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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 | import OAI.Computability.DegreeRigidity.Syntax.BooleanExpressionAssembly
namespace OAI
namespace TuringRigidity.BooleanExpressionCertificate
open TransitiveNameModel BoundedSetTheory InternalBooleanSyntax InternalGeneratedAlgebra
open InternalRegularOperations InternalRegularAlgebra
theorem sup_certified (M c B Q S F : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) (hBM : B ∈ M) (hQM : Q ∈ M) (hSM : S ∈ M)
(hB : ∀ U, U ∈ B ↔ U ∈ M ∧ IsCode c U)
(hQ : ∀ a, a ∈ Q ↔ a ∈ M ∧ a ⊆ B) (hFQ : F ∈ Q)
(hFC : F ⊆ certified M c B Q S) : supCode c F ∈ certified M c B Q S := by
obtain ⟨hFM,hFB⟩ := (hQ F).mp hFQ
have he := parameters_mem M c B Q S hM hT hc hBM hQM hSM
have hColl := sigma_collection M hM hT.pairing hT.union hT.powerSet
hT.separation.finitePrefix hT.replacement.finitePrefix hT.infinity hT.choice
obtain ⟨W,hWM,hW⟩ := hColl (.bounded witnessBody) (parameters c B Q S) he F hFM (by
intro U hUF
obtain ⟨_,K,hKM,hK⟩ := ZFSet.mem_sep.mp (hFC hUF)
exact ⟨K,hKM,(witnessBody_realize M c B Q S K U hM he hKM (hM F hFM U hUF)).mpr hK⟩)
obtain ⟨L,hLM,hL,hWL⟩ := internal_transitive_container M hM hT.pairing hT.union
hT.separation.finitePrefix.bounded hT.replacement.finitePrefix hT.infinity hWM
have hKL (K : ZFSet.{0}) (hK : K ∈ W) : K ⊆ L :=
fun t ht => hL K (hL W hWL K hK) t ht
let a := goodGraphs (ZFSet.sUnion W) c B Q S L
have haM : a ∈ M := goodGraphs_mem M _ c B Q S L hM hT
(union_mem M hM hT.union hWM) hc hBM hQM hSM hLM
let g := ZFSet.sUnion a
have hgM : g ∈ M := union_mem M hM hT.union haM
have hg : Certificate c B Q S L g := Certificate.sUnion (fun f hf => (ZFSet.mem_sep.mp hf).2)
have htotal (U : ZFSet.{0}) (hUF : U ∈ F) : ∃ t ∈ L, ZFSet.pair t U ∈ g := by
obtain ⟨K,hKW,hK⟩ := hW U hUF
obtain ⟨_,f,hfK,hf,t,htK,hp⟩ :=
(witnessBody_realize M c B Q S K U hM he (hM W hWM K hKW) (hM F hFM U hUF)).mp hK
have hfa : f ∈ a := ZFSet.mem_sep.mpr ⟨ZFSet.mem_sUnion.mpr ⟨K,hKW,hfK⟩,hf.enlarge (hKL K hKW)⟩
exact ⟨t,hKL K hKW htK,ZFSet.mem_sUnion.mpr ⟨f,hfa,hp⟩⟩
let T := children L F g
have hTM : T ∈ M := children_mem M L F g hM hT hLM hFM hgM
let s := FiniteTerm.tag 2 T
have hsM : s ∈ M := tag_mem M hM hT 2 hTM
let D := L ∪ ({T,s} : ZFSet.{0})
have hLD : L ⊆ D := fun _ h => ZFSet.mem_union.mpr (Or.inl h)
have hTD : T ∈ D := ZFSet.mem_union.mpr (Or.inr (ZFSet.mem_pair.mpr (Or.inl rfl)))
have hsD : s ∈ D := ZFSet.mem_union.mpr (Or.inr (ZFSet.mem_pair.mpr (Or.inr rfl)))
have hDM : D ∈ M := binary_union_mem M hM hT.pairing hT.union hLM (pair_mem M hM hT.pairing hTM hsM)
have hVB := internal_join M c B F hM hT hc hB hFM
have hVM := ((hB _).mp hVB).1
have hs : Step c B Q S D g s (supCode c F) := by
refine Or.inr (Or.inr ⟨T,hTD,F,hFQ,rfl,rfl,?_,?_⟩)
· exact fun t ht => (ZFSet.mem_sep.mp ht).2
· intro U hUF
obtain ⟨t,htL,hp⟩ := htotal U hUF
exact ⟨t,ZFSet.mem_sep.mpr ⟨htL,U,hUF,hp⟩,hp⟩
let f := g ∪ ({ZFSet.pair s (supCode c F)} : ZFSet.{0})
have hfM : f ∈ M := binary_union_mem M hM hT.pairing hT.union hgM
(singleton_mem M hM hT.pairing (orderedPair_mem M hM hT.pairing hsM hVM))
have hf := (hg.enlarge hLD).insert hsD hVB hs
exact ZFSet.mem_sep.mpr ⟨hVB,witness_of_certificate M c B Q S D f s _ hM hT hDM hfM hf
(ZFSet.mem_union.mpr (Or.inr (ZFSet.mem_singleton.mpr rfl)))⟩
theorem certified_closed (M c B Q S : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) (hBM : B ∈ M) (hQM : Q ∈ M) (hSM : S ∈ M)
(hB : ∀ U, U ∈ B ↔ U ∈ M ∧ IsCode c U)
(hQ : ∀ a, a ∈ Q ↔ a ∈ M ∧ a ⊆ B) : Closed c B Q (certified M c B Q S) :=
⟨fun _ h => (ZFSet.mem_sep.mp h).1,
fun U hU => neg_certified M c B Q S U hM hT hc hB hU,
fun F hF hFC => sup_certified M c B Q S F hM hT hc hBM hQM hSM hB hQ hF hFC⟩
theorem generated_certificate (M c B Q S U : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) (hBM : B ∈ M) (hQM : Q ∈ M) (hSM : S ∈ M)
(hB : ∀ V, V ∈ B ↔ V ∈ M ∧ IsCode c V)
(hQ : ∀ a, a ∈ Q ↔ a ∈ M ∧ a ⊆ B) (hSB : S ⊆ B)
(hU : U ∈ generated c B Q S) : ∃ K ∈ M, Witness c B Q S K U := by
have hC := certified_closed M c B Q S hM hT hc hBM hQM hSM hB hQ
have hCM := certified_mem M c B Q S hM hT hc hBM hQM hSM
have hCQ := (hQ _).mpr ⟨hCM,hC.1⟩
have hSC : S ⊆ certified M c B Q S := by
intro V hV
exact seed_certified M c B Q S V hM hT (hM S hSM V hV) (hSB hV) hV
exact (ZFSet.mem_sep.mp (generated_le c B Q S _ hCQ hSC hC hU)).2
theorem certified_eq_generated (M c B Q S : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) (hBM : B ∈ M) (hQM : Q ∈ M) (hSM : S ∈ M)
(hB : ∀ V, V ∈ B ↔ V ∈ M ∧ IsCode c V)
(hQ : ∀ a, a ∈ Q ↔ a ∈ M ∧ a ⊆ B) (hSB : S ⊆ B) :
certified M c B Q S = generated c B Q S := by
have hA := generated_closed c B Q S (algebra_closed M c B Q hM hT hc hB hQ)
apply ZFSet.ext; intro U
constructor
· intro hU
obtain ⟨_,K,_,_,g,_,hg,t,_,hp⟩ := ZFSet.mem_sep.mp hU
exact hg.induction (fun V => V ∈ generated c B Q S) (fun V hV => seeds_subset c B Q S hSB hV)
(fun V _ hV => hA.2.1 V hV) (fun F hF hFA => hA.2.2 F hF (fun V hV => hFA V hV)) t U hp
· intro hU
exact ZFSet.mem_sep.mpr ⟨generated_subset c B Q S hU,
generated_certificate M c B Q S U hM hT hc hBM hQM hSM hB hQ hSB hU⟩
end TuringRigidity.BooleanExpressionCertificate
end OAI
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