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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 | import OAI.Computability.DegreeRigidity.SetModels.CofinalPaddedDecoder
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel OrdinalCoding
universe u
attribute [local irreducible] Construction.ownRealsMembership SentenceForm.bound
noncomputable def Construction.ordinalVector (t : Construction.{u}) :
Fin t.ownRealsMembership.bound → Ordinal.{u} := fun k => (t.ordinalData k).getD 0
theorem Construction.ordinalVector_internal (t : Construction.{u}) (M : ZFSet.{u})
(hM : Transitive M) (hT : SourceT M) (hi : t.Indexed M) :
∀ k, (t.ordinalVector k).toZFSet ∈ M := by
intro k
cases ho : t.ordinalData k with
| none => simpa only [Construction.ordinalVector,ho,Option.getD_none] using internal_ordinal_zero M hM hT
| some o => simpa only [Construction.ordinalVector,ho,Option.getD_some] using t.ordinalData_mem M hi k o ho
noncomputable def Construction.decodedOrdinalInput (t : Construction.{u}) (k : ℕ) : SentenceForm :=
if hk : k < t.ownRealsMembership.bound then
match t.ordinalData k with
| none => .equal 0 2
| some _ => largestPaddedEntrySentence t.ownRealsMembership.bound ⟨k,hk⟩
else .equal 0 2
noncomputable def Construction.decodedOrdinalBody (t : Construction.{u}) (q : SentenceForm) : SentenceForm :=
bindComputed t.ownRealsMembership.bound t.decodedOrdinalInput
(q.rename (ordinalBodySlots t.ownRealsMembership.bound))
theorem bindComputed_bound (n m : ℕ) (f : ℕ → SentenceForm) (p : SentenceForm)
(hp : p.bound ≤ n+m) (hf : ∀ i < n, (f i).bound ≤ m+1) :
(bindComputed n f p).bound ≤ m := by
induction n generalizing m f with
| zero => simpa only [bindComputed,zero_add] using hp
| succ n ih =>
have hinner := ih (m+1) (fun i => (f i).rename skipValue) (by omega) (by
intro i hi
apply code_rename_bound (f i) skipValue (m+2)
intro j hj
cases j with
| zero => change 0 < m+2; omega
| succ j => change j+2 < m+2; have h := hf i (by omega); omega)
have hn := hf n (by omega)
simpa only [bindComputed,SentenceForm.bound] using
(show max (f n).bound (bindComputed n (fun i => (f i).rename skipValue) p).bound - 1 ≤ m by omega)
theorem Construction.decodedOrdinalBody_bound (t : Construction.{u}) (q : SentenceForm) :
(t.decodedOrdinalBody q).bound ≤ 2 := by
apply bindComputed_bound
· apply code_rename_bound
intro i _
rcases i with _|_|i
· change t.ownRealsMembership.bound < t.ownRealsMembership.bound+2; omega
· change t.ownRealsMembership.bound+1 < t.ownRealsMembership.bound+2; omega
· dsimp only [ordinalBodySlots]; split <;> omega
· intro i _
by_cases hi : i < t.ownRealsMembership.bound
· rw [Construction.decodedOrdinalInput,dite_eq_left hi]
cases t.ordinalData i with
| none => simp only [SentenceForm.bound]; omega
| some o => exact (largestPaddedEntrySentence_bound _ _).trans (by omega)
· rw [Construction.decodedOrdinalInput,dite_eq_right hi]
simp only [SentenceForm.bound]; omega
theorem Construction.decodedOrdinalBody_spec (t : Construction.{u}) (q : SentenceForm)
(M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M) (P : Oracle)
(β : Ordinal.{u}) (hoff : realSeedOffset ≤ β)
(hcert : PaddedCodeCertificates
(level (groundReals M) (heightDomainIndex (paddedCode t.ordinalVector β))) t.ordinalVector β)
(he : ∀ i, t.ordinalInputs P i ∈
level (groundReals M) (heightDomainIndex (paddedCode t.ordinalVector β)))
(z : ZFSet.{u}) :
(t.decodedOrdinalBody q).Sat
(level (groundReals M) (heightDomainIndex (paddedCode t.ordinalVector β)) : Set ZFSet)
(cons z (fun _ => realCode P)) ↔
q.Sat (level (groundReals M) (heightDomainIndex (paddedCode t.ordinalVector β)) : Set ZFSet)
(cons z (t.ordinalInputs P)) := by
let n := t.ownRealsMembership.bound
let w := fun k => t.ordinalInputs P (k+1)
let A := level (groundReals M) (heightDomainIndex (paddedCode t.ordinalVector β))
have hf (k : ℕ) (hk : k < n) (x : ZFSet.{u}) (hx : x ∈ A) :
(t.decodedOrdinalInput k).Sat (A : Set ZFSet) (cons x (cons z (fun _ => realCode P))) ↔ x = w k := by
have hk' : k < t.ownRealsMembership.bound := hk
rw [Construction.decodedOrdinalInput,dite_eq_left hk']
cases ho : t.ordinalData k with
| none => simp only [SentenceForm.Sat,w,Construction.ordinalInputs,cons_succ,cons_zero,ite_eq_left hk',ho]
| some o =>
have h := largestPaddedEntrySentence_of_certificates M hM hT n ⟨k,hk⟩ t.ordinalVector β hoff
(cons x (cons z (fun _ => realCode P))) hx hcert
simpa only [cons_zero,w,Construction.ordinalInputs,cons_succ,ite_eq_left hk',
Construction.ordinalVector,ho,Option.getD_some] using h
rw [Construction.decodedOrdinalBody,bindComputed_spec A n t.decodedOrdinalInput _
(cons z (fun _ => realCode P)) w (fun i _ => he (i+1)) hf,SentenceForm.sat_rename]
have henv : (fun i => prependValues w n (cons z (fun _ => realCode P)) (ordinalBodySlots n i)) =
cons z (t.ordinalInputs P) := by
funext i
rcases i with _|_|k
· exact prependValues_tail w n _ 0
· exact prependValues_tail w n _ 1
· by_cases hk : k < n
· simpa only [ordinalBodySlots,ite_eq_left hk,cons_succ,w] using prependValues_lt w n _ k hk
· have hk' : ¬ k < t.ownRealsMembership.bound := hk
simp only [ordinalBodySlots,ite_eq_right hk,cons_succ,Construction.ordinalInputs,ite_eq_right hk']
exact prependValues_tail w n _ 1
rw [henv]
end TuringRigidity.RelativeConstructible
end OAI
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