File size: 3,432 Bytes
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 | import OAI.Computability.DegreeRigidity.Constructibility.RelativeHierarchyInternal
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding
open BoundedDefinability SetModelFunctions
universe u
def HierarchyRequest (Q B s f r x c : ZFSet.{u}) : Prop :=
∃ A ∈ c, ∃ v ∈ r, ∃ G ∈ c, ∃ T ∈ c, ∃ H ∈ c, ∃ Z ∈ c,
ZFSet.pair x v ∈ f ∧ StageStep s f r x A ∧ DefSystem Q B A G T H Z v
theorem hierarchyRequest_definable {M Q B : ZFSet.{u}} (C : Context M)
(hQ : Q ∈ M) (hB : B ∈ M) (s f r x c : ℕ) :
Definable M (fun e => HierarchyRequest Q B (e s) (e f) (e r) (e x) (e c)) := by
have h := (defPairMem C (x+6) 4 (f+6)).and
((stageStep_definable C (s+6) (f+6) (r+6) (x+6) 5).and
(defSystem_definable C hQ hB 5 3 2 1 0 4))
exact (((((h.existsMem (c+5)).existsMem (c+4)).existsMem (c+3)).existsMem (c+2)).existsMem (r+1)).existsMem c
theorem hierarchyRequest_binary {M Q B s f r : ZFSet.{u}} (C : Context M)
(hQ : Q ∈ M) (hB : B ∈ M) (hs : s ∈ M) (hf : f ∈ M) (hr : r ∈ M) :
∃ p : SigmaFormula, ∃ e : ℕ → ZFSet.{u}, (∀ i, e i ∈ M) ∧
∀ x ∈ M, ∀ c ∈ M, p.Realize M (cons c (cons x e)) ↔
HierarchyRequest Q B s f r x c := by
have h := (hierarchyRequest_definable C hQ hB 2 1 0 4 3).toSigma C.transitive
have eq := ((equal_param hr 0).and ((equal_param hf 1).and (equal_param hs 2))).toSigma C.transitive
have hc := (eq.and h).existsSet.existsSet.existsSet
apply sigma_binary_relation _ (hc.congr ?_)
intro e _
simp only [cons_zero,cons_succ]
constructor
· rintro ⟨s',_,f',_,r',_,⟨rfl,rfl,rfl⟩,h⟩
exact h
· intro h
exact ⟨s,hs,f,hf,r,hr,⟨rfl,rfl,rfl⟩,h⟩
theorem hierarchy_witness_collection {M Q B s d r f : ZFSet.{u}} (C : Context M)
(hColl : SigmaCollection M) (hQ : Q ∈ M) (hB : B ∈ M)
(hq : ∀ p : SentenceForm, family p ∈ Q)
(hb : ∀ p : SentenceForm, supportGraph p ∈ B)
(hs : s ∈ M) (hd : d ∈ M) (hr : r ∈ M) (hf : f ∈ M)
(hg : HierarchyGraph M s d r f) :
∃ W ∈ M, ∀ x ∈ d, ∃ c ∈ W, HierarchyRequest Q B s f r x c := by
obtain ⟨p,e,he,hp⟩ := hierarchyRequest_binary C hQ hB hs hf hr
obtain ⟨W,hW,hw⟩ := hColl p e he d hd (by
intro x hx
obtain ⟨v,hv,hfv,_⟩ := hg.2.1.2 x hx
obtain ⟨A,hA,hstep,hvA⟩ := hg.2.2 x hx v hv hfv
obtain ⟨G,hG,T,hT,H,hH,Z,hZ,hdef⟩ := internal_defSystem_of_context M C Q B A hA hq hb
let a : Fin 5 → ZFSet.{u} := fun i => match i.val with
| 0 => A | 1 => G | 2 => T | 3 => H | _ => Z
have ha : ∀ i, a i ∈ M := by
intro i; dsimp [a]; split <;> assumption
have hc := finite_range_mem M C.transitive C.pairing C.union (C.nat_mem 0) 5 a ha
refine ⟨ZFSet.range a,hc,(hp x (C.transitive d hd x hx) _ hc).mpr ?_⟩
refine ⟨A,ZFSet.mem_range.mpr ⟨⟨0,by decide⟩,rfl⟩,v,hv,
G,ZFSet.mem_range.mpr ⟨⟨1,by decide⟩,rfl⟩,
T,ZFSet.mem_range.mpr ⟨⟨2,by decide⟩,rfl⟩,
H,ZFSet.mem_range.mpr ⟨⟨3,by decide⟩,rfl⟩,
Z,ZFSet.mem_range.mpr ⟨⟨4,by decide⟩,rfl⟩,hfv,hstep,?_⟩
rwa [hvA])
exact ⟨W,hW,fun x hx => by
obtain ⟨c,hc,h⟩ := hw x hx
exact ⟨c,hc,(hp x (C.transitive d hd x hx) c (C.transitive W hW c hc)).mp h⟩⟩
end TuringRigidity.RelativeConstructible
end OAI
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