File size: 6,055 Bytes
9dbc64e | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | import OAI.Computability.DegreeRigidity.SetModels.ElementaryAxioms
namespace OAI
namespace TuringRigidity.ElementaryModel
open BoundedSetTheory TransitiveNameModel SentenceForm
universe u
private def sepMap : ℕ → ℕ | 0 => 0 | i+1 => i+3
private def relationMap : ℕ → ℕ | 0 => 0 | 1 => 1 | i+2 => i+3
private def uniqueMap : ℕ → ℕ | 0 => 0 | 1 => 2 | i+2 => i+4
private def imageMap : ℕ → ℕ | 0 => 1 | 1 => 0 | i+2 => i+4
private theorem sat_sepMap (q : SentenceForm) (D : Set ZFSet.{u})
(x b a : ZFSet.{u}) (e : ℕ → ZFSet.{u}) :
(q.rename sepMap).Sat D (cons x (cons b (cons a e))) ↔ q.Sat D (cons x e) := by
rw [sat_rename]
have he : (fun i => cons x (cons b (cons a e)) (sepMap i)) = cons x e := by
funext i; cases i <;> rfl
rw [he]
private theorem sat_relationMap (q : SentenceForm) (D : Set ZFSet.{u})
(y x a : ZFSet.{u}) (e : ℕ → ZFSet.{u}) :
(q.rename relationMap).Sat D (cons y (cons x (cons a e))) ↔ q.Sat D (cons y (cons x e)) := by
rw [sat_rename]
have he : (fun i => cons y (cons x (cons a e)) (relationMap i)) = cons y (cons x e) := by
funext i; rcases i with _|_|i <;> rfl
rw [he]
private theorem sat_uniqueMap (q : SentenceForm) (D : Set ZFSet.{u})
(z y x a : ZFSet.{u}) (e : ℕ → ZFSet.{u}) :
(q.rename uniqueMap).Sat D (cons z (cons y (cons x (cons a e)))) ↔ q.Sat D (cons z (cons x e)) := by
rw [sat_rename]
have he : (fun i => cons z (cons y (cons x (cons a e))) (uniqueMap i)) = cons z (cons x e) := by
funext i; rcases i with _|_|i <;> rfl
rw [he]
private theorem sat_imageMap (q : SentenceForm) (D : Set ZFSet.{u})
(x y b a : ZFSet.{u}) (e : ℕ → ZFSet.{u}) :
(q.rename imageMap).Sat D (cons x (cons y (cons b (cons a e)))) ↔ q.Sat D (cons y (cons x e)) := by
rw [sat_rename]
have he : (fun i => cons x (cons y (cons b (cons a e))) (imageMap i)) = cons y (cons x e) := by
funext i; rcases i with _|_|i <;> rfl
rw [he]
theorem Valid.separate {M : ZFSet.{u}} (hv : Valid M) (q : SentenceForm)
(e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) (a : ZFSet.{u}) (ha : a ∈ M) :
∃ b ∈ M, ∀ x ∈ M, x ∈ b ↔ x ∈ a ∧ q.Sat (M : Set ZFSet) (cons x e) := by
let p := ex (all (iff (.member 0 1) (.conj (.member 0 2) (q.rename sepMap))))
have hp : ∀ v : ℕ → ZFSet.{u}, p.Sat Set.univ v := by
intro v
let e' := fun i => v (i+1)
have hv' : v = cons (v 0) e' := by funext i; cases i <;> rfl
rw [hv']
simp only [p,Sat,sat_all,sat_iff,sat_sepMap,cons_zero,cons_succ,Set.mem_univ,true_and,forall_const]
exact ⟨ZFSet.sep (fun x => q.Sat Set.univ (cons x e')) (v 0),fun x => ZFSet.mem_sep⟩
have h := hv p hp (cons a e) (by intro i; cases i <;> simp [cons,he,ha])
simpa only [p,Sat,sat_all,sat_iff,sat_sepMap,cons_zero,cons_succ,SetLike.mem_coe] using h
theorem ambient_image (a : ZFSet.{u}) (R : ZFSet.{u} → ZFSet.{u} → Prop)
(ht : ∀ x ∈ a, ∃ y, R x y ∧ ∀ z, R x z → z = y) :
∃ b : ZFSet.{u}, ∀ y, y ∈ b ↔ ∃ x ∈ a, R x y := by
classical
let f (i : Conditions a) := Classical.choose (ht (label a i) (label_mem a i))
have hf (i : Conditions a) : R (label a i) (f i) ∧ ∀ z, R (label a i) z → z = f i :=
Classical.choose_spec (ht _ (label_mem a i))
refine ⟨ZFSet.range f,?_⟩
intro y; rw [ZFSet.mem_range]
constructor
· rintro ⟨i,rfl⟩; exact ⟨_,label_mem a i,(hf i).1⟩
· rintro ⟨x,hx,hr⟩
obtain ⟨i,rfl⟩ := label_surjective a hx
exact ⟨i,((hf i).2 y hr).symm⟩
theorem Valid.replace {M : ZFSet.{u}} (hv : Valid M) (hM : Transitive M) (q : SentenceForm)
(e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) (a : ZFSet.{u}) (ha : a ∈ M)
(ht : ∀ x ∈ a, ∃ y ∈ M, q.Sat (M : Set ZFSet) (cons y (cons x e)) ∧
∀ z ∈ M, q.Sat (M : Set ZFSet) (cons z (cons x e)) → z = y) :
∃ b ∈ M, ∀ y ∈ M, y ∈ b ↔ ∃ x ∈ a, q.Sat (M : Set ZFSet) (cons y (cons x e)) := by
let p := imp
(all (imp (.member 0 1) (ex (.conj (q.rename relationMap)
(all (imp (q.rename uniqueMap) (.equal 0 1)))))))
(ex (all (iff (.member 0 1) (ex (.conj (.member 0 3) (q.rename imageMap))))))
have hp : ∀ v : ℕ → ZFSet.{u}, p.Sat Set.univ v := by
intro v
let e' := fun i => v (i+1)
have hv' : v = cons (v 0) e' := by funext i; cases i <;> rfl
rw [hv']
simp only [p,Sat,sat_all,sat_iff,sat_imp,sat_relationMap,sat_uniqueMap,sat_imageMap,
cons_zero,cons_succ,Set.mem_univ,true_and,forall_const]
exact ambient_image (v 0) (fun x y => q.Sat Set.univ (cons y (cons x e')))
have h := hv p hp (cons a e) (by intro i; cases i <;> simp [cons,he,ha])
simp only [p,Sat,sat_all,sat_iff,sat_imp,sat_relationMap,sat_uniqueMap,sat_imageMap,
cons_zero,cons_succ] at h
obtain ⟨b,hb,hbdef⟩ := h (fun x _ hx => ht x hx)
refine ⟨b,hb,?_⟩
intro y hy
rw [hbdef y hy]
exact ⟨fun ⟨x,_,hx,hr⟩ => ⟨x,hx,hr⟩,fun ⟨x,hx,hr⟩ => ⟨x,hM a ha x hx,hx,hr⟩⟩
theorem Valid.sigmaSeparation {M : ZFSet.{u}} (hv : Valid M) : SigmaSeparation M := by
intro p e he a ha
obtain ⟨b,hb,h⟩ := hv.separate (fromSigma p) e he a ha
exact ⟨b,hb,fun x hx => (h x hx).trans (and_congr_right (fun _ => sigma_sat p M _))⟩
theorem Valid.sigmaReplacement {M : ZFSet.{u}} (hv : Valid M) (hM : Transitive M) : SigmaReplacement M := by
intro p e he a ha ht
obtain ⟨b,hb,h⟩ := hv.replace hM (fromSigma p) e he a ha (by
intro x hx
obtain ⟨y,hy,hp,hu⟩ := ht x hx
exact ⟨y,hy,(sigma_sat p M _).mpr hp,fun z hz hpz => hu z hz ((sigma_sat p M _).mp hpz)⟩)
refine ⟨b,hb,?_⟩
intro y hy
rw [h y hy]
apply exists_congr; intro x
exact and_congr_right (fun _ => sigma_sat p M _)
theorem Valid.sourceT {M : ZFSet.{u}} (hv : Valid M) (hM : Transitive M)
(hne : ∃ a, a ∈ M) : SourceT M :=
sourceT_of_finite_schemas M hM hv.pairing hv.union hv.powerSet hv.sigmaSeparation
(hv.sigmaReplacement hM) (hv.infinity hne) (hv.choice hM)
end TuringRigidity.ElementaryModel
end OAI
|