File size: 2,955 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 | import OAI.Computability.DegreeRigidity.SetModels.OrderedNameCode
namespace OAI
namespace TuringRigidity.TransitiveNameModel
open RecursiveNames BoundedSetTheory
universe u
def evaluationOutputFormula : SigmaFormula := .existsSet
(.andBounded (.conj (.member 0 4) (.pairMem 2 0 3))
(NameCodeFormula.orderedName 5 0 2 1))
theorem internal_evaluation_name_codes (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : Separation M) (hR : SigmaReplacement M) (hI : Infinity M)
{V t : ZFSet.{u}} (hV : V ∈ M) (ht : t ∈ M) :
∃ B ∈ M, ∀ z, z ∈ B ↔ ∃ x ∈ V,
orderedNameCode t (checkedCode t x) x = z := by
obtain ⟨d,hdM,hd,hVd⟩ := internal_transitive_container M hM hP hU hS hR hI hV
obtain ⟨q,hqM,hq⟩ := internal_power M hM hPow hdM
obtain ⟨F,hF,hfg⟩ := internal_checkGraph M d q t hM hP hU hPow hS hR
hdM hqM ht hd hq V hVd
let r := iterUnion 2 F
have hr : r ∈ M := iterUnion_mem M hM hU hF 2
have hxdom (x : ZFSet.{u}) (hx : x ∈ V) : x ∈ hull d q V :=
(hull_transitive d q V hd) V (self_mem_hull d q hVd) x hx
have check (x : ZFSet.{u}) (hx : x ∈ V) :
checkedCode t x ∈ r ∧ ZFSet.pair x (checkedCode t x) ∈ F := by
obtain ⟨y,hy,hxy,_⟩ := hfg.2.2.1 x (hxdom x hx)
have he := hfg.correct x (hxdom x hx) y hy hxy
exact he ▸ ⟨hy,hxy⟩
let e := cons F (cons r (fun _ => t))
have he : ∀ i, e i ∈ M := by
intro i; rcases i with _|i; exact hF
rcases i with _|i; exact hr
exact ht
apply replacement_image M hM hR evaluationOutputFormula e he hV
(fun x => orderedNameCode t (checkedCode t x) x)
· intro x hx
exact orderedNameCode_mem M hM hP ht (hM r hr _ (check x hx).1) (hM V hV x hx)
· intro x hx z hz
have hxM := hM V hV x hx
have matrix (w : ZFSet.{u}) (hw : w ∈ M) :
(SigmaFormula.andBounded (.conj (.member 0 4) (.pairMem 2 0 3))
(NameCodeFormula.orderedName 5 0 2 1)).Realize M (cons w (cons z (cons x e))) ↔
w ∈ r ∧ ZFSet.pair x w ∈ F ∧ z = orderedNameCode t w x := by
have he' : ∀ i, cons w (cons z (cons x e)) i ∈ M := by
intro i; rcases i with _|i; exact hw
rcases i with _|i; exact hz
rcases i with _|i; exact hxM
exact he i
rw [SigmaFormula.realize_andBounded,Formula.absolute _ M hM _ he',
NameCodeFormula.realize_orderedName M hM hP _ _ _ _ _ he']
simp only [Formula.Eval,Formula.eval_pairMem,cons_zero,cons_succ,e]
exact and_assoc
change (∃ w ∈ M, _) ↔ _
constructor
· rintro ⟨w,hw,h⟩
obtain ⟨hwr,hxw,hz⟩ := (matrix w hw).mp h
rw [hfg.correct x (hxdom x hx) w hwr hxw] at hz
exact hz
· intro hz
have hwM := hM r hr _ (check x hx).1
exact ⟨_,hwM,(matrix _ hwM).mpr ⟨(check x hx).1,(check x hx).2,hz⟩⟩
end TuringRigidity.TransitiveNameModel
end OAI
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