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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.BooleanExpressionCoverage
import OAI.Computability.DegreeRigidity.Syntax.BoundedTruthRecursion
namespace OAI
namespace TuringRigidity.BooleanExpressionTruth
open TransitiveNameModel BoundedSetTheory
theorem rankWF : WellFounded (fun a b : ZFSet.{0} => ZFSet.rank a < ZFSet.rank b) :=
InvImage.wf ZFSet.rank Ordinal.lt_wf
def body (B R t : ZFSet.{0}) (rec : ∀ s : ZFSet.{0}, ZFSet.rank s < ZFSet.rank t → Prop) : Prop :=
(∃ U ∈ B, t = FiniteTerm.tag 0 U ∧ U ∈ R) ∨
(∃ s, ∃ h : t = FiniteTerm.tag 1 s,
¬ rec s (by rw [h]; exact FiniteTerm.pair_rank_right _ _)) ∨
(∃ T, ∃ h : t = FiniteTerm.tag 2 T, ∃ s, ∃ hs : s ∈ T,
rec s (by rw [h]; exact (ZFSet.rank_lt_of_mem hs).trans (FiniteTerm.pair_rank_right _ _)))
noncomputable def value (B R : ZFSet.{0}) : ZFSet.{0} → Prop := rankWF.fix (body B R)
theorem value_unfold (B R t : ZFSet.{0}) : value B R t ↔
(∃ U ∈ B, t = FiniteTerm.tag 0 U ∧ U ∈ R) ∨
(∃ s, t = FiniteTerm.tag 1 s ∧ ¬ value B R s) ∨
(∃ T, t = FiniteTerm.tag 2 T ∧ ∃ s ∈ T, value B R s) := by
change rankWF.fix (body B R) t ↔ _
rw [WellFounded.fix_eq]
simp only [body,value,exists_prop]
theorem payload_mem (D : ZFSet.{0}) (hD : Transitive D) (n : ℕ) (s : ZFSet.{0})
(ht : FiniteTerm.tag n s ∈ D) : s ∈ D := by
have hp := hD _ ht _ (ZFSet.mem_pair.mpr (Or.inr rfl))
exact hD _ hp _ (ZFSet.mem_pair.mpr (Or.inr rfl))
def step (K B R Y t : ZFSet.{0}) : Prop :=
(∃ U ∈ B, t = FiniteTerm.tag 0 U ∧ U ∈ R) ∨
(∃ s ∈ K, t = FiniteTerm.tag 1 s ∧ s ∉ Y) ∨
(∃ T ∈ K, t = FiniteTerm.tag 2 T ∧ ∃ s ∈ T, s ∈ Y)
def formula : Formula :=
.disj (.existsMem 3 (.conj (.orderedPair 1 6 0) (.member 0 5)))
(.disj (.existsMem 2 (.conj (.orderedPair 1 7 0) (.neg (.member 0 2))))
(.existsMem 2 (.conj (.orderedPair 1 8 0) (.existsMem 0 (.member 0 3)))))
def parameters (K B R : ZFSet.{0}) : ℕ → ZFSet.{0} :=
cons K (cons B (cons R (cons (natSet 0) (cons (natSet 1) (fun _ => natSet 2)))))
theorem formula_spec (K B R Y t : ZFSet.{0}) :
formula.Eval (cons t (cons Y (parameters K B R))) ↔ step K B R Y t := by
simp only [formula,Formula.eval_disj,Formula.Eval,Formula.eval_orderedPair,
parameters,cons_zero,cons_succ,step,FiniteTerm.tag]
theorem formula_local (K B R : ZFSet.{0}) : BoundedTruthRecursion.Local formula (parameters K B R) := by
intro D hD t ht Y Z ha
rw [formula_spec,formula_spec]
apply or_congr Iff.rfl
apply or_congr
· apply exists_congr; intro s
apply and_congr_right; intro _
apply and_congr_right; intro he
have hsD := payload_mem D hD 1 s (he ▸ ht)
have hlt : ZFSet.rank s < ZFSet.rank t := by rw [he]; exact FiniteTerm.pair_rank_right _ _
exact not_congr (ha s hsD hlt)
· apply exists_congr; intro T
apply and_congr_right; intro _
apply and_congr_right; intro he
have hTD := payload_mem D hD 2 T (he ▸ ht)
apply exists_congr; intro s
apply and_congr_right; intro hs
have hlt : ZFSet.rank s < ZFSet.rank t := by
rw [he]; exact (ZFSet.rank_lt_of_mem hs).trans (FiniteTerm.pair_rank_right _ _)
exact ha s (hD T hTD s hs) hlt
noncomputable def valueSet (K B R : ZFSet.{0}) : ZFSet.{0} := K.sep (value B R)
theorem valueSet_fixed (K B R : ZFSet.{0}) (hK : Transitive K) :
∀ t ∈ K, t ∈ valueSet K B R ↔ formula.Eval (cons t (cons (valueSet K B R) (parameters K B R))) := by
intro t ht
rw [valueSet,ZFSet.mem_sep,and_iff_right ht,value_unfold,formula_spec]
apply or_congr Iff.rfl
apply or_congr
· constructor
· rintro ⟨s,he,hs⟩
have hsK := payload_mem K hK 1 s (he ▸ ht)
exact ⟨s,hsK,he,fun h => hs (ZFSet.mem_sep.mp h).2⟩
· rintro ⟨s,hsK,he,hs⟩
exact ⟨s,he,fun hv => hs (ZFSet.mem_sep.mpr ⟨hsK,hv⟩)⟩
· constructor
· rintro ⟨T,he,s,hs,hv⟩
have hTK := payload_mem K hK 2 T (he ▸ ht)
exact ⟨T,hTK,he,s,hs,ZFSet.mem_sep.mpr ⟨hK T hTK s hs,hv⟩⟩
· rintro ⟨T,_,he,s,hs,hv⟩
exact ⟨T,he,s,hs,(ZFSet.mem_sep.mp hv).2⟩
theorem valueSet_mem (N K B R : ZFSet.{0}) (hN : Transitive N) (hT : SourceT N)
(hKN : K ∈ N) (hK : Transitive K) (hB : B ∈ N) (hR : R ∈ N) : valueSet K B R ∈ N := by
have he : ∀ i, parameters K B R i ∈ N := by
intro i; rcases i with _|_|_|_|_|i
exact hKN; exact hB; exact hR
all_goals exact hN _ (sourceT_omega_mem N hN hT) _ ((mem_omega _).mpr ⟨_,rfl⟩)
have hi := BoundedTruthRecursion.truth_internal N K (valueSet K B R) hN hT hKN hK
formula (parameters K B R) he (formula_local K B R) (valueSet_fixed K B R hK)
have eq : K.sep (fun t => t ∈ valueSet K B R) = valueSet K B R := by
apply ZFSet.ext; intro t
simp only [valueSet,ZFSet.mem_sep,and_self_left]
exact eq ▸ hi
end TuringRigidity.BooleanExpressionTruth
end OAI
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