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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 | import OAI.Computability.DegreeRigidity.Effective.FiniteCodeBound
import Mathlib.SetTheory.ZFC.Rank
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel
universe u
namespace FiniteTerm
noncomputable def tag (n : ℕ) (x : ZFSet.{u}) : ZFSet.{u} := ZFSet.pair (natSet n) x
theorem tag_inj (n m : ℕ) (x y : ZFSet.{u}) : tag n x = tag m y ↔ n = m ∧ x = y := by
rw [tag,tag,ZFSet.pair_inj]
exact and_congr ⟨fun h => natSet_injective h,fun h => congrArg natSet h⟩ Iff.rfl
inductive Eval (A B L O D : ZFSet.{u}) : ZFSet.{u} → ZFSet.{u} → Prop
| leaf {a x : ZFSet.{u}} : a ∈ A → x ∈ D → ZFSet.pair a x ∈ L → Eval A B L O D (tag 0 a) x
| app {o c x y : ZFSet.{u}} : o ∈ B → Eval A B L O D c x → y ∈ D →
ZFSet.pair (ZFSet.pair o x) y ∈ O → Eval A B L O D (tag 1 (ZFSet.pair o c)) y
| pair {a b x y : ZFSet.{u}} : Eval A B L O D a x → Eval A B L O D b y →
ZFSet.pair x y ∈ D → Eval A B L O D (tag 2 (ZFSet.pair a b)) (ZFSet.pair x y)
theorem Eval.value_mem {A B L O D c x : ZFSet.{u}} (h : Eval A B L O D c x) : x ∈ D := by
cases h with
| leaf _ hx _ => exact hx
| app _ _ hx _ => exact hx
| pair _ _ hx => exact hx
theorem Eval.code_finite {A B L O D t c x : ZFSet.{u}}
(hA : A ⊆ t) (hB : B ⊆ t) (hn : ∀ n, natSet.{u} n ∈ t)
(h : Eval A B L O D c x) : FiniteTuple.FiniteCode t c := by
induction h with
| leaf ha _ _ => exact (FiniteTuple.FiniteCode.atom _ (hn 0)).orderedPair (.atom _ (hA ha))
| app ho _ _ _ ih =>
exact (FiniteTuple.FiniteCode.atom _ (hn 1)).orderedPair
((FiniteTuple.FiniteCode.atom _ (hB ho)).orderedPair ih)
| pair _ _ _ iha ihb => exact (FiniteTuple.FiniteCode.atom _ (hn 2)).orderedPair (iha.orderedPair ihb)
def Step (A B L O K D f c v : ZFSet.{u}) : Prop :=
(∃ a ∈ A, c = tag 0 a ∧ ZFSet.pair a v ∈ L) ∨
(∃ o ∈ B, ∃ a ∈ K, ∃ x ∈ D, c = tag 1 (ZFSet.pair o a) ∧
ZFSet.pair a x ∈ f ∧ ZFSet.pair (ZFSet.pair o x) v ∈ O) ∨
(∃ a ∈ K, ∃ b ∈ K, ∃ x ∈ D, ∃ y ∈ D, c = tag 2 (ZFSet.pair a b) ∧
ZFSet.pair a x ∈ f ∧ ZFSet.pair b y ∈ f ∧ v = ZFSet.pair x y)
def Certificate (A B L O K D f : ZFSet.{u}) : Prop :=
∀ z ∈ f, ∃ c ∈ K, ∃ v ∈ D, z = ZFSet.pair c v ∧ Step A B L O K D f c v
theorem Step.mono {A B L O K D f g c v : ZFSet.{u}} (hfg : f ⊆ g)
(h : Step A B L O K D f c v) : Step A B L O K D g c v := by
rcases h with h|h|h
· exact Or.inl h
· obtain ⟨o,ho,a,ha,x,hx,hc,hax,hox⟩ := h
exact Or.inr (Or.inl ⟨o,ho,a,ha,x,hx,hc,hfg hax,hox⟩)
· obtain ⟨a,ha,b,hb,x,hx,y,hy,hc,hax,hby,hv⟩ := h
exact Or.inr (Or.inr ⟨a,ha,b,hb,x,hx,y,hy,hc,hfg hax,hfg hby,hv⟩)
theorem Certificate.subset {A B L O K D f : ZFSet.{u}} (hf : Certificate A B L O K D f) :
f ⊆ ZFSet.prod K D := by
intro z hz
obtain ⟨c,hc,v,hv,hz,_⟩ := hf z hz
exact ZFSet.mem_prod.mpr ⟨c,hc,v,hv,hz⟩
theorem Certificate.step {A B L O K D f c v : ZFSet.{u}}
(hf : Certificate A B L O K D f) (hp : ZFSet.pair c v ∈ f) :
c ∈ K ∧ v ∈ D ∧ Step A B L O K D f c v := by
obtain ⟨c',hc,v',hv,he,hs⟩ := hf _ hp
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp he
exact ⟨hc,hv,hs⟩
theorem pair_rank_left (x y : ZFSet.{u}) : ZFSet.rank x < ZFSet.rank (ZFSet.pair x y) := by
have ha : x ∈ ({x,y} : ZFSet.{u}) := ZFSet.mem_pair.mpr (Or.inl rfl)
have hb : ({x,y} : ZFSet.{u}) ∈ ZFSet.pair x y := ZFSet.mem_pair.mpr (Or.inr rfl)
exact (ZFSet.rank_lt_of_mem ha).trans (ZFSet.rank_lt_of_mem hb)
theorem pair_rank_right (x y : ZFSet.{u}) : ZFSet.rank y < ZFSet.rank (ZFSet.pair x y) := by
have ha : y ∈ ({x,y} : ZFSet.{u}) := ZFSet.mem_pair.mpr (Or.inr rfl)
have hb : ({x,y} : ZFSet.{u}) ∈ ZFSet.pair x y := ZFSet.mem_pair.mpr (Or.inr rfl)
exact (ZFSet.rank_lt_of_mem ha).trans (ZFSet.rank_lt_of_mem hb)
theorem Certificate.sound {A B L O K D f : ZFSet.{u}}
(hf : Certificate A B L O K D f) (c v : ZFSet.{u})
(hp : ZFSet.pair c v ∈ f) : Eval A B L O D c v := by
have wf : WellFounded (fun a b : ZFSet.{u} => ZFSet.rank a < ZFSet.rank b) :=
InvImage.wf ZFSet.rank Ordinal.lt_wf
induction c using wf.induction generalizing v with
| h c ih =>
have hv := (hf.step hp).2.1
rcases (hf.step hp).2.2 with h|h|h
· obtain ⟨a,ha,rfl,hav⟩ := h
exact .leaf ha hv hav
· obtain ⟨o,ho,a,_,x,_,rfl,hax,hov⟩ := h
have hlt := (pair_rank_right o a).trans (pair_rank_right (natSet 1) (ZFSet.pair o a))
exact .app ho (ih a hlt x hax) hv hov
· obtain ⟨a,_,b,_,x,_,y,_,rfl,hax,hby,rfl⟩ := h
have ha := (pair_rank_left a b).trans (pair_rank_right (natSet 2) (ZFSet.pair a b))
have hb := (pair_rank_right a b).trans (pair_rank_right (natSet 2) (ZFSet.pair a b))
exact .pair (ih a ha x hax) (ih b hb y hby) hv
end FiniteTerm
end TuringRigidity.BoundedSetTheory
end OAI
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