model_theory.ultraproducts | 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

f1ae6206

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

0b7c740e

bump to nightly-2024-03-17-08

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

22f01ce4

Diff

@@ -57,7 +57,7 @@ instance setoidPrestructure : L.Prestructure ((u : Filter α).productSetoid M) :
     fun_equiv := fun n f x y xy =>
       by
       refine' mem_of_superset (Inter_mem.2 xy) fun a ha => _
-      simp only [Set.mem_iInter, Set.mem_setOf_eq] at ha 
+      simp only [Set.mem_iInter, Set.mem_setOf_eq] at ha
       simp only [Set.mem_setOf_eq, ha]
     rel_equiv := fun n r x y xy => by
       rw [← iff_eq_eq]
@@ -143,7 +143,7 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
     simp only [← Fin.comp_snoc, ih, h']
     refine' ⟨fun h => _, fun h m => _⟩
     · contrapose! h
-      simp_rw [← Ultrafilter.eventually_not, Classical.not_forall] at h 
+      simp_rw [← Ultrafilter.eventually_not, Classical.not_forall] at h
       refine'
         ⟨fun a : α =>
           Classical.epsilon fun m : M a =>

0b7c740e

bump to nightly-2023-10-21-06

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

a2dc86f8

Diff

@@ -143,7 +143,7 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
     simp only [← Fin.comp_snoc, ih, h']
     refine' ⟨fun h => _, fun h m => _⟩
     · contrapose! h
-      simp_rw [← Ultrafilter.eventually_not, not_forall] at h 
+      simp_rw [← Ultrafilter.eventually_not, Classical.not_forall] at h 
       refine'
         ⟨fun a : α =>
           Classical.epsilon fun m : M a =>

0b7c740e

bump to nightly-2023-09-24-06

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

5655435f

Diff

@@ -3,9 +3,9 @@ Copyright (c) 2022 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson
 -/
-import Mathbin.ModelTheory.Quotients
-import Mathbin.Order.Filter.Germ
-import Mathbin.Order.Filter.Ultrafilter
+import ModelTheory.Quotients
+import Order.Filter.Germ
+import Order.Filter.Ultrafilter
 
 #align_import model_theory.ultraproducts from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"

0b7c740e

bump to nightly-2023-09-20-06

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

e28e6964

Diff

@@ -131,7 +131,7 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
       ∀ m : ∀ a : α, M a,
         φ.realize (fun i : β => (x i : (u : Filter α).product M))
           (Fin.snoc (coe ∘ v) (↑m : (u : Filter α).product M))
-    · exact forall_quotient_iff
+    · exact Quotient.forall
     have h' :
       ∀ (m : ∀ a, M a) (a : α),
         (fun i : Fin (k + 1) => (Fin.snoc v m : _ → ∀ a, M a) i a) =

0b7c740e

bump to nightly-2023-07-20-09

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

9c56d0dd

Diff

@@ -2,16 +2,13 @@
 Copyright (c) 2022 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson
-
-! This file was ported from Lean 3 source module model_theory.ultraproducts
-! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.ModelTheory.Quotients
 import Mathbin.Order.Filter.Germ
 import Mathbin.Order.Filter.Ultrafilter
 
+#align_import model_theory.ultraproducts from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
+
 /-! # Ultraproducts and Łoś's Theorem
 
 > THIS FILE IS SYNCHRONIZED WITH MATHLIB4.

0b7c740e

bump to nightly-2023-07-16-17

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

6db63fa6

Diff

@@ -142,7 +142,7 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
       by
       refine' fun m a => funext (Fin.reverseInduction _ fun i hi => _)
       · simp only [Fin.snoc_last]
-      · simp only [Fin.snoc_castSuccEmb]
+      · simp only [Fin.snoc_castSucc]
     simp only [← Fin.comp_snoc, ih, h']
     refine' ⟨fun h => _, fun h m => _⟩
     · contrapose! h

0b7c740e

bump to nightly-2023-07-06-11

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

a2f50eda

Diff

@@ -142,7 +142,7 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
       by
       refine' fun m a => funext (Fin.reverseInduction _ fun i hi => _)
       · simp only [Fin.snoc_last]
-      · simp only [Fin.snoc_castSucc]
+      · simp only [Fin.snoc_castSuccEmb]
     simp only [← Fin.comp_snoc, ih, h']
     refine' ⟨fun h => _, fun h m => _⟩
     · contrapose! h

0b7c740e

bump to nightly-2023-06-13-21

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

829520ac

Diff

@@ -86,11 +86,14 @@ instance structure : L.Structure ((u : Filter α).product M) :=
 #align first_order.language.ultraproduct.Structure FirstOrder.Language.Ultraproduct.structure
 -/
 
+#print FirstOrder.Language.Ultraproduct.funMap_cast /-
 theorem funMap_cast {n : ℕ} (f : L.Functions n) (x : Fin n → ∀ a, M a) :
     (funMap f fun i => (x i : (u : Filter α).product M)) = fun a => funMap f fun i => x i a := by
   apply fun_map_quotient_mk
 #align first_order.language.ultraproduct.fun_map_cast FirstOrder.Language.Ultraproduct.funMap_cast
+-/
 
+#print FirstOrder.Language.Ultraproduct.term_realize_cast /-
 theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.term β) :
     (t.realize fun i => (x i : (u : Filter α).product M)) = fun a => t.realize fun i => x i a :=
   by
@@ -103,9 +106,11 @@ theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.term β)
   · simp only [term.realize, t_ih]
     rfl
 #align first_order.language.ultraproduct.term_realize_cast FirstOrder.Language.Ultraproduct.term_realize_cast
+-/
 
 variable [∀ a : α, Nonempty (M a)]
 
+#print FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast /-
 theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormula β n)
     (x : β → ∀ a, M a) (v : Fin n → ∀ a, M a) :
     (φ.realize (fun i : β => (x i : (u : Filter α).product M)) fun i => v i) ↔
@@ -152,7 +157,9 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
     · rw [Filter.eventually_iff] at *
       exact Filter.mem_of_superset h fun a ha => ha (m a)
 #align first_order.language.ultraproduct.bounded_formula_realize_cast FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast
+-/
 
+#print FirstOrder.Language.Ultraproduct.realize_formula_cast /-
 theorem realize_formula_cast {β : Type _} (φ : L.Formula β) (x : β → ∀ a, M a) :
     (φ.realize fun i => (x i : (u : Filter α).product M)) ↔
       ∀ᶠ a : α in u, φ.realize fun i => x i a :=
@@ -160,9 +167,11 @@ theorem realize_formula_cast {β : Type _} (φ : L.Formula β) (x : β → ∀ a
   simp_rw [formula.realize, ← bounded_formula_realize_cast φ x, iff_eq_eq]
   exact congr rfl (Subsingleton.elim _ _)
 #align first_order.language.ultraproduct.realize_formula_cast FirstOrder.Language.Ultraproduct.realize_formula_cast
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print FirstOrder.Language.Ultraproduct.sentence_realize /-
 /-- Łoś's Theorem : A sentence is true in an ultraproduct if and only if the set of structures it is
   true in is in the ultrafilter. -/
 theorem sentence_realize (φ : L.Sentence) : (u : Filter α).product M ⊨ φ ↔ ∀ᶠ a : α in u, M a ⊨ φ :=
@@ -170,6 +179,7 @@ theorem sentence_realize (φ : L.Sentence) : (u : Filter α).product M ⊨ φ 
   simp_rw [sentence.realize, ← realize_formula_cast φ, iff_eq_eq]
   exact congr rfl (Subsingleton.elim _ _)
 #align first_order.language.ultraproduct.sentence_realize FirstOrder.Language.Ultraproduct.sentence_realize
+-/
 
 instance : Nonempty ((u : Filter α).product M) :=
   letI : ∀ a, Inhabited (M a) := fun _ => Classical.inhabited_of_nonempty'

0b7c740e

bump to nightly-2023-06-04-18

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

3fb4268b

Diff

@@ -94,7 +94,8 @@ theorem funMap_cast {n : ℕ} (f : L.Functions n) (x : Fin n → ∀ a, M a) :
 theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.term β) :
     (t.realize fun i => (x i : (u : Filter α).product M)) = fun a => t.realize fun i => x i a :=
   by
-  convert@term.realize_quotient_mk L _ ((u : Filter α).productSetoid M)
+  convert
+    @term.realize_quotient_mk L _ ((u : Filter α).productSetoid M)
       (ultraproduct.setoid_prestructure M u) _ t x
   ext a
   induction t

0b7c740e

bump to nightly-2023-06-01-16

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

b9c87c70

Diff

@@ -60,7 +60,7 @@ instance setoidPrestructure : L.Prestructure ((u : Filter α).productSetoid M) :
     fun_equiv := fun n f x y xy =>
       by
       refine' mem_of_superset (Inter_mem.2 xy) fun a ha => _
-      simp only [Set.mem_iInter, Set.mem_setOf_eq] at ha
+      simp only [Set.mem_iInter, Set.mem_setOf_eq] at ha 
       simp only [Set.mem_setOf_eq, ha]
     rel_equiv := fun n r x y xy => by
       rw [← iff_eq_eq]
@@ -140,7 +140,7 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
     simp only [← Fin.comp_snoc, ih, h']
     refine' ⟨fun h => _, fun h m => _⟩
     · contrapose! h
-      simp_rw [← Ultrafilter.eventually_not, not_forall] at h
+      simp_rw [← Ultrafilter.eventually_not, not_forall] at h 
       refine'
         ⟨fun a : α =>
           Classical.epsilon fun m : M a =>

0b7c740e

bump to nightly-2023-05-28-18

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

8d353152

Diff

@@ -35,7 +35,7 @@ universe u v
 
 variable {α : Type _} (M : α → Type _) (u : Ultrafilter α)
 
-open FirstOrder Filter
+open scoped FirstOrder Filter
 
 open Filter

0b7c740e

bump to nightly-2023-05-28-06

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

ba5d8b97

Diff

@@ -86,23 +86,11 @@ instance structure : L.Structure ((u : Filter α).product M) :=
 #align first_order.language.ultraproduct.Structure FirstOrder.Language.Ultraproduct.structure
 -/
 
-/- warning: first_order.language.ultraproduct.fun_map_cast -> FirstOrder.Language.Ultraproduct.funMap_cast is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u2}} {M : α -> Type.{u1}} {u : Ultrafilter.{u2} α} {L : FirstOrder.Language.{u3, u4}} [_inst_1 : forall (a : α), FirstOrder.Language.Structure.{u3, u4, u1} L (M a)] {n : Nat} (f : FirstOrder.Language.Functions.{u3, u4} L n) (x : (Fin n) -> (forall (a : α), M a)), Eq.{max (succ u2) (succ u1)} (Quotient.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a))) (FirstOrder.Language.Structure.funMap.{u3, u4, max u2 u1} L (Quotient.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a))) (FirstOrder.Language.quotientStructure.{u3, u4, max u2 u1} L (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (FirstOrder.Language.Ultraproduct.setoidPrestructure.{u3, u4, u2, u1} α (fun (a : α) => M a) u L (fun (a : α) => _inst_1 a))) n f (fun (i : Fin n) => Quotient.mk'.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (x i))) (Quotient.mk'.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (fun (a : α) => FirstOrder.Language.Structure.funMap.{u3, u4, u1} L (M a) (_inst_1 a) n f (fun (i : Fin n) => x i a)))
-Case conversion may be inaccurate. Consider using '#align first_order.language.ultraproduct.fun_map_cast FirstOrder.Language.Ultraproduct.funMap_castₓ'. -/
 theorem funMap_cast {n : ℕ} (f : L.Functions n) (x : Fin n → ∀ a, M a) :
     (funMap f fun i => (x i : (u : Filter α).product M)) = fun a => funMap f fun i => x i a := by
   apply fun_map_quotient_mk
 #align first_order.language.ultraproduct.fun_map_cast FirstOrder.Language.Ultraproduct.funMap_cast
 
-/- warning: first_order.language.ultraproduct.term_realize_cast -> FirstOrder.Language.Ultraproduct.term_realize_cast is a dubious translation:
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 theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.term β) :
     (t.realize fun i => (x i : (u : Filter α).product M)) = fun a => t.realize fun i => x i a :=
   by
@@ -117,9 +105,6 @@ theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.term β)
 
 variable [∀ a : α, Nonempty (M a)]
 
-/- warning: first_order.language.ultraproduct.bounded_formula_realize_cast -> FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align first_order.language.ultraproduct.bounded_formula_realize_cast FirstOrder.Language.Ultraproduct.boundedFormula_realize_castₓ'. -/
 theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormula β n)
     (x : β → ∀ a, M a) (v : Fin n → ∀ a, M a) :
     (φ.realize (fun i : β => (x i : (u : Filter α).product M)) fun i => v i) ↔
@@ -167,12 +152,6 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
       exact Filter.mem_of_superset h fun a ha => ha (m a)
 #align first_order.language.ultraproduct.bounded_formula_realize_cast FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast
 
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 theorem realize_formula_cast {β : Type _} (φ : L.Formula β) (x : β → ∀ a, M a) :
     (φ.realize fun i => (x i : (u : Filter α).product M)) ↔
       ∀ᶠ a : α in u, φ.realize fun i => x i a :=
@@ -181,12 +160,6 @@ theorem realize_formula_cast {β : Type _} (φ : L.Formula β) (x : β → ∀ a
   exact congr rfl (Subsingleton.elim _ _)
 #align first_order.language.ultraproduct.realize_formula_cast FirstOrder.Language.Ultraproduct.realize_formula_cast
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- Łoś's Theorem : A sentence is true in an ultraproduct if and only if the set of structures it is

0b7c740e

bump to nightly-2023-05-28-03

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

6c6011cf

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson
 
 ! This file was ported from Lean 3 source module model_theory.ultraproducts
-! leanprover-community/mathlib commit f1ae620609496a37534c2ab3640b641d5be8b6f0
+! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Order.Filter.Ultrafilter
 
 /-! # Ultraproducts and Łoś's Theorem
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 ## Main Definitions
 * `first_order.language.ultraproduct.Structure` is the ultraproduct structure on `filter.product`.
 
@@ -115,10 +118,7 @@ theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.term β)
 variable [∀ a : α, Nonempty (M a)]
 
 /- warning: first_order.language.ultraproduct.bounded_formula_realize_cast -> FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align first_order.language.ultraproduct.bounded_formula_realize_cast FirstOrder.Language.Ultraproduct.boundedFormula_realize_castₓ'. -/
 theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormula β n)
     (x : β → ∀ a, M a) (v : Fin n → ∀ a, M a) :

f1ae6206

bump to nightly-2023-05-25-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

94ad6350

Diff

@@ -46,6 +46,7 @@ variable {L : Language.{u, v}} [∀ a, L.Structure (M a)]
 
 namespace Ultraproduct
 
+#print FirstOrder.Language.Ultraproduct.setoidPrestructure /-
 instance setoidPrestructure : L.Prestructure ((u : Filter α).productSetoid M) :=
   {
     (u : Filter α).productSetoid
@@ -72,18 +73,33 @@ instance setoidPrestructure : L.Prestructure ((u : Filter α).productSetoid M) :
         rw [funext ha2]
         exact ha1 }
 #align first_order.language.ultraproduct.setoid_prestructure FirstOrder.Language.Ultraproduct.setoidPrestructure
+-/
 
 variable {M} {u}
 
+#print FirstOrder.Language.Ultraproduct.structure /-
 instance structure : L.Structure ((u : Filter α).product M) :=
   Language.quotientStructure
 #align first_order.language.ultraproduct.Structure FirstOrder.Language.Ultraproduct.structure
+-/
 
+/- warning: first_order.language.ultraproduct.fun_map_cast -> FirstOrder.Language.Ultraproduct.funMap_cast is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align first_order.language.ultraproduct.fun_map_cast FirstOrder.Language.Ultraproduct.funMap_castₓ'. -/
 theorem funMap_cast {n : ℕ} (f : L.Functions n) (x : Fin n → ∀ a, M a) :
     (funMap f fun i => (x i : (u : Filter α).product M)) = fun a => funMap f fun i => x i a := by
   apply fun_map_quotient_mk
 #align first_order.language.ultraproduct.fun_map_cast FirstOrder.Language.Ultraproduct.funMap_cast
 
+/- warning: first_order.language.ultraproduct.term_realize_cast -> FirstOrder.Language.Ultraproduct.term_realize_cast is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u3}} {M : α -> Type.{u4}} {u : Ultrafilter.{u3} α} {L : FirstOrder.Language.{u1, u2}} [_inst_1 : forall (a : α), FirstOrder.Language.Structure.{u1, u2, u4} L (M a)] {β : Type.{u5}} (x : β -> (forall (a : α), M a)) (t : FirstOrder.Language.Term.{u1, u2, u5} L β), Eq.{succ (max u3 u4)} (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (FirstOrder.Language.Term.realize.{u1, u2, max u3 u4, u5} L (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (FirstOrder.Language.Ultraproduct.structure.{u1, u2, u3, u4} α M u L (fun (a : α) => _inst_1 a)) β (fun (i : β) => (fun (a : Sort.{max (succ u3) (succ u4)}) (b : Type.{max u3 u4}) [self : HasLiftT.{max (succ u3) (succ u4), succ (max u3 u4)} a b] => self.0) (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (HasLiftT.mk.{max (succ u3) (succ u4), succ (max u3 u4)} (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (CoeTCₓ.coe.{max (succ u3) (succ u4), succ (max u3 u4)} (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (Filter.Product.hasCoeT.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) (fun (a : α) => M a)))) (x i)) t) ((fun (a : Sort.{max (succ u3) (succ u4)}) (b : Type.{max u3 u4}) [self : HasLiftT.{max (succ u3) (succ u4), succ (max u3 u4)} a b] => self.0) (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (HasLiftT.mk.{max (succ u3) (succ u4), succ (max u3 u4)} (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (CoeTCₓ.coe.{max (succ u3) (succ u4), succ (max u3 u4)} (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (Filter.Product.hasCoeT.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) (fun (a : α) => M a)))) (fun (a : α) => FirstOrder.Language.Term.realize.{u1, u2, u4, u5} L (M a) (_inst_1 a) β (fun (i : β) => x i a) t))
+but is expected to have type
+  forall {α : Type.{u2}} {M : α -> Type.{u1}} {u : Ultrafilter.{u2} α} {L : FirstOrder.Language.{u4, u5}} [_inst_1 : forall (a : α), FirstOrder.Language.Structure.{u4, u5, u1} L (M a)] {β : Type.{u3}} (x : β -> (forall (a : α), M a)) (t : FirstOrder.Language.Term.{u4, u5, u3} L β), Eq.{max (succ u2) (succ u1)} (Quotient.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a))) (FirstOrder.Language.Term.realize.{u4, u5, max u2 u1, u3} L (Quotient.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a))) (FirstOrder.Language.quotientStructure.{u4, u5, max u2 u1} L (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (FirstOrder.Language.Ultraproduct.setoidPrestructure.{u4, u5, u2, u1} α (fun (a : α) => M a) u L (fun (a : α) => _inst_1 a))) β (fun (i : β) => Quotient.mk'.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (x i)) t) (Quotient.mk'.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (fun (a : α) => FirstOrder.Language.Term.realize.{u4, u5, u1, u3} L (M a) (_inst_1 a) β (fun (i : β) => x i a) t))
+Case conversion may be inaccurate. Consider using '#align first_order.language.ultraproduct.term_realize_cast FirstOrder.Language.Ultraproduct.term_realize_castₓ'. -/
 theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.term β) :
     (t.realize fun i => (x i : (u : Filter α).product M)) = fun a => t.realize fun i => x i a :=
   by
@@ -98,6 +114,12 @@ theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.term β)
 
 variable [∀ a : α, Nonempty (M a)]
 
+/- warning: first_order.language.ultraproduct.bounded_formula_realize_cast -> FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u3}} {M : α -> Type.{u4}} {u : Ultrafilter.{u3} α} {L : FirstOrder.Language.{u1, u2}} [_inst_1 : forall (a : α), FirstOrder.Language.Structure.{u1, u2, u4} L (M a)] [_inst_2 : forall (a : α), Nonempty.{succ u4} (M a)] {β : Type.{u5}} {n : Nat} (φ : FirstOrder.Language.BoundedFormula.{u1, u2, u5} L β n) (x : β -> (forall (a : α), M a)) (v : (Fin n) -> (forall (a : α), M a)), Iff (FirstOrder.Language.BoundedFormula.Realize.{u1, u2, max u3 u4, u5} L (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (FirstOrder.Language.Ultraproduct.structure.{u1, u2, u3, u4} α M u L (fun (a : α) => _inst_1 a)) β n φ (fun (i : β) => (fun (a : Sort.{max (succ u3) (succ u4)}) (b : Type.{max u3 u4}) [self : HasLiftT.{max (succ u3) (succ u4), succ (max u3 u4)} a b] => self.0) (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (HasLiftT.mk.{max (succ u3) (succ u4), succ (max u3 u4)} (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (CoeTCₓ.coe.{max (succ u3) (succ u4), succ (max u3 u4)} (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (Filter.Product.hasCoeT.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) (fun (a : α) => M a)))) (x i)) (fun (i : Fin n) => (fun (a : Sort.{max (succ u3) (succ u4)}) (b : Type.{max u3 u4}) [self : HasLiftT.{max (succ u3) (succ u4), succ (max u3 u4)} a b] => self.0) (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (HasLiftT.mk.{max (succ u3) (succ u4), succ (max u3 u4)} (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (CoeTCₓ.coe.{max (succ u3) (succ u4), succ (max u3 u4)} (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (Filter.Product.hasCoeT.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) (fun (a : α) => M a)))) (v i))) (Filter.Eventually.{u3} α (fun (a : α) => FirstOrder.Language.BoundedFormula.Realize.{u1, u2, u4, u5} L (M a) (_inst_1 a) β n φ (fun (i : β) => x i a) (fun (i : Fin n) => v i a)) ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u))
+but is expected to have type
+  forall {α : Type.{u2}} {M : α -> Type.{u1}} {u : Ultrafilter.{u2} α} {L : FirstOrder.Language.{u4, u5}} [_inst_1 : forall (a : α), FirstOrder.Language.Structure.{u4, u5, u1} L (M a)] [_inst_2 : forall (a : α), Nonempty.{succ u1} (M a)] {β : Type.{u3}} {n : Nat} (φ : FirstOrder.Language.BoundedFormula.{u4, u5, u3} L β n) (x : β -> (forall (a : α), M a)) (v : (Fin n) -> (forall (a : α), M a)), Iff (FirstOrder.Language.BoundedFormula.Realize.{u4, u5, max u2 u1, u3} L (Quotient.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a))) (FirstOrder.Language.quotientStructure.{u4, u5, max u2 u1} L (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (FirstOrder.Language.Ultraproduct.setoidPrestructure.{u4, u5, u2, u1} α (fun (a : α) => M a) u L (fun (a : α) => _inst_1 a))) β n φ (fun (i : β) => Quotient.mk'.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (x i)) (fun (i : Fin n) => Quotient.mk'.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (v i))) (Filter.Eventually.{u2} α (fun (a : α) => FirstOrder.Language.BoundedFormula.Realize.{u4, u5, u1, u3} L (M a) (_inst_1 a) β n φ (fun (i : β) => x i a) (fun (i : Fin n) => v i a)) (Ultrafilter.toFilter.{u2} α u))
+Case conversion may be inaccurate. Consider using '#align first_order.language.ultraproduct.bounded_formula_realize_cast FirstOrder.Language.Ultraproduct.boundedFormula_realize_castₓ'. -/
 theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormula β n)
     (x : β → ∀ a, M a) (v : Fin n → ∀ a, M a) :
     (φ.realize (fun i : β => (x i : (u : Filter α).product M)) fun i => v i) ↔
@@ -145,6 +167,12 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
       exact Filter.mem_of_superset h fun a ha => ha (m a)
 #align first_order.language.ultraproduct.bounded_formula_realize_cast FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast
 
+/- warning: first_order.language.ultraproduct.realize_formula_cast -> FirstOrder.Language.Ultraproduct.realize_formula_cast is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u3}} {M : α -> Type.{u4}} {u : Ultrafilter.{u3} α} {L : FirstOrder.Language.{u1, u2}} [_inst_1 : forall (a : α), FirstOrder.Language.Structure.{u1, u2, u4} L (M a)] [_inst_2 : forall (a : α), Nonempty.{succ u4} (M a)] {β : Type.{u5}} (φ : FirstOrder.Language.Formula.{u1, u2, u5} L β) (x : β -> (forall (a : α), M a)), Iff (FirstOrder.Language.Formula.Realize.{u1, u2, max u3 u4, u5} L (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (FirstOrder.Language.Ultraproduct.structure.{u1, u2, u3, u4} α M u L (fun (a : α) => _inst_1 a)) β φ (fun (i : β) => (fun (a : Sort.{max (succ u3) (succ u4)}) (b : Type.{max u3 u4}) [self : HasLiftT.{max (succ u3) (succ u4), succ (max u3 u4)} a b] => self.0) (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (HasLiftT.mk.{max (succ u3) (succ u4), succ (max u3 u4)} (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (CoeTCₓ.coe.{max (succ u3) (succ u4), succ (max u3 u4)} (forall (a : α), M a) (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (Filter.Product.hasCoeT.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) (fun (a : α) => M a)))) (x i))) (Filter.Eventually.{u3} α (fun (a : α) => FirstOrder.Language.Formula.Realize.{u1, u2, u4, u5} L (M a) (_inst_1 a) β φ (fun (i : β) => x i a)) ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u))
+but is expected to have type
+  forall {α : Type.{u2}} {M : α -> Type.{u1}} {u : Ultrafilter.{u2} α} {L : FirstOrder.Language.{u4, u5}} [_inst_1 : forall (a : α), FirstOrder.Language.Structure.{u4, u5, u1} L (M a)] [_inst_2 : forall (a : α), Nonempty.{succ u1} (M a)] {β : Type.{u3}} (φ : FirstOrder.Language.Formula.{u4, u5, u3} L β) (x : β -> (forall (a : α), M a)), Iff (FirstOrder.Language.Formula.Realize.{u4, u5, max u2 u1, u3} L (Quotient.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a))) (FirstOrder.Language.quotientStructure.{u4, u5, max u2 u1} L (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (FirstOrder.Language.Ultraproduct.setoidPrestructure.{u4, u5, u2, u1} α (fun (a : α) => M a) u L (fun (a : α) => _inst_1 a))) β φ (fun (i : β) => Quotient.mk'.{max (succ u2) (succ u1)} (forall (a : α), (fun (a : α) => M a) a) (Filter.productSetoid.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) (fun (a : α) => M a)) (x i))) (Filter.Eventually.{u2} α (fun (a : α) => FirstOrder.Language.Formula.Realize.{u4, u5, u1, u3} L (M a) (_inst_1 a) β φ (fun (i : β) => x i a)) (Ultrafilter.toFilter.{u2} α u))
+Case conversion may be inaccurate. Consider using '#align first_order.language.ultraproduct.realize_formula_cast FirstOrder.Language.Ultraproduct.realize_formula_castₓ'. -/
 theorem realize_formula_cast {β : Type _} (φ : L.Formula β) (x : β → ∀ a, M a) :
     (φ.realize fun i => (x i : (u : Filter α).product M)) ↔
       ∀ᶠ a : α in u, φ.realize fun i => x i a :=
@@ -153,6 +181,12 @@ theorem realize_formula_cast {β : Type _} (φ : L.Formula β) (x : β → ∀ a
   exact congr rfl (Subsingleton.elim _ _)
 #align first_order.language.ultraproduct.realize_formula_cast FirstOrder.Language.Ultraproduct.realize_formula_cast
 
+/- warning: first_order.language.ultraproduct.sentence_realize -> FirstOrder.Language.Ultraproduct.sentence_realize is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u3}} {M : α -> Type.{u4}} {u : Ultrafilter.{u3} α} {L : FirstOrder.Language.{u1, u2}} [_inst_1 : forall (a : α), FirstOrder.Language.Structure.{u1, u2, u4} L (M a)] [_inst_2 : forall (a : α), Nonempty.{succ u4} (M a)] (φ : FirstOrder.Language.Sentence.{u1, u2} L), Iff (FirstOrder.Language.Sentence.Realize.{u1, u2, max u3 u4} L (Filter.Product.{u3, u4} α ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u) M) (FirstOrder.Language.Ultraproduct.structure.{u1, u2, u3, u4} α M u L (fun (a : α) => _inst_1 a)) φ) (Filter.Eventually.{u3} α (fun (a : α) => FirstOrder.Language.Sentence.Realize.{u1, u2, u4} L (M a) (_inst_1 a) φ) ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Ultrafilter.{u3} α) (Filter.{u3} α) (HasLiftT.mk.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (CoeTCₓ.coe.{succ u3, succ u3} (Ultrafilter.{u3} α) (Filter.{u3} α) (Ultrafilter.Filter.hasCoeT.{u3} α))) u))
+but is expected to have type
+  forall {α : Type.{u2}} {M : α -> Type.{u1}} {u : Ultrafilter.{u2} α} {L : FirstOrder.Language.{u3, u4}} [_inst_1 : forall (a : α), FirstOrder.Language.Structure.{u3, u4, u1} L (M a)] [_inst_2 : forall (a : α), Nonempty.{succ u1} (M a)] (φ : FirstOrder.Language.Sentence.{u3, u4} L), Iff (FirstOrder.Language.Sentence.Realize.{u3, u4, max u2 u1} L (Filter.Product.{u2, u1} α (Ultrafilter.toFilter.{u2} α u) M) (FirstOrder.Language.Ultraproduct.structure.{u3, u4, u2, u1} α M u L (fun (a : α) => _inst_1 a)) φ) (Filter.Eventually.{u2} α (fun (a : α) => FirstOrder.Language.Sentence.Realize.{u3, u4, u1} L (M a) (_inst_1 a) φ) (Ultrafilter.toFilter.{u2} α u))
+Case conversion may be inaccurate. Consider using '#align first_order.language.ultraproduct.sentence_realize FirstOrder.Language.Ultraproduct.sentence_realizeₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- Łoś's Theorem : A sentence is true in an ultraproduct if and only if the set of structures it is

f1ae6206

bump to nightly-2023-05-14-08

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

d31da313

Diff

@@ -56,19 +56,19 @@ instance setoidPrestructure : L.Prestructure ((u : Filter α).productSetoid M) :
     fun_equiv := fun n f x y xy =>
       by
       refine' mem_of_superset (Inter_mem.2 xy) fun a ha => _
-      simp only [Set.mem_interᵢ, Set.mem_setOf_eq] at ha
+      simp only [Set.mem_iInter, Set.mem_setOf_eq] at ha
       simp only [Set.mem_setOf_eq, ha]
     rel_equiv := fun n r x y xy => by
       rw [← iff_eq_eq]
       refine' ⟨fun hx => _, fun hy => _⟩
       · refine' mem_of_superset (inter_mem hx (Inter_mem.2 xy)) _
         rintro a ⟨ha1, ha2⟩
-        simp only [Set.mem_interᵢ, Set.mem_setOf_eq] at *
+        simp only [Set.mem_iInter, Set.mem_setOf_eq] at *
         rw [← funext ha2]
         exact ha1
       · refine' mem_of_superset (inter_mem hy (Inter_mem.2 xy)) _
         rintro a ⟨ha1, ha2⟩
-        simp only [Set.mem_interᵢ, Set.mem_setOf_eq] at *
+        simp only [Set.mem_iInter, Set.mem_setOf_eq] at *
         rw [funext ha2]
         exact ha1 }
 #align first_order.language.ultraproduct.setoid_prestructure FirstOrder.Language.Ultraproduct.setoidPrestructure

f1ae6206

bump to nightly-2023-05-09-13

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

6ff25244

Diff

@@ -129,7 +129,7 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
       by
       refine' fun m a => funext (Fin.reverseInduction _ fun i hi => _)
       · simp only [Fin.snoc_last]
-      · simp only [Fin.snoc_cast_succ]
+      · simp only [Fin.snoc_castSucc]
     simp only [← Fin.comp_snoc, ih, h']
     refine' ⟨fun h => _, fun h m => _⟩
     · contrapose! h

f1ae6206

bump to nightly-2023-03-17-00

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

c85864d4

Diff

@@ -87,8 +87,7 @@ theorem funMap_cast {n : ℕ} (f : L.Functions n) (x : Fin n → ∀ a, M a) :
 theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.term β) :
     (t.realize fun i => (x i : (u : Filter α).product M)) = fun a => t.realize fun i => x i a :=
   by
-  convert
-    @term.realize_quotient_mk L _ ((u : Filter α).productSetoid M)
+  convert@term.realize_quotient_mk L _ ((u : Filter α).productSetoid M)
       (ultraproduct.setoid_prestructure M u) _ t x
   ext a
   induction t

f1ae6206

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

f1ae6206

chore: avoid automatically generated instance names (#12270)

47e1ef4b

Diff

@@ -162,7 +162,7 @@ theorem sentence_realize (φ : L.Sentence) :
 
 nonrec instance Product.instNonempty : Nonempty ((u : Filter α).Product M) :=
   letI : ∀ a, Inhabited (M a) := fun _ => Classical.inhabited_of_nonempty'
-  instNonempty
+  inferInstance
 #align first_order.language.ultraproduct.product.nonempty FirstOrder.Language.Ultraproduct.Product.instNonempty
 
 end Ultraproduct

f1ae6206

style: use cases x with | ... instead of cases x; case => ... (#9321)

This converts usages of the pattern

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

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

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

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

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

is replaced by:

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

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

e04a464d

Diff

@@ -86,12 +86,9 @@ theorem term_realize_cast {β : Type*} (x : β → ∀ a, M a) (t : L.Term β) :
   convert @Term.realize_quotient_mk' L _ ((u : Filter α).productSetoid M)
       (Ultraproduct.setoidPrestructure M u) _ t x using 2
   ext a
-  induction t
-  case var =>
-    rfl
-  case func _ _ _ t_ih =>
-    simp only [Term.realize, t_ih]
-    rfl
+  induction t with
+  | var => rfl
+  | func _ _ t_ih => simp only [Term.realize, t_ih]; rfl
 #align first_order.language.ultraproduct.term_realize_cast FirstOrder.Language.Ultraproduct.term_realize_cast
 
 variable [∀ a : α, Nonempty (M a)]

f1ae6206

doc: Mark named theorems including Compactness Theorem, Halting Problem, Tarski Vaught, Łoś (#8743)

b8e4d751

Diff

@@ -154,8 +154,8 @@ theorem realize_formula_cast {β : Type*} (φ : L.Formula β) (x : β → ∀ a,
   exact congr rfl (Subsingleton.elim _ _)
 #align first_order.language.ultraproduct.realize_formula_cast FirstOrder.Language.Ultraproduct.realize_formula_cast
 
-/-- Łoś's Theorem : A sentence is true in an ultraproduct if and only if the set of structures it is
-  true in is in the ultrafilter. -/
+/-- **Łoś's Theorem**: A sentence is true in an ultraproduct if and only if the set of structures
+it is true in is in the ultrafilter. -/
 theorem sentence_realize (φ : L.Sentence) :
     (u : Filter α).Product M ⊨ φ ↔ ∀ᶠ a : α in u, M a ⊨ φ := by
   simp_rw [Sentence.Realize]

f1ae6206

feat: add lemma Quotient.exists (#7220)

Also forall_quotient_iff {α : Type*} [r : Setoid α] ... -> Quotient.forall {α : Sort*} {s : Setoid α} ...

9ebe3593

Diff

@@ -123,7 +123,7 @@ theorem boundedFormula_realize_cast {β : Type*} {n : ℕ} (φ : L.BoundedFormul
       φ.Realize (fun i : β => (x i : (u : Filter α).Product M))
         (Fin.snoc (((↑) : (∀ a, M a) → (u : Filter α).Product M) ∘ v)
           (m : (u : Filter α).Product M)))
-    · exact forall_quotient_iff
+    · exact Quotient.forall
     have h' :
       ∀ (m : ∀ a, M a) (a : α),
         (fun i : Fin (k + 1) => (Fin.snoc v m : _ → ∀ a, M a) i a) =

f1ae6206

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

@@ -27,7 +27,7 @@ ultraproduct, Los's theorem
 
 universe u v
 
-variable {α : Type _} (M : α → Type _) (u : Ultrafilter α)
+variable {α : Type*} (M : α → Type*) (u : Ultrafilter α)
 
 open FirstOrder Filter
 
@@ -80,7 +80,7 @@ theorem funMap_cast {n : ℕ} (f : L.Functions n) (x : Fin n → ∀ a, M a) :
   apply funMap_quotient_mk'
 #align first_order.language.ultraproduct.fun_map_cast FirstOrder.Language.Ultraproduct.funMap_cast
 
-theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.Term β) :
+theorem term_realize_cast {β : Type*} (x : β → ∀ a, M a) (t : L.Term β) :
     (t.realize fun i => (x i : (u : Filter α).Product M)) =
       (fun a => t.realize fun i => x i a : (u : Filter α).Product M) := by
   convert @Term.realize_quotient_mk' L _ ((u : Filter α).productSetoid M)
@@ -96,7 +96,7 @@ theorem term_realize_cast {β : Type _} (x : β → ∀ a, M a) (t : L.Term β)
 
 variable [∀ a : α, Nonempty (M a)]
 
-theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormula β n)
+theorem boundedFormula_realize_cast {β : Type*} {n : ℕ} (φ : L.BoundedFormula β n)
     (x : β → ∀ a, M a) (v : Fin n → ∀ a, M a) :
     (φ.Realize (fun i : β => (x i : (u : Filter α).Product M))
         (fun i => (v i : (u : Filter α).Product M))) ↔
@@ -147,7 +147,7 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
       exact Filter.mem_of_superset h fun a ha => ha (m a)
 #align first_order.language.ultraproduct.bounded_formula_realize_cast FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast
 
-theorem realize_formula_cast {β : Type _} (φ : L.Formula β) (x : β → ∀ a, M a) :
+theorem realize_formula_cast {β : Type*} (φ : L.Formula β) (x : β → ∀ a, M a) :
     (φ.Realize fun i => (x i : (u : Filter α).Product M)) ↔
       ∀ᶠ a : α in u, φ.Realize fun i => x i a := by
   simp_rw [Formula.Realize, ← boundedFormula_realize_cast φ x, iff_eq_eq]

f1ae6206

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

depends on: #5966

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

89aa3a3b

Diff

@@ -2,16 +2,13 @@
 Copyright (c) 2022 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson
-
-! This file was ported from Lean 3 source module model_theory.ultraproducts
-! leanprover-community/mathlib commit f1ae620609496a37534c2ab3640b641d5be8b6f0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.ModelTheory.Quotients
 import Mathlib.Order.Filter.Germ
 import Mathlib.Order.Filter.Ultrafilter
 
+#align_import model_theory.ultraproducts from "leanprover-community/mathlib"@"f1ae620609496a37534c2ab3640b641d5be8b6f0"
+
 /-! # Ultraproducts and Łoś's Theorem
 
 ## Main Definitions

f1ae6206

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

@@ -133,7 +133,7 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
           Fin.snoc (fun i : Fin k => v i a) (m a) := by
       refine' fun m a => funext (Fin.reverseInduction _ fun i _ => _)
       · simp only [Fin.snoc_last]
-      · simp only [Fin.snoc_castSuccEmb]
+      · simp only [Fin.snoc_castSucc]
     simp only [← Fin.comp_snoc]
     simp only [Function.comp, ih, h']
     refine' ⟨fun h => _, fun h m => _⟩

f1ae6206

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

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

f4e42872

Diff

@@ -133,7 +133,7 @@ theorem boundedFormula_realize_cast {β : Type _} {n : ℕ} (φ : L.BoundedFormu
           Fin.snoc (fun i : Fin k => v i a) (m a) := by
       refine' fun m a => funext (Fin.reverseInduction _ fun i _ => _)
       · simp only [Fin.snoc_last]
-      · simp only [Fin.snoc_castSucc]
+      · simp only [Fin.snoc_castSuccEmb]
     simp only [← Fin.comp_snoc]
     simp only [Function.comp, ih, h']
     refine' ⟨fun h => _, fun h m => _⟩

f1ae6206

feat: port ModelTheory.Ultraproducts (#3976)

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

141c9c36

`model_theory.ultraproducts` ⟷ `Mathlib.ModelTheory.Ultraproducts`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 7 + 330

model_theory.ultraproducts ⟷ Mathlib.ModelTheory.Ultraproducts

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 7 + 330

`model_theory.ultraproducts` ⟷ `Mathlib.ModelTheory.Ultraproducts`