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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 | import OAI.Computability.DegreeRigidity.Representation.AutomorphismOutputSentence
namespace OAI
namespace TuringRigidity.BoundedForcing
open TransitiveNameModel BoundedSetTheory SetDegreeDecoding PersistentRestrictions
universe u
def boundedOutputFormula : Formula :=
.existsMem 3 (.existsMem 4 (.conj (.member 3 1)
(.conj (.member 4 0) (.pairMem 1 0 2))))
theorem boundedOutputFormula_raw (f x y D : ZFSet.{u}) :
boundedOutputFormula.Eval (cons f (cons x (cons y (fun _ => D)))) ↔
∃ d ∈ D, ∃ b ∈ D, x ∈ d ∧ y ∈ b ∧ ZFSet.pair d b ∈ f := by
simp only [boundedOutputFormula,Formula.Eval,Formula.eval_pairMem,cons_zero,cons_succ]
theorem boundedOutputFormula_spec (I : CountableIdeal) (σ : I ≃o I)
(A : Oracle) (hA : degree A ∈ I.carrier) (y : ZFSet.{u}) :
boundedOutputFormula.Eval
(cons (automorphismSet σ) (cons (realCode A) (cons y (fun _ => idealSet I)))) ↔
∃ B : Oracle, realCode B = y ∧ degree B = (σ ⟨degree A,hA⟩).val := by
rw [boundedOutputFormula_raw]
constructor
· rintro ⟨d,_,b,_,had,hbb,hf⟩
obtain ⟨a,hpair⟩ := (mem_automorphismSet σ _).mp hf
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp hpair
have ha : degree A = a.val := (real_mem_degreeSet A a.val).mp had
have ha' : (⟨degree A,hA⟩ : I) = a := Subtype.ext ha
obtain ⟨B,hB,hdeg⟩ := (mem_degreeSet (σ a).val y).mp hbb
exact ⟨B,hB.symm,by rwa [ha']⟩
· rintro ⟨B,hB,hdeg⟩
have hi (a : I) : degreeSet.{u} a.val ∈ idealSet I :=
(mem_idealSet I _).mpr ⟨a.val,a.property,rfl⟩
refine ⟨degreeSet (degree A),hi ⟨degree A,hA⟩,
degreeSet (σ ⟨degree A,hA⟩).val,hi (σ ⟨degree A,hA⟩),
(real_mem_degreeSet A _).mpr rfl,?_,?_⟩
· rw [←hB]; exact (real_mem_degreeSet B _).mpr hdeg
· exact (mem_automorphismSet σ _).mpr ⟨⟨degree A,hA⟩,rfl⟩
end TuringRigidity.BoundedForcing
end OAI
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