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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 | import OAI.Computability.DegreeRigidity.Syntax.CheckedCode
import OAI.Computability.DegreeRigidity.SetModels.NameUnionCode
namespace OAI
namespace TuringRigidity.TransitiveNameModel
open RecursiveNames BoundedSetTheory
universe u
theorem second_mem_doubleUnion {a b f : ZFSet.{u}} (h : ZFSet.pair a b ∈ f) :
b ∈ iterUnion 2 f := by
apply ZFSet.mem_sUnion.mpr
refine ⟨({a,b} : ZFSet.{u}),?_,ZFSet.mem_pair.mpr (Or.inr rfl)⟩
exact ZFSet.mem_sUnion.mpr ⟨_,h,ZFSet.mem_pair.mpr (Or.inr rfl)⟩
def taggedImageFormula : Formula :=
.existsMem 1 (.existsMem 4 (.conj (.pairMem 1 0 4) (.orderedPair 2 0 6)))
theorem eval_taggedImageFormula (z x f r t : ZFSet.{u}) (e : ℕ → ZFSet.{u}) :
taggedImageFormula.Eval (cons z (cons x (cons f (cons r (cons t e))))) ↔
∃ y ∈ x, ∃ w ∈ r, ZFSet.pair y w ∈ f ∧ z = ZFSet.pair w t := by
simp only [taggedImageFormula,Formula.Eval,Formula.eval_pairMem,
Formula.eval_orderedPair,cons_zero,cons_succ]
noncomputable def taggedImage (x f t : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun z => ∃ y ∈ x, ∃ w ∈ iterUnion 2 f,
ZFSet.pair y w ∈ f ∧ z = ZFSet.pair w t)
(ZFSet.prod (iterUnion 2 f) ({t} : ZFSet.{u}))
theorem mem_taggedImage (x f t z : ZFSet.{u}) :
z ∈ taggedImage x f t ↔ ∃ y ∈ x, ∃ w, ZFSet.pair y w ∈ f ∧ z = ZFSet.pair w t := by
rw [taggedImage,ZFSet.mem_sep]
constructor
· rintro ⟨_,y,hy,w,_,hfw,hz⟩
exact ⟨y,hy,w,hfw,hz⟩
· rintro ⟨y,hy,w,hfw,rfl⟩
have hw := second_mem_doubleUnion hfw
exact ⟨ZFSet.mem_prod.mpr ⟨w,hw,t,ZFSet.mem_singleton.mpr rfl,rfl⟩,
y,hy,w,hw,hfw,rfl⟩
theorem taggedImage_mem (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M)
(hU : BoundedSetTheory.Union M) (hPow : PowerSet M) (hS : Separation M)
{x f t : ZFSet.{u}} (hx : x ∈ M) (hf : f ∈ M) (ht : t ∈ M) :
taggedImage x f t ∈ M := by
have hr := iterUnion_mem M hM hU hf 2
let e := cons x (cons f (cons (iterUnion 2 f) (cons t (fun _ => x))))
have he : ∀ i, e i ∈ M := by
intro i; cases i with
| zero => exact hx
| succ i => cases i with
| zero => exact hf
| succ i => cases i with
| zero => exact hr
| succ i => cases i <;> assumption
have hs := sep_mem M hM hS taggedImageFormula e he
(product_mem M hM hP hU hPow hS hr (singleton_mem M hM hP ht))
have heq : ZFSet.sep (fun z => taggedImageFormula.Eval (cons z e))
(ZFSet.prod (iterUnion 2 f) ({t} : ZFSet.{u})) = taggedImage x f t := by
apply ZFSet.ext
intro z
simp only [taggedImage,ZFSet.mem_sep]
exact and_congr_right (fun _ => eval_taggedImageFormula z x f _ t _)
exact heq ▸ hs
theorem CheckGraph.mem_iff {d r f t : ZFSet.{u}} (h : CheckGraph d r f t)
(z : ZFSet.{u}) :
z ∈ f ↔ ∃ x ∈ d, z = ZFSet.pair x (checkedCode t x) := by
constructor
· intro hz
obtain ⟨x,hx,y,hy,rfl⟩ := h.2.1 z hz
exact ⟨x,hx,by rw [h.correct x hx y hy hz]⟩
· rintro ⟨x,hx,rfl⟩
obtain ⟨y,hy,hfy,_⟩ := h.2.2.1 x hx
rw [←h.correct x hx y hy hfy]
exact hfy
theorem checkGraph_of_mem_iff (d f t : ZFSet.{u}) (hd : Transitive d)
(hf : ∀ z, z ∈ f ↔ ∃ x ∈ d, z = ZFSet.pair x (checkedCode t x)) :
CheckGraph d (iterUnion 2 f) f t := by
have hpair (x v : ZFSet.{u}) : ZFSet.pair x v ∈ f ↔ x ∈ d ∧ v = checkedCode t x := by
rw [hf]
constructor
· rintro ⟨y,hy,he⟩
obtain ⟨rfl,hv⟩ := ZFSet.pair_inj.mp he
exact ⟨hy,hv⟩
· rintro ⟨hx,rfl⟩
exact ⟨x,hx,rfl⟩
refine ⟨hd,?_,?_,?_⟩
· intro z hz
obtain ⟨x,hx,rfl⟩ := (hf z).mp hz
exact ⟨x,hx,checkedCode t x,second_mem_doubleUnion hz,rfl⟩
· intro x hx
have hp := (hpair x _).mpr ⟨hx,rfl⟩
exact ⟨_,second_mem_doubleUnion hp,hp,fun v _ hv => (hpair x v).mp hv |>.2⟩
· intro x hx v _ hv
have he := (hpair x v).mp hv |>.2
subst v
constructor
· intro z hz
obtain ⟨y,hy,rfl⟩ := (mem_checkedCode t x z).mp hz
have hp := (hpair y _).mpr ⟨hd x hx y hy,rfl⟩
exact ⟨y,hy,_,second_mem_doubleUnion hp,hp,rfl⟩
· intro y hy w _ hw
rw [(hpair y w).mp hw |>.2]
exact (mem_checkedCode t x _).mpr ⟨y,hy,rfl⟩
end TuringRigidity.TransitiveNameModel
end OAI
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