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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 | import OAI.Computability.DegreeRigidity.Syntax.BooleanTruthReading
namespace OAI
namespace TuringRigidity.BooleanExpressionTruth
open TransitiveNameModel BoundedSetTheory BooleanExpressionCertificate
def GoodTruth (K B R Y : ZFSet.{0}) : Prop :=
Y ⊆ K ∧ ∀ t ∈ K, t ∈ Y ↔ step K B R Y t
theorem GoodTruth.agrees {K B R Y : ZFSet.{0}} (hK : Transitive K) (hy : GoodTruth K B R Y) :
∀ t ∈ K, t ∈ Y ↔ value B R t := by
have hp : BoundedTruthRecursion.Partial formula (parameters K B R) K K Y :=
⟨hK,fun _ h => h,hy.1,fun t ht => (hy.2 t ht).trans (formula_spec K B R Y t).symm⟩
intro t ht
exact (BoundedTruthRecursion.partial_sound formula (parameters K B R) K K Y (valueSet K B R)
(formula_local K B R) hp (valueSet_fixed K B R hK) t ht).trans
(ZFSet.mem_sep.trans (and_iff_right ht))
theorem valueSet_good (K B R : ZFSet.{0}) (hK : Transitive K) : GoodTruth K B R (valueSet K B R) :=
⟨fun _ h => (ZFSet.mem_sep.mp h).1,
fun t ht => (valueSet_fixed K B R hK t ht).trans (formula_spec K B R _ t)⟩
def Evaluated (c B Q S R K Y U : ZFSet.{0}) : Prop :=
Transitive K ∧ GoodTruth K B R Y ∧ ∃ g ∈ K, Certificate c B Q S K g ∧
∃ t ∈ K, ZFSet.pair t U ∈ g ∧ t ∈ Y
def truthSlots (t Y K B R z0 z1 z2 : ℕ) : ℕ → ℕ
| 0 => t
| 1 => Y
| 2 => K
| 3 => B
| 4 => R
| 5 => z0
| 6 => z1
| _ => z2
def truthAt (t Y K B R z0 z1 z2 : ℕ) : Formula := formula.rename (truthSlots t Y K B R z0 z1 z2)
theorem truthAt_spec (t Y K B R z0 z1 z2 : ℕ) (e : ℕ → ZFSet.{0})
(h0 : e z0 = natSet 0) (h1 : e z1 = natSet 1) (h2 : e z2 = natSet 2) :
(truthAt t Y K B R z0 z1 z2).Eval e ↔ step (e K) (e B) (e R) (e Y) (e t) := by
have eq : e ∘ truthSlots t Y K B R z0 z1 z2 = cons (e t) (cons (e Y) (parameters (e K) (e B) (e R))) := by
funext i
rcases i with _|_|_|_|_|_|_|i
rfl; rfl; rfl; rfl; rfl; exact h0; exact h1; exact h2
rw [truthAt,Formula.eval_rename_comp,eq,formula_spec]
def goodTruthFormula (K B R Y z0 z1 z2 : ℕ) : Formula :=
.conj (.subset Y K) (.allMem K (.iff (.member 0 (Y+1))
(truthAt 0 (Y+1) (K+1) (B+1) (R+1) (z0+1) (z1+1) (z2+1))))
theorem goodTruthFormula_spec (K B R Y z0 z1 z2 : ℕ) (e : ℕ → ZFSet.{0})
(h0 : e z0 = natSet 0) (h1 : e z1 = natSet 1) (h2 : e z2 = natSet 2) :
(goodTruthFormula K B R Y z0 z1 z2).Eval e ↔ GoodTruth (e K) (e B) (e R) (e Y) := by
simp only [goodTruthFormula,Formula.Eval,Formula.eval_subset,Formula.eval_allMem,Formula.eval_iff,
cons_zero,cons_succ,GoodTruth]
apply and_congr Iff.rfl
apply forall_congr'; intro t
apply imp_congr_right; intro _
exact iff_congr Iff.rfl (truthAt_spec _ _ _ _ _ _ _ _ _ h0 h1 h2)
def evaluatedFormula (c B Q S R K Y U z0 z1 z2 : ℕ) : Formula :=
.conj (.transitive K) (.conj (goodTruthFormula K B R Y z0 z1 z2)
(.existsMem K (.conj (certificateFormula (c+1) (B+1) (Q+1) (S+1) (K+1) 0 (z0+1) (z1+1) (z2+1))
(.existsMem (K+1) (.conj (.pairMem 0 (U+2) 1) (.member 0 (Y+2)))))))
theorem evaluatedFormula_spec (c B Q S R K Y U z0 z1 z2 : ℕ) (e : ℕ → ZFSet.{0})
(h0 : e z0 = natSet 0) (h1 : e z1 = natSet 1) (h2 : e z2 = natSet 2) :
(evaluatedFormula c B Q S R K Y U z0 z1 z2).Eval e ↔
Evaluated (e c) (e B) (e Q) (e S) (e R) (e K) (e Y) (e U) := by
simp only [evaluatedFormula,Formula.Eval,Formula.eval_transitive,Formula.eval_pairMem,
cons_zero,cons_succ,Evaluated]
apply and_congr Iff.rfl
apply and_congr (goodTruthFormula_spec _ _ _ _ _ _ _ _ h0 h1 h2)
apply exists_congr; intro g
apply and_congr_right; intro _
exact and_congr (certificateFormula_spec _ _ _ _ _ _ _ _ _ _ h0 h1 h2) Iff.rfl
def evaluationParameters (W c B Q S R : ZFSet.{0}) : ℕ → ZFSet.{0} :=
cons W (cons c (cons B (cons Q (cons S (cons R
(cons (natSet 0) (cons (natSet 1) (fun _ => natSet 2))))))))
def evaluationBody : Formula := .existsMem 2 (evaluatedFormula 4 5 6 7 8 0 1 2 9 10 11)
theorem evaluationBody_realize (N W c B Q S R Y U : ZFSet.{0}) (hN : Transitive N)
(he : ∀ i, evaluationParameters W c B Q S R i ∈ N) (hY : Y ∈ N) (hU : U ∈ N) :
evaluationBody.Realize N (cons Y (cons U (evaluationParameters W c B Q S R))) ↔
∃ K ∈ W, Evaluated c B Q S R K Y U := by
rw [Formula.absolute _ N hN _ (by
intro i; rcases i with _|_|i; exact hY; exact hU; exact he i)]
change (∃ K ∈ W, (evaluatedFormula 4 5 6 7 8 0 1 2 9 10 11).Eval
(cons K (cons Y (cons U (evaluationParameters W c B Q S R))))) ↔ _
apply exists_congr; intro K
apply and_congr_right; intro _
exact evaluatedFormula_spec _ _ _ _ _ _ _ _ _ _ _ _ rfl rfl rfl
end TuringRigidity.BooleanExpressionTruth
end OAI
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