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e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | import OAI.SetTheory.PartitionPrinciple.Models.GroundCountableQuotients
namespace OAI
namespace PartitionPilot.Forcing
universe u
noncomputable def equivalenceClass (A E y : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun x => ZFSet.pair x y ∈ E) A
theorem mem_equivalenceClass (A E y x : ZFSet.{u}) :
x ∈ equivalenceClass A E y ↔ x ∈ A ∧ ZFSet.pair x y ∈ E :=
ZFSet.mem_sep
theorem equivalenceClass_subset (A E y : ZFSet.{u}) : equivalenceClass A E y ⊆ A :=
fun _ h => (mem_equivalenceClass A E y _).1 h |>.1
noncomputable def equivalenceClasses (A E : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun C => ∃ y ∈ A, C = equivalenceClass A E y) (ZFSet.powerset A)
theorem mem_equivalenceClasses (A E C : ZFSet.{u}) :
C ∈ equivalenceClasses A E ↔ ∃ y ∈ A, C = equivalenceClass A E y := by
rw [equivalenceClasses, ZFSet.mem_sep, ZFSet.mem_powerset]
exact ⟨And.right, fun ⟨y, hy, equal⟩ =>
⟨equal ▸ equivalenceClass_subset A E y, y, hy, equal⟩⟩
theorem equivalenceClass_mem {A E y : ZFSet.{u}} (hy : y ∈ A) :
equivalenceClass A E y ∈ equivalenceClasses A E :=
(mem_equivalenceClasses A E _).2 ⟨y, hy, rfl⟩
noncomputable def equivalenceClassifier (A E : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun z => ∃ y ∈ A, z = ZFSet.pair y (equivalenceClass A E y))
(ZFSet.prod A (equivalenceClasses A E))
theorem mem_equivalenceClassifier (A E z : ZFSet.{u}) :
z ∈ equivalenceClassifier A E ↔
∃ y ∈ A, z = ZFSet.pair y (equivalenceClass A E y) := by
rw [equivalenceClassifier, ZFSet.mem_sep]
refine ⟨And.right, ?_⟩
rintro ⟨y, hy, rfl⟩
exact ⟨ZFSet.pair_mem_prod.2 ⟨hy, equivalenceClass_mem hy⟩, y, hy, rfl⟩
theorem pair_mem_equivalenceClassifier (A E y C : ZFSet.{u}) :
ZFSet.pair y C ∈ equivalenceClassifier A E ↔
y ∈ A ∧ C = equivalenceClass A E y := by
rw [mem_equivalenceClassifier]
constructor
· rintro ⟨x, hx, equal⟩
obtain ⟨rfl, equal⟩ := ZFSet.pair_inj.1 equal
exact ⟨hx, equal⟩
· rintro ⟨hy, rfl⟩
exact ⟨y, hy, rfl⟩
theorem equivalenceClassifier_function (A E : ZFSet.{u}) :
ZFSet.IsFunc A (equivalenceClasses A E) (equivalenceClassifier A E) := by
refine ⟨fun _ h => (ZFSet.mem_sep.1 h).1, ?_⟩
intro y hy
exact ⟨equivalenceClass A E y,
(pair_mem_equivalenceClassifier A E y _).2 ⟨hy, rfl⟩,
fun C hC => ((pair_mem_equivalenceClassifier A E y C).1 hC).2⟩
theorem equivalenceClassifier_onto {A E C : ZFSet.{u}}
(hC : C ∈ equivalenceClasses A E) :
∃ y ∈ A, ZFSet.pair y C ∈ equivalenceClassifier A E := by
obtain ⟨y, hy, rfl⟩ := (mem_equivalenceClasses A E C).1 hC
exact ⟨y, hy, (pair_mem_equivalenceClassifier A E y _).2 ⟨hy, rfl⟩⟩
namespace EquivalenceRelationCode
theorem class_self {A E : ZFSet.{u}} (equivalence : EquivalenceRelationCode A E)
{x : ZFSet.{u}} (hx : x ∈ A) : x ∈ equivalenceClass A E x :=
(mem_equivalenceClass A E x x).2 ⟨hx, equivalence.refl x hx⟩
theorem class_eq_iff {A E : ZFSet.{u}} (equivalence : EquivalenceRelationCode A E)
{x y : ZFSet.{u}} (hx : x ∈ A) (_hy : y ∈ A) :
equivalenceClass A E x = equivalenceClass A E y ↔ ZFSet.pair x y ∈ E := by
constructor
· intro equal
exact ((mem_equivalenceClass A E y x).1 (equal ▸ equivalence.class_self hx)).2
· intro related
apply ZFSet.ext
intro z
rw [mem_equivalenceClass, mem_equivalenceClass]
exact and_congr_right (fun _ => ⟨
fun h => equivalence.trans z x y h related,
fun h => equivalence.trans z y x h (equivalence.symm x y related)⟩)
theorem classifier_kernel {A E : ZFSet.{u}} (equivalence : EquivalenceRelationCode A E)
{x y C D : ZFSet.{u}}
(first : ZFSet.pair x C ∈ equivalenceClassifier A E)
(second : ZFSet.pair y D ∈ equivalenceClassifier A E) :
C = D ↔ ZFSet.pair x y ∈ E := by
obtain ⟨hx, rfl⟩ := (pair_mem_equivalenceClassifier A E x C).1 first
obtain ⟨hy, rfl⟩ := (pair_mem_equivalenceClassifier A E y D).1 second
exact equivalence.class_eq_iff hx hy
theorem class_nonempty {A E C : ZFSet.{u}} (equivalence : EquivalenceRelationCode A E)
(hC : C ∈ equivalenceClasses A E) : ∃ x, x ∈ C := by
obtain ⟨x, hx, rfl⟩ := (mem_equivalenceClasses A E C).1 hC
exact ⟨x, equivalence.class_self hx⟩
end EquivalenceRelationCode
end PartitionPilot.Forcing
end OAI
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