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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 | import OAI.Computability.DegreeRigidity.Syntax.BooleanExpressionTruth
namespace OAI
namespace TuringRigidity.BooleanExpressionTruth
open TransitiveNameModel BoundedSetTheory CountableForcing
open InternalRegularOperations InternalRegularAlgebra InternalBooleanBits InternalBooleanGeneric
open BooleanExpressionCertificate
attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder
theorem value_seed (B R U : ZFSet.{0}) (hU : U ∈ B) :
value B R (FiniteTerm.tag 0 U) ↔ U ∈ R := by
rw [value_unfold]
simp [FiniteTerm.tag_inj,hU]
theorem value_neg (B R t : ZFSet.{0}) :
value B R (FiniteTerm.tag 1 t) ↔ ¬ value B R t := by
rw [value_unfold]
simp [FiniteTerm.tag_inj]
theorem value_sup (B R T : ZFSet.{0}) :
value B R (FiniteTerm.tag 2 T) ↔ ∃ t ∈ T, value B R t := by
rw [value_unfold]
simp [FiniteTerm.tag_inj]
theorem certificate_value (M c B Q S K g R : ZFSet.{0})
(hM : Transitive M) (hT : SourceT M) (hc : c ∈ M)
(hB : ∀ U, U ∈ B ↔ U ∈ M ∧ IsCode c U)
(hQ : ∀ F, F ∈ Q ↔ F ∈ M ∧ F ⊆ B)
(hg : Certificate c B Q S K g)
(G : GenericFilter (Conditions c)) (hG : AtomicForcing.GroundGeneric M G)
(hseed : ∀ U ∈ S, U ∈ R ↔ Hit G U)
(t U : ZFSet.{0}) (hp : ZFSet.pair t U ∈ g) : value B R t ↔ Hit G U := by
induction t using rankWF.induction generalizing U with
| h t ih =>
have hUB := (hg.step hp).2.1
rcases (hg.step hp).2.2 with hs | hs | hs
· obtain ⟨hUS,rfl⟩ := hs
exact (value_seed B R U hUB).trans (hseed U hUS)
· obtain ⟨s,_,V,hVB,rfl,hchild,rfl⟩ := hs
obtain ⟨hVM,hVc⟩ := (hB V).mp hVB
rw [value_neg,hit_complement_iff M c V hM hT hc hVM hVc G hG,
ih s (FiniteTerm.pair_rank_right _ _) V hchild]
· obtain ⟨T,_,F,hFQ,rfl,rfl,hTF,hFT⟩ := hs
obtain ⟨hFM,hFB⟩ := (hQ F).mp hFQ
rw [value_sup,hit_supCode_iff M c F hM hT hc hFM (fun V hV => ((hB V).mp (hFB hV)).2) G hG]
constructor
· rintro ⟨s,hs,hv⟩
obtain ⟨V,hVF,hchild⟩ := hTF s hs
exact ⟨V,hVF,(ih s ((ZFSet.rank_lt_of_mem hs).trans (FiniteTerm.pair_rank_right _ _)) V hchild).mp hv⟩
· rintro ⟨V,hVF,hv⟩
obtain ⟨s,hs,hchild⟩ := hFT V hVF
exact ⟨s,hs,(ih s ((ZFSet.rank_lt_of_mem hs).trans (FiniteTerm.pair_rank_right _ _)) V hchild).mpr hv⟩
noncomputable def seedTruth (c B : ZFSet.{0}) (E : ℕ → ZFSet.{0}) (X : ZFSet.{0}) : ZFSet.{0} :=
B.sep (fun U => ∃ n, natSet n ∈ X ∧ U = bitCode c (E n))
theorem seedTruth_mem (N c B : ZFSet.{0}) (hN : Transitive N) (hT : SourceT N)
(hc : c ∈ N) (hB : B ∈ N) (E : ℕ → ZFSet.{0})
(hE : ∀ n, E n ∈ N ∧ E n ⊆ c) (hgraph : orbitGraph E ∈ N)
(X : ZFSet.{0}) (hX : X ∈ N) : seedTruth c B E X ∈ N := by
obtain ⟨Q,hQ,hQdef⟩ := internal_power N hN hT.powerSet hc
let e := cons c (cons ZFSet.omega (cons Q (cons (orbitGraph E) (fun _ => X))))
have he : ∀ i, e i ∈ N := by
intro i; rcases i with _|_|_|_|i
exact hc; exact sourceT_omega_mem N hN hT; exact hQ; exact hgraph; exact hX
let φ : Formula := .existsMem 2 (.conj (.member 0 6)
(.existsMem 4 (.conj (.pairMem 1 0 6) (bitFormula 3 0 2))))
have spec (U : ZFSet.{0}) : φ.Eval (cons U e) ↔ ∃ n, natSet n ∈ X ∧ U = bitCode c (E n) := by
simp only [φ,Formula.Eval,Formula.eval_pairMem,bitFormula_spec,e,cons_zero,cons_succ]
constructor
· rintro ⟨n,hn,hnX,V,_,hp,hU⟩
obtain ⟨n,rfl⟩ := (mem_omega n).mp hn
have hV := (orbitGraph_pair E n V).mp hp
exact ⟨n,hnX,hV ▸ hU⟩
· rintro ⟨n,hnX,hU⟩
exact ⟨natSet n,(mem_omega _).mpr ⟨n,rfl⟩,hnX,E n,(hQdef _).mpr (hE n),
(orbitGraph_pair E n _).mpr rfl,hU⟩
have hs := sep_mem N hN hT.separation.finitePrefix.bounded φ e he hB
simpa only [spec,seedTruth] using hs
theorem seedTruth_hit (M c B : ZFSet.{0}) [Top (Conditions c)]
(hM : Transitive M) (hT : SourceT M) (hc : c ∈ M)
(E : ℕ → ZFSet.{0}) (hE : ∀ n, E n ∈ M ∧ E n ⊆ c)
(G : GenericFilter (Conditions c)) (hG : AtomicForcing.GroundGeneric M G)
(hGt : ⊤ ∈ G.carrier) (U : ZFSet.{0}) (hUS : U ∈ seeds c B E) :
U ∈ seedTruth c B E ((InternalNiceName.nice E : RecursiveNames.Name (Conditions c)).val G.carrier) ↔ Hit G U := by
constructor
· intro h
obtain ⟨_,n,hn,rfl⟩ := ZFSet.mem_sep.mp h
exact (hit_bit_iff M c hM hT hc E hE G hG hGt n).mpr hn
· intro h
obtain ⟨hUB,n,rfl⟩ := ZFSet.mem_sep.mp hUS
exact ZFSet.mem_sep.mpr ⟨hUB,n,(hit_bit_iff M c hM hT hc E hE G hG hGt n).mp h,rfl⟩
end TuringRigidity.BooleanExpressionTruth
end OAI
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