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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 | import OAI.Computability.DegreeRigidity.Syntax.BooleanExpressionCertificate
namespace OAI
namespace TuringRigidity.BooleanExpressionCertificate
open TransitiveNameModel BoundedSetTheory InternalBooleanSyntax
def stepFormula (c B Q S K g t U z0 z1 z2 : ℕ) : Formula :=
.disj (.conj (.member U S) (.orderedPair t z0 U))
(.disj
(.existsMem K (.existsMem (B+1)
(.conj (.orderedPair (t+2) (z1+2) 1)
(.conj (.pairMem 1 0 (g+2)) (negFormula (c+2) 0 (U+2))))))
(.existsMem K (.existsMem (Q+1)
(.conj (.orderedPair (t+2) (z2+2) 1)
(.conj (supFormula (c+2) 0 (U+2))
(.conj (.allMem 1 (.existsMem 1 (.pairMem 1 0 (g+4))))
(.allMem 0 (.existsMem 2 (.pairMem 0 1 (g+4))))))))))
theorem stepFormula_spec (c B Q S K g t U z0 z1 z2 : ℕ) (e : ℕ → ZFSet.{0})
(h0 : e z0 = natSet 0) (h1 : e z1 = natSet 1) (h2 : e z2 = natSet 2) :
(stepFormula c B Q S K g t U z0 z1 z2).Eval e ↔
Step (e c) (e B) (e Q) (e S) (e K) (e g) (e t) (e U) := by
simp only [stepFormula,Formula.eval_disj,Formula.Eval,Formula.eval_orderedPair,
Formula.eval_pairMem,Formula.eval_allMem,negFormula_spec,supFormula_spec,
cons_zero,cons_succ,h0,h1,h2,Step,FiniteTerm.tag]
def certificateFormula (c B Q S K g z0 z1 z2 : ℕ) : Formula :=
.allMem g (.existsMem (K+1) (.existsMem (B+2)
(.conj (.orderedPair 2 1 0)
(stepFormula (c+3) (B+3) (Q+3) (S+3) (K+3) (g+3) 1 0 (z0+3) (z1+3) (z2+3)))))
theorem certificateFormula_spec (c B Q S K g z0 z1 z2 : ℕ) (e : ℕ → ZFSet.{0})
(h0 : e z0 = natSet 0) (h1 : e z1 = natSet 1) (h2 : e z2 = natSet 2) :
(certificateFormula c B Q S K g z0 z1 z2).Eval e ↔
Certificate (e c) (e B) (e Q) (e S) (e K) (e g) := by
simp only [certificateFormula,Formula.eval_allMem,Formula.Eval,
Formula.eval_orderedPair,cons_zero,cons_succ,Certificate]
apply forall_congr'; intro z
apply imp_congr_right; intro _
apply exists_congr; intro t
apply and_congr_right; intro _
apply exists_congr; intro U
apply and_congr_right; intro _
apply and_congr_right; intro _
exact stepFormula_spec _ _ _ _ _ _ _ _ _ _ _ _ h0 h1 h2
def Witness (c B Q S K U : ZFSet.{0}) : Prop :=
Transitive K ∧ ∃ g ∈ K, Certificate c B Q S K g ∧
∃ t ∈ K, ZFSet.pair t U ∈ g
def witnessFormula (c B Q S K U z0 z1 z2 : ℕ) : Formula :=
.conj (.allMem K (.subset 0 (K+1)))
(.existsMem K (.conj
(certificateFormula (c+1) (B+1) (Q+1) (S+1) (K+1) 0 (z0+1) (z1+1) (z2+1))
(.existsMem (K+1) (.pairMem 0 (U+2) 1))))
theorem witnessFormula_spec (c B Q S K U z0 z1 z2 : ℕ) (e : ℕ → ZFSet.{0})
(h0 : e z0 = natSet 0) (h1 : e z1 = natSet 1) (h2 : e z2 = natSet 2) :
(witnessFormula c B Q S K U z0 z1 z2).Eval e ↔
Witness (e c) (e B) (e Q) (e S) (e K) (e U) := by
simp only [witnessFormula,Formula.Eval,Formula.eval_allMem,Formula.eval_subset,
Formula.eval_pairMem,cons_zero,cons_succ,Witness,TransitiveNameModel.Transitive]
apply and_congr Iff.rfl
apply exists_congr; intro g
apply and_congr_right; intro _
exact and_congr (certificateFormula_spec _ _ _ _ _ _ _ _ _ _ h0 h1 h2) Iff.rfl
def parameters (c B Q S : ZFSet.{0}) : ℕ → ZFSet.{0} :=
cons c (cons B (cons Q (cons S (cons (natSet 0) (cons (natSet 1) (fun _ => natSet 2))))))
theorem parameters_mem (M c B Q S : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) (hB : B ∈ M) (hQ : Q ∈ M) (hS : S ∈ M) :
∀ i, parameters c B Q S i ∈ M := by
intro i
rcases i with _|_|_|_|_|_|i
exact hc; exact hB; exact hQ; exact hS
all_goals exact hM _ (sourceT_omega_mem M hM hT) _ ((mem_omega _).mpr ⟨_,rfl⟩)
def witnessBody : Formula := witnessFormula 2 3 4 5 0 1 6 7 8
theorem witnessBody_realize (M c B Q S K U : ZFSet.{0}) (hM : Transitive M)
(he : ∀ i, parameters c B Q S i ∈ M) (hK : K ∈ M) (hU : U ∈ M) :
witnessBody.Realize M (cons K (cons U (parameters c B Q S))) ↔ Witness c B Q S K U := by
rw [Formula.absolute _ M hM _ (by
intro i; rcases i with _|_|i; exact hK; exact hU; exact he i)]
exact witnessFormula_spec _ _ _ _ _ _ _ _ _ _ rfl rfl rfl
noncomputable def certified (M c B Q S : ZFSet.{0}) : ZFSet.{0} :=
B.sep (fun U => ∃ K ∈ M, Witness c B Q S K U)
theorem certified_mem (M c B Q S : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) (hB : B ∈ M) (hQ : Q ∈ M) (hS : S ∈ M) : certified M c B Q S ∈ M := by
have he := parameters_mem M c B Q S hM hT hc hB hQ hS
obtain ⟨a,ha,hspec⟩ := hT.separation.finitePrefix
(.existsSet (.bounded witnessBody)) (parameters c B Q S) he B hB
have eq : a = certified M c B Q S := by
apply ZFSet.ext; intro U
rw [certified,ZFSet.mem_sep]
constructor
· intro hU
obtain ⟨hUB,K,hKM,hK⟩ := (hspec U (hM _ ha _ hU)).mp hU
exact ⟨hUB,K,hKM,(witnessBody_realize M c B Q S K U hM he hKM (hM _ hB _ hUB)).mp hK⟩
· rintro ⟨hUB,K,hKM,hK⟩
have hUM := hM _ hB _ hUB
exact (hspec U hUM).mpr ⟨hUB,K,hKM,(witnessBody_realize M c B Q S K U hM he hKM hUM).mpr hK⟩
exact eq ▸ ha
end TuringRigidity.BooleanExpressionCertificate
end OAI
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