b3df

chore: rename Fin.castSucc to Fin.castSuccEmb (#5729)

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

f4e42872

Diff

@@ -428,22 +428,22 @@ theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M
       · rw [if_neg h]
         refine' (congr rfl (ext _)).trans (snoc_last _ _)
         simp
-    · simp only [Function.comp_apply, Fin.snoc_castSucc]
-      refine' (congr rfl (ext _)).trans (snoc_castSucc _ _ _)
-      simp only [coe_castSucc, coe_orderIso_apply]
+    · simp only [Function.comp_apply, Fin.snoc_castSuccEmb]
+      refine' (congr rfl (ext _)).trans (snoc_castSuccEmb _ _ _)
+      simp only [coe_castSuccEmb, coe_orderIso_apply]
       split_ifs <;> simp
 #align first_order.language.bounded_formula.realize_lift_at FirstOrder.Language.BoundedFormula.realize_liftAt
 
 theorem realize_liftAt_one {n m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + 1) → M}
     (hmn : m ≤ n) :
     (φ.liftAt 1 m).Realize v xs ↔
-      φ.Realize v (xs ∘ fun i => if ↑i < m then castSucc i else i.succ) := by
-  simp [realize_liftAt (add_le_add_right hmn 1), castSucc]
+      φ.Realize v (xs ∘ fun i => if ↑i < m then castSuccEmb i else i.succ) := by
+  simp [realize_liftAt (add_le_add_right hmn 1), castSuccEmb]
 #align first_order.language.bounded_formula.realize_lift_at_one FirstOrder.Language.BoundedFormula.realize_liftAt_one
 
 @[simp]
 theorem realize_liftAt_one_self {n : ℕ} {φ : L.BoundedFormula α n} {v : α → M}
-    {xs : Fin (n + 1) → M} : (φ.liftAt 1 n).Realize v xs ↔ φ.Realize v (xs ∘ castSucc) := by
+    {xs : Fin (n + 1) → M} : (φ.liftAt 1 n).Realize v xs ↔ φ.Realize v (xs ∘ castSuccEmb) := by
   rw [realize_liftAt_one (refl n), iff_eq_eq]
   refine' congr rfl (congr rfl (funext fun i => _))
   rw [if_pos i.is_lt]
@@ -514,12 +514,12 @@ theorem realize_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQF φ)
   induction' hψ with _ _ hψ _ _ _hψ ih _ _ _hψ ih
   · rw [hψ.toPrenexImpRight]
   · refine' _root_.trans (forall_congr' fun _ => ih hφ.liftAt) _
-    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_all]
+    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSuccEmb, realize_all]
     exact ⟨fun h1 a h2 => h1 h2 a, fun h1 h2 a => h1 a h2⟩
   · unfold toPrenexImpRight
     rw [realize_ex]
     refine' _root_.trans (exists_congr fun _ => ih hφ.liftAt) _
-    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_ex]
+    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSuccEmb, realize_ex]
     refine' ⟨_, fun h' => _⟩
     · rintro ⟨a, ha⟩ h
       exact ⟨a, ha h⟩
@@ -539,7 +539,7 @@ theorem realize_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ)
   · unfold toPrenexImp
     rw [realize_ex]
     refine' _root_.trans (exists_congr fun _ => ih hψ.liftAt) _
-    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_all]
+    simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSuccEmb, realize_all]
     refine' ⟨_, fun h' => _⟩
     · rintro ⟨a, ha⟩ h
       exact ha (h a)
@@ -933,9 +933,9 @@ theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M
         · rw [Sum.elim_inr, Sum.elim_inr,
             finSumFinEquiv_symm_last, Sum.map_inr, Sum.elim_inr]
           simp [Fin.snoc]
-        · simp only [castSucc, Function.comp_apply, Sum.elim_inr,
+        · simp only [castSuccEmb, Function.comp_apply, Sum.elim_inr,
             finSumFinEquiv_symm_apply_castAdd, Sum.map_inl, Sum.elim_inl]
-          rw [← castSucc]
+          rw [← castSuccEmb]
           simp
     · exact Fin.elim0 x
 #align first_order.language.bounded_formula.realize_to_formula FirstOrder.Language.BoundedFormula.realize_toFormula

d565b3df

chore: rename Fin.cast to Fin.castIso (#5584)

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

b0bb2972

Diff

@@ -356,9 +356,9 @@ theorem realize_iff : (φ.iff ψ).Realize v xs ↔ (φ.Realize v xs ↔ ψ.Reali
 #align first_order.language.bounded_formula.realize_iff FirstOrder.Language.BoundedFormula.realize_iff
 
 theorem realize_castLe_of_eq {m n : ℕ} (h : m = n) {h' : m ≤ n} {φ : L.BoundedFormula α m}
-    {v : α → M} {xs : Fin n → M} : (φ.castLE h').Realize v xs ↔ φ.Realize v (xs ∘ Fin.cast h) := by
+    {v : α → M} {xs : Fin n → M} : (φ.castLE h').Realize v xs ↔ φ.Realize v (xs ∘ castIso h) := by
   subst h
-  simp only [castLE_rfl, cast_refl, OrderIso.coe_refl, Function.comp.right_id]
+  simp only [castLE_rfl, castIso_refl, OrderIso.coe_refl, Function.comp.right_id]
 #align first_order.language.bounded_formula.realize_cast_le_of_eq FirstOrder.Language.BoundedFormula.realize_castLe_of_eq
 
 theorem realize_mapTermRel_id [L'.Structure M]
@@ -679,7 +679,7 @@ theorem realize_relabel {φ : L.Formula α} {g : α → β} {v : β → M} :
 theorem realize_relabel_sum_inr (φ : L.Formula (Fin n)) {v : Empty → M} {x : Fin n → M} :
     (BoundedFormula.relabel Sum.inr φ).Realize v x ↔ φ.Realize x := by
   rw [BoundedFormula.realize_relabel, Formula.Realize, Sum.elim_comp_inr, Fin.castAdd_zero,
-    cast_refl, OrderIso.coe_refl, Function.comp.right_id,
+    castIso_refl, OrderIso.coe_refl, Function.comp.right_id,
     Subsingleton.elim (x ∘ (natAdd n : Fin 0 → Fin n)) default]
 #align first_order.language.formula.realize_relabel_sum_inr FirstOrder.Language.Formula.realize_relabel_sum_inr

d565b3df

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

@@ -185,8 +185,8 @@ theorem realize_varsToConstants [L[[α]].Structure M] [(lhomWithConstants L α).
   induction' t with ab n f ts ih
   · cases' ab with a b
     --Porting note: both cases were `simp [Language.con]`
-    . simp [Language.con, realize, constantMap, funMap_eq_coe_constants]
-    . simp [realize, constantMap]
+    · simp [Language.con, realize, constantMap, funMap_eq_coe_constants]
+    · simp [realize, constantMap]
   · simp only [realize, constantsOn, mk₂_Functions, ih]
     --Porting note: below lemma does not work with simp for some reason
     rw [withConstants_funMap_sum_inl]
@@ -335,8 +335,7 @@ theorem realize_foldr_sup (l : List (L.BoundedFormula α n)) (v : α → M) (xs
     (l.foldr (· ⊔ ·) ⊥).Realize v xs ↔ ∃ φ ∈ l, BoundedFormula.Realize φ v xs := by
   induction' l with φ l ih
   · simp
-  ·
-    simp_rw [List.foldr_cons, realize_sup, ih, exists_prop, List.mem_cons, or_and_right, exists_or,
+  · simp_rw [List.foldr_cons, realize_sup, ih, exists_prop, List.mem_cons, or_and_right, exists_or,
       exists_eq_left]
 #align first_order.language.bounded_formula.realize_foldr_sup FirstOrder.Language.BoundedFormula.realize_foldr_sup
 
@@ -557,9 +556,9 @@ theorem realize_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ)
 theorem realize_toPrenex (φ : L.BoundedFormula α n) {v : α → M} :
     ∀ {xs : Fin n → M}, φ.toPrenex.Realize v xs ↔ φ.Realize v xs := by
   induction' φ with _ _ _ _ _ _ _ _ _ f1 f2 h1 h2 _ _ h
-  . exact Iff.rfl
-  . exact Iff.rfl
-  . exact Iff.rfl
+  · exact Iff.rfl
+  · exact Iff.rfl
+  · exact Iff.rfl
   · intros
     rw [toPrenex, realize_toPrenexImp f1.toPrenex_isPrenex f2.toPrenex_isPrenex, realize_imp,
       realize_imp, h1, h2]
@@ -741,8 +740,8 @@ theorem realize_equivSentence_symm_con [L[[α]].Structure M]
   refine' _root_.trans _ BoundedFormula.realize_constantsVarsEquiv
   rw [iff_iff_eq]
   congr with (_ | a)
-  . simp
-  . cases a
+  · simp
+  · cases a
 #align first_order.language.formula.realize_equiv_sentence_symm_con FirstOrder.Language.Formula.realize_equivSentence_symm_con
 
 @[simp]
@@ -919,8 +918,7 @@ theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M
   · rfl
   · simp [BoundedFormula.Realize]
   · simp [BoundedFormula.Realize]
-  ·
-    rw [toFormula, Formula.Realize, realize_imp, ← Formula.Realize, ih1, ← Formula.Realize, ih2,
+  · rw [toFormula, Formula.Realize, realize_imp, ← Formula.Realize, ih1, ← Formula.Realize, ih2,
       realize_imp]
   · rw [toFormula, Formula.Realize, realize_all, realize_all]
     refine' forall_congr' fun a => _

d565b3df

fix: precedence of # (#5623)

d450fcd7

Diff

@@ -1036,7 +1036,7 @@ section Cardinality
 
 variable (L)
 @[simp]
-theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ (#M) := by
+theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ #M := by
   rw [← lift_mk_fin, ← lift_le.{0}, lift_lift, lift_mk_le, Sentence.cardGe, Sentence.Realize,
     BoundedFormula.realize_exs]
   simp_rw [BoundedFormula.realize_foldr_inf]
@@ -1089,7 +1089,7 @@ theorem model_distinctConstantsTheory {M : Type w} [L[[α]].Structure M] (s : Se
 #align first_order.language.model_distinct_constants_theory FirstOrder.Language.model_distinctConstantsTheory
 
 theorem card_le_of_model_distinctConstantsTheory (s : Set α) (M : Type w) [L[[α]].Structure M]
-    [h : M ⊨ L.distinctConstantsTheory s] : Cardinal.lift.{w} (#s) ≤ Cardinal.lift.{u'} (#M) :=
+    [h : M ⊨ L.distinctConstantsTheory s] : Cardinal.lift.{w} #s ≤ Cardinal.lift.{u'} #M :=
   lift_mk_le'.2 ⟨⟨_, Set.injOn_iff_injective.1 ((L.model_distinctConstantsTheory s).1 h)⟩⟩
 #align first_order.language.card_le_of_model_distinct_constants_theory FirstOrder.Language.card_le_of_model_distinctConstantsTheory

d565b3df

chore: reorder universe variables in Cardinal.lift_le and Cardinal.lift_mk_le (#5325)

Cardinal.lift_le and Cardinal.lift_mk_le have their universes out of order, in the sense that persistently through the rest of the library we need to specify the 2nd universe (resp 3rd), while the others are solved by unification.

This PR reorders the universes so it's easier to specify the thing you need to specify!

(This PR doesn't get rid of all the occurrences of \.\{_, in the library, but I'd like to do that later.)

I do have a hidden agenda here, which is that I've been experimenting with solutions to the dreaded "Can't solve max u v = max v ?u" universe unification issue (which is making life hellish forward porting https://github.com/leanprover-community/mathlib/pull/19153), and my favourite (but still hacky) solution doesn't like some of the occasions where we reference a lemma filling in some of its universe arguments with _ but then fully specify a later one. (e.g. rw [← lift_le.{_, max u v}, lift_lift, lift_mk_le.{_, _, v}] in ModelTheory/Skolem.lean). Hence the cleanup proposed in this PR makes my life easier working on these experiments. :-)

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

719a2170

Diff

@@ -1037,7 +1037,7 @@ section Cardinality
 variable (L)
 @[simp]
 theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ (#M) := by
-  rw [← lift_mk_fin, ← lift_le.{w, 0}, lift_lift, lift_mk_le, Sentence.cardGe, Sentence.Realize,
+  rw [← lift_mk_fin, ← lift_le.{0}, lift_lift, lift_mk_le, Sentence.cardGe, Sentence.Realize,
     BoundedFormula.realize_exs]
   simp_rw [BoundedFormula.realize_foldr_inf]
   simp only [Function.comp_apply, List.mem_map, Prod.exists, Ne.def, List.mem_product,

d565b3df

chore: fix backtick in docs (#5077)

I wrote a script to find lines that contain an odd number of backticks

878d95c4

Diff

@@ -184,7 +184,7 @@ theorem realize_varsToConstants [L[[α]].Structure M] [(lhomWithConstants L α).
     t.varsToConstants.realize v = t.realize (Sum.elim (fun a => ↑(L.con a)) v) := by
   induction' t with ab n f ts ih
   · cases' ab with a b
-    --Porting note: both cases were `simp [Language.con]
+    --Porting note: both cases were `simp [Language.con]`
     . simp [Language.con, realize, constantMap, funMap_eq_coe_constants]
     . simp [realize, constantMap]
   · simp only [realize, constantsOn, mk₂_Functions, ih]

d565b3df

doc: fix the docs on ModelTheory/* (#3968)

a2ab29ce

Diff

@@ -19,27 +19,27 @@ in a style inspired by the [Flypitch project](https://flypitch.github.io/).
 ## Main Definitions
 * `FirstOrder.Language.Term.realize` is defined so that `t.realize v` is the term `t` evaluated at
 variables `v`.
-* `FirstOrder.Language.BoundedFormula.Realize` is defined so that `φ.realize v xs` is the bounded
+* `FirstOrder.Language.BoundedFormula.Realize` is defined so that `φ.Realize v xs` is the bounded
 formula `φ` evaluated at tuples of variables `v` and `xs`.
-* `FirstOrder.Language.Formula.Realize` is defined so that `φ.realize v` is the formula `φ`
+* `FirstOrder.Language.Formula.Realize` is defined so that `φ.Realize v` is the formula `φ`
 evaluated at variables `v`.
-* `FirstOrder.Language.Sentence.Realize` is defined so that `φ.realize M` is the sentence `φ`
+* `FirstOrder.Language.Sentence.Realize` is defined so that `φ.Realize M` is the sentence `φ`
 evaluated in the structure `M`. Also denoted `M ⊨ φ`.
-* `FirstOrder.Language.Theory.Model` is defined so that `T.model M` is true if and only if every
+* `FirstOrder.Language.Theory.Model` is defined so that `T.Model M` is true if and only if every
 sentence of `T` is realized in `M`. Also denoted `T ⊨ φ`.
 
 ## Main Results
 * `FirstOrder.Language.BoundedFormula.realize_toPrenex` shows that the prenex normal form of a
 formula has the same realization as the original formula.
-* Several results in this file show that syntactic constructions such as `relabel`, `castLe`,
-`lift_at`, `subst`, and the actions of language maps commute with realization of terms, formulas,
+* Several results in this file show that syntactic constructions such as `relabel`, `castLE`,
+`liftAt`, `subst`, and the actions of language maps commute with realization of terms, formulas,
 sentences, and theories.
 
 ## Implementation Notes
-* Formulas use a modified version of de Bruijn variables. Specifically, a `L.boundedFormula α n`
+* Formulas use a modified version of de Bruijn variables. Specifically, a `L.BoundedFormula α n`
 is a formula with some variables indexed by a type `α`, which cannot be quantified over, and some
-indexed by `Fin n`, which can. For any `φ : L.boundedFormula α (n + 1)`, we define the formula
-`∀' φ : L.boundedFormula α n` by universally quantifying over the variable indexed by
+indexed by `Fin n`, which can. For any `φ : L.BoundedFormula α (n + 1)`, we define the formula
+`∀' φ : L.BoundedFormula α n` by universally quantifying over the variable indexed by
 `n : Fin (n + 1)`.
 
 ## References

d565b3df

refactor: rename FirstOrder.Language.Theory.CompleteTheory.subset to FirstOrder.Language.Theory.completeTheory.subset (#3990)

0de17db2

Diff

@@ -864,10 +864,10 @@ theorem model_iff_subset_completeTheory : M ⊨ T ↔ T ⊆ L.completeTheory M :
 set_option linter.uppercaseLean3 false in
 #align first_order.language.Theory.model_iff_subset_complete_theory FirstOrder.Language.Theory.model_iff_subset_completeTheory
 
-theorem CompleteTheory.subset [MT : M ⊨ T] : T ⊆ L.completeTheory M :=
+theorem completeTheory.subset [MT : M ⊨ T] : T ⊆ L.completeTheory M :=
   model_iff_subset_completeTheory.1 MT
 set_option linter.uppercaseLean3 false in
-#align first_order.language.Theory.complete_theory.subset FirstOrder.Language.Theory.CompleteTheory.subset
+#align first_order.language.Theory.complete_theory.subset FirstOrder.Language.Theory.completeTheory.subset
 
 end Theory

d565b3df

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

@@ -143,7 +143,7 @@ theorem realize_restrictVar [DecidableEq α] {t : L.Term α} {s : Set α} (h : 
     {v : α → M} : (t.restrictVar (Set.inclusion h)).realize (v ∘ (↑)) = t.realize v := by
   induction' t with _ _ _ _ ih
   · rfl
-  · simp_rw [varFinset, Finset.coe_bunionᵢ, Set.unionᵢ_subset_iff] at h
+  · simp_rw [varFinset, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var FirstOrder.Language.Term.realize_restrictVar
 
@@ -154,7 +154,7 @@ theorem realize_restrictVarLeft [DecidableEq α] {γ : Type _} {t : L.Term (Sum
       t.realize (Sum.elim v xs) := by
   induction' t with a _ _ _ ih
   · cases a <;> rfl
-  · simp_rw [varFinsetLeft, Finset.coe_bunionᵢ, Set.unionᵢ_subset_iff] at h
+  · simp_rw [varFinsetLeft, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 #align first_order.language.term.realize_restrict_var_left FirstOrder.Language.Term.realize_restrictVarLeft

d565b3df

feat: port ModelTheory.Semantics (#3196)

Co-authored-by: ChrisHughes24 <chrishughes24@gmail.com>

017781e6

`model_theory.semantics` ⟷ `Mathlib.ModelTheory.Semantics`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Renames

Dependencies 7 + 306

model_theory.semantics ⟷ Mathlib.ModelTheory.Semantics

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Renames

Dependencies 7 + 306

`model_theory.semantics` ⟷ `Mathlib.ModelTheory.Semantics`