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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import OAI.Computability.DegreeRigidity.Syntax.BoundedRelations
namespace OAI
namespace TuringRigidity.TransitiveNameModel
open RecursiveNames BoundedSetTheory
universe u
noncomputable def checkedCode (t x : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.range (fun y : Shrink x =>
ZFSet.pair (checkedCode t ((equivShrink x).symm y).val) t)
termination_by x
decreasing_by exact ((equivShrink x).symm y).property
theorem mem_checkedCode (t x z : ZFSet.{u}) :
z ∈ checkedCode t x ↔ ∃ y ∈ x, z = ZFSet.pair (checkedCode t y) t := by
rw [checkedCode, ZFSet.mem_range]
constructor
· rintro ⟨y,hy⟩
exact ⟨_,((equivShrink x).symm y).property,hy.symm⟩
· rintro ⟨y,hy,hz⟩
refine ⟨equivShrink x ⟨y,hy⟩,?_⟩
simpa using hz.symm
theorem encode_checkPSet {P : Type u} [Top P] (l : P → ZFSet.{u}) (x : PSet.{u}) :
(Name.checkPSet x : Name P).encode l = checkedCode (l ⊤) (ZFSet.mk x) := by
induction x with
| mk ι child ih =>
apply ZFSet.ext
intro z
change z ∈ ZFSet.range _ ↔ _
rw [ZFSet.mem_range,mem_checkedCode]
constructor
· rintro ⟨i,hi⟩
refine ⟨ZFSet.mk (child i),?_,by simpa only [ih] using hi.symm⟩
change ∃ j, PSet.Equiv (child i) (child j)
exact ⟨i,PSet.Equiv.refl _⟩
· rintro ⟨y,hy,hz⟩
induction y using Quotient.inductionOn with
| h y =>
change ∃ i, PSet.Equiv y (child i) at hy
obtain ⟨i,hi⟩ := hy
have he : ZFSet.mk y = ZFSet.mk (child i) := Quotient.sound hi
change z = ZFSet.pair (checkedCode (l ⊤) (ZFSet.mk y)) (l ⊤) at hz
rw [he] at hz
exact ⟨i,by simpa only [ih] using hz.symm⟩
theorem encode_check {P : Type u} [Top P] (l : P → ZFSet.{u}) (x : ZFSet.{u}) :
(Name.check x : Name P).encode l = checkedCode (l ⊤) x := by
rw [Name.check,encode_checkPSet,ZFSet.mk_out]
def RecursionStep (f r t x v : ZFSet.{u}) : Prop :=
(∀ z ∈ v, ∃ y ∈ x, ∃ w ∈ r, ZFSet.pair y w ∈ f ∧ z = ZFSet.pair w t) ∧
(∀ y ∈ x, ∀ w ∈ r, ZFSet.pair y w ∈ f → ZFSet.pair w t ∈ v)
def CheckGraph (d r f t : ZFSet.{u}) : Prop :=
Transitive d ∧
(∀ z ∈ f, ∃ x ∈ d, ∃ y ∈ r, z = ZFSet.pair x y) ∧
(∀ x ∈ d, ∃ y ∈ r, ZFSet.pair x y ∈ f ∧
∀ z ∈ r, ZFSet.pair x z ∈ f → z = y) ∧
(∀ x ∈ d, ∀ y ∈ r, ZFSet.pair x y ∈ f → RecursionStep f r t x y)
theorem CheckGraph.correct {d r f t : ZFSet.{u}} (h : CheckGraph d r f t)
(x : ZFSet.{u}) (hx : x ∈ d) (v : ZFSet.{u}) (hv : v ∈ r)
(hfv : ZFSet.pair x v ∈ f) : v = checkedCode t x := by
induction x using ZFSet.inductionOn generalizing v with
| h x ih =>
have hs := h.2.2.2 x hx v hv hfv
apply ZFSet.ext
intro z
rw [mem_checkedCode]
constructor
· intro hz
obtain ⟨y,hy,w,hw,hfw,rfl⟩ := hs.1 z hz
exact ⟨y,hy,by rw [ih y hy (h.1 x hx y hy) w hw hfw]⟩
· rintro ⟨y,hy,rfl⟩
obtain ⟨w,hw,hfw,_⟩ := h.2.2.1 y (h.1 x hx y hy)
rw [←ih y hy (h.1 x hx y hy) w hw hfw]
exact hs.2 y hy w hw hfw
theorem CheckGraph.unique {d r s f g t : ZFSet.{u}}
(hf : CheckGraph d r f t) (hg : CheckGraph d s g t) : f = g := by
apply ZFSet.ext
intro z
have forward {r s f g : ZFSet.{u}} (hf : CheckGraph d r f t)
(hg : CheckGraph d s g t) (hz : z ∈ f) : z ∈ g := by
obtain ⟨x,hx,y,hy,rfl⟩ := hf.2.1 z hz
obtain ⟨v,hv,hgv,_⟩ := hg.2.2.1 x hx
have he := (hf.correct x hx y hy hz).trans (hg.correct x hx v hv hgv).symm
exact he ▸ hgv
exact ⟨forward hf hg,forward hg hf⟩
namespace CheckFormula
open BoundedSetTheory.Formula
def step (f r t x v : ℕ) : Formula :=
.conj (allMem v (.existsMem (x+1) (.existsMem (r+2)
(.conj (pairMem 1 0 (f+3)) (orderedPair 2 0 (t+3))))))
(allMem x (allMem (r+1) (imp (pairMem 1 0 (f+2))
(.existsMem (v+2) (orderedPair 0 1 (t+3))))))
theorem eval_step (f r t x v : ℕ) (e : ℕ → ZFSet.{u}) :
(step f r t x v).Eval e ↔ RecursionStep (e f) (e r) (e t) (e x) (e v) := by
simp only [step, Formula.Eval, eval_allMem, eval_pairMem, eval_orderedPair,
eval_imp, cons_zero, cons_succ, RecursionStep]
simp only [exists_eq_right]
def graph (d r f t : ℕ) : Formula :=
.conj (transitive d) (.conj (functionGraph f d r)
(allMem d (allMem (r+1) (imp (pairMem 1 0 (f+2)) (step (f+2) (r+2) (t+2) 1 0)))))
theorem eval_graph (d r f t : ℕ) (e : ℕ → ZFSet.{u}) :
(graph d r f t).Eval e ↔ CheckGraph (e d) (e r) (e f) (e t) := by
simp only [graph, Formula.Eval, eval_transitive, eval_functionGraph,
eval_allMem, eval_imp, eval_pairMem, eval_step, cons_zero, cons_succ, CheckGraph]
exact and_congr_right (fun _ => and_assoc)
end CheckFormula
end TuringRigidity.TransitiveNameModel
end OAI
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