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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import OAI.Computability.DegreeRigidity.SetModels.InternalBooleanGeneric
import OAI.Computability.DegreeRigidity.Syntax.FiniteTerm
namespace OAI
namespace TuringRigidity.BooleanExpressionCertificate
open TransitiveNameModel BoundedSetTheory InternalRegularOperations InternalRegularAlgebra
open InternalBooleanGeneric CountableForcing
attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder
def Step (c B Q S K g t U : ZFSet.{0}) : Prop :=
(U ∈ S ∧ t = FiniteTerm.tag 0 U) ∨
(∃ s ∈ K, ∃ V ∈ B, t = FiniteTerm.tag 1 s ∧
ZFSet.pair s V ∈ g ∧ U = neg c V) ∨
(∃ T ∈ K, ∃ F ∈ Q, t = FiniteTerm.tag 2 T ∧ U = supCode c F ∧
(∀ s ∈ T, ∃ V ∈ F, ZFSet.pair s V ∈ g) ∧
(∀ V ∈ F, ∃ s ∈ T, ZFSet.pair s V ∈ g))
def Certificate (c B Q S K g : ZFSet.{0}) : Prop :=
∀ z ∈ g, ∃ t ∈ K, ∃ U ∈ B, z = ZFSet.pair t U ∧ Step c B Q S K g t U
theorem Step.mono {c B Q S K L g h t U : ZFSet.{0}}
(hKL : K ⊆ L) (hgh : g ⊆ h) (hs : Step c B Q S K g t U) :
Step c B Q S L h t U := by
rcases hs with hs | hs | hs
· exact Or.inl hs
· obtain ⟨s,hs,V,hV,ht,hp,hU⟩ := hs
exact Or.inr (Or.inl ⟨s,hKL hs,V,hV,ht,hgh hp,hU⟩)
· obtain ⟨T,hT,F,hF,ht,hU,hTF,hFT⟩ := hs
exact Or.inr (Or.inr ⟨T,hKL hT,F,hF,ht,hU,
fun s hs => (hTF s hs).imp (fun V h => ⟨h.1,hgh h.2⟩),
fun V hV => (hFT V hV).imp (fun s h => ⟨h.1,hgh h.2⟩)⟩)
theorem Certificate.step {c B Q S K g t U : ZFSet.{0}}
(hg : Certificate c B Q S K g) (hp : ZFSet.pair t U ∈ g) :
t ∈ K ∧ U ∈ B ∧ Step c B Q S K g t U := by
obtain ⟨s,hs,V,hV,he,hstep⟩ := hg _ hp
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp he
exact ⟨hs,hV,hstep⟩
theorem Certificate.enlarge {c B Q S K L g : ZFSet.{0}}
(hg : Certificate c B Q S K g) (hKL : K ⊆ L) : Certificate c B Q S L g := by
intro z hz
obtain ⟨t,ht,U,hU,he,hs⟩ := hg z hz
exact ⟨t,hKL ht,U,hU,he,hs.mono hKL (fun _ h => h)⟩
theorem Certificate.union {c B Q S K g h : ZFSet.{0}}
(hg : Certificate c B Q S K g) (hh : Certificate c B Q S K h) :
Certificate c B Q S K (g ∪ h) := by
intro z hz
rcases ZFSet.mem_union.mp hz with hz | hz
· obtain ⟨t,ht,U,hU,he,hs⟩ := hg z hz
exact ⟨t,ht,U,hU,he,hs.mono (fun _ h => h) (fun _ h => ZFSet.mem_union.mpr (Or.inl h))⟩
· obtain ⟨t,ht,U,hU,he,hs⟩ := hh z hz
exact ⟨t,ht,U,hU,he,hs.mono (fun _ h => h) (fun _ h => ZFSet.mem_union.mpr (Or.inr h))⟩
theorem Certificate.insert {c B Q S K g t U : ZFSet.{0}}
(hg : Certificate c B Q S K g) (ht : t ∈ K) (hU : U ∈ B)
(hs : Step c B Q S K g t U) :
Certificate c B Q S K (g ∪ ({ZFSet.pair t U} : ZFSet.{0})) := by
have hgg : g ⊆ g ∪ ({ZFSet.pair t U} : ZFSet.{0}) :=
fun _ h => ZFSet.mem_union.mpr (Or.inl h)
intro z hz
rcases ZFSet.mem_union.mp hz with hz | hz
· obtain ⟨s,hs,V,hV,he,hstep⟩ := hg z hz
exact ⟨s,hs,V,hV,he,hstep.mono (fun _ h => h) hgg⟩
· have he := ZFSet.mem_singleton.mp hz
exact ⟨t,ht,U,hU,he,hs.mono (fun _ h => h) hgg⟩
theorem Certificate.induction {c B Q S K g : ZFSet.{0}}
(hg : Certificate c B Q S K g) (P : ZFSet.{0} → Prop)
(hseed : ∀ U ∈ S, P U)
(hneg : ∀ U ∈ B, P U → P (neg c U))
(hsup : ∀ F ∈ Q, (∀ U ∈ F, P U) → P (supCode c F))
(t U : ZFSet.{0}) (hp : ZFSet.pair t U ∈ g) : P U := by
have wf : WellFounded (fun a b : ZFSet.{0} => ZFSet.rank a < ZFSet.rank b) :=
InvImage.wf ZFSet.rank Ordinal.lt_wf
induction t using wf.induction generalizing U with
| h t ih =>
rcases (hg.step hp).2.2 with hs | hs | hs
· exact hseed U hs.1
· obtain ⟨s,_,V,hV,rfl,hchild,rfl⟩ := hs
exact hneg V hV (ih s (FiniteTerm.pair_rank_right _ _) V hchild)
· obtain ⟨T,_,F,hF,rfl,rfl,_,hFT⟩ := hs
apply hsup F hF
intro V hV
obtain ⟨s,hs,hp⟩ := hFT V hV
exact ih s ((ZFSet.rank_lt_of_mem hs).trans (FiniteTerm.pair_rank_right _ _)) V hp
theorem Certificate.generic_agreement (M c B Q S K g : 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 H : GenericFilter (Conditions c))
(hG : AtomicForcing.GroundGeneric M G) (hH : AtomicForcing.GroundGeneric M H)
(hseed : ∀ U ∈ S, Hit G U ↔ Hit H U)
(t U : ZFSet.{0}) (hp : ZFSet.pair t U ∈ g) : Hit G U ↔ Hit H U := by
apply hg.induction (fun U => Hit G U ↔ Hit H U) hseed ?_ ?_ t U hp
· intro V hV ih
obtain ⟨hVM,hVc⟩ := (hB V).mp hV
rw [hit_complement_iff M c V hM hT hc hVM hVc G hG,
hit_complement_iff M c V hM hT hc hVM hVc H hH,ih]
· intro F hF ih
obtain ⟨hFM,hFB⟩ := (hQ F).mp hF
have hFc := fun U hU => ((hB U).mp (hFB hU)).2
rw [hit_supCode_iff M c F hM hT hc hFM hFc G hG,
hit_supCode_iff M c F hM hT hc hFM hFc H hH]
exact exists_congr (fun U => and_congr_right (fun hU => ih U hU))
end TuringRigidity.BooleanExpressionCertificate
end OAI
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