File size: 4,604 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 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 | import OAI.Computability.DegreeRigidity.SetModels.ElementaryCollapsePoset
import OAI.Computability.DegreeRigidity.Syntax.CheckGraphConstruction
namespace OAI
namespace TuringRigidity.ElementaryModel
open BoundedSetTheory TransitiveNameModel RelationCollapse SentenceForm
universe u
theorem ambient_transitive_container (x : ZFSet.{u}) :
∃ d : ZFSet.{u}, TransitiveNameModel.Transitive d ∧ x ∈ d := by
let b := ZFSet.range (fun n : ℕ => iterUnion n x)
let d := ({x} : ZFSet.{u}) ∪ ZFSet.sUnion b
refine ⟨d,?_,ZFSet.mem_union.mpr (Or.inl (ZFSet.mem_singleton.mpr rfl))⟩
intro y hy z hz
apply ZFSet.mem_union.mpr
apply Or.inr
rcases ZFSet.mem_union.mp hy with hy|hy
· have hyx : y = x := ZFSet.mem_singleton.mp hy
subst y
exact ZFSet.mem_sUnion.mpr ⟨x,ZFSet.mem_range.mpr ⟨0,rfl⟩,hz⟩
· obtain ⟨c,hc,hyc⟩ := ZFSet.mem_sUnion.mp hy
obtain ⟨n,rfl⟩ := ZFSet.mem_range.mp hc
exact ZFSet.mem_sUnion.mpr ⟨iterUnion (n+1) x,ZFSet.mem_range.mpr ⟨n+1,rfl⟩,
ZFSet.mem_sUnion.mpr ⟨y,hyc,hz⟩⟩
theorem ambient_checkGraph (d t : ZFSet.{u}) (hd : Transitive d) :
∃ r f, CheckGraph d r f t := by
let f := ZFSet.range (fun i : Conditions d => ZFSet.pair (label d i) (checkedCode t (label d i)))
refine ⟨iterUnion 2 f,f,checkGraph_of_mem_iff d f t hd ?_⟩
intro z
rw [ZFSet.mem_range]
constructor
· rintro ⟨i,hi⟩
exact ⟨label d i,label_mem d i,hi.symm⟩
· rintro ⟨x,hx,hz⟩
obtain ⟨i,rfl⟩ := label_surjective d hx
exact ⟨i,hz.symm⟩
noncomputable def checkedValueFormula : SentenceForm :=
.ex (.ex (.ex (fromBounded (.conj (CheckFormula.graph 2 1 0 5)
(.conj (.member 4 2) (.conj (.member 3 1) (.pairMem 4 3 0)))))))
theorem checkedValue_univ (v x t : ZFSet.{u}) :
checkedValueFormula.Sat Set.univ (cons v (cons x (fun _ => t))) ↔ v = checkedCode t x := by
simp only [checkedValueFormula,Sat,bounded_univ,Formula.Eval,CheckFormula.eval_graph,
Formula.eval_pairMem,cons_zero,cons_succ,Set.mem_univ,true_and]
constructor
· rintro ⟨d,r,f,hg,hx,hv,hf⟩
exact hg.correct x hx v hv hf
· intro hv
obtain ⟨d,hd,hx⟩ := ambient_transitive_container x
obtain ⟨r,f,hg⟩ := ambient_checkGraph d t hd
obtain ⟨w,hw,hf,_⟩ := hg.2.2.1 x hx
have he : w = v := (hg.correct x hx w hw hf).trans hv.symm
exact ⟨d,r,f,hg,hx,he ▸ hw,he ▸ hf⟩
theorem checkedValue_sound (M : ZFSet.{u}) (hM : Transitive M)
(e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M)
(h : checkedValueFormula.Sat (M : Set ZFSet) e) : e 0 = checkedCode (e 2) (e 1) := by
obtain ⟨d,hd,r,hr,f,hf,h⟩ := h
have hi : ∀ i, cons f (cons r (cons d e)) i ∈ M := by
intro i; rcases i with _|_|_|i
exact hf
exact hr
exact hd
exact he i
rw [bounded_sat,Formula.absolute _ M hM _ hi] at h
simp only [Formula.Eval,CheckFormula.eval_graph,Formula.eval_pairMem,cons_zero,cons_succ] at h
exact h.1.correct (e 1) h.2.1 (e 0) h.2.2.1 h.2.2.2
theorem checkedCode_mem_hull (a : ℕ → ZFSet.{u}) {t x : ZFSet.{u}}
(ht : t ∈ hullSet a) (hx : x ∈ hullSet a) : checkedCode t x ∈ hullSet a := by
exact unique_definable_mem a checkedValueFormula (cons x (fun _ => t))
(by intro i; cases i <;> assumption) _ ((checkedValue_univ _ x t).mpr rfl)
(fun v hv => (checkedValue_univ v x t).mp hv)
theorem collapse_checkedCode (a : ℕ → ZFSet.{u}) {t x : ZFSet.{u}}
(ht : t ∈ hullSet a) (hx : x ∈ hullSet a) :
structureMap (hullSet a) (checkedCode t x) =
checkedCode (structureMap (hullSet a) t) (structureMap (hullSet a) x) := by
let e := cons (checkedCode t x) (cons x (fun _ => t))
have he : ∀ i, e i ∈ hullSet a := by
intro i; rcases i with _|_|i
exact checkedCode_mem_hull a ht hx
exact hx
exact ht
have hs := (collapsed_hull_elementary a checkedValueFormula e he).mpr
((checkedValue_univ _ x t).mpr rfl)
exact checkedValue_sound _ (collapsed_transitive _) _
(fun i => (mem_collapsed _ _).mpr ⟨e i,he i,rfl⟩) hs
theorem collapse_encoded_check (a : ℕ → ZFSet.{u})
{c c' : ZFSet.{u}} [Top (Conditions c)] [Top (Conditions c')]
(ht : label c ⊤ ∈ hullSet a)
(ht' : structureMap (hullSet a) (label c ⊤) = label c' ⊤)
{x : ZFSet.{u}} (hx : x ∈ hullSet a) :
structureMap (hullSet a) ((RecursiveNames.Name.check x).encode (label c)) =
(RecursiveNames.Name.check (structureMap (hullSet a) x)).encode (label c') := by
rw [encode_check,encode_check,collapse_checkedCode a ht hx,ht']
end TuringRigidity.ElementaryModel
end OAI
|