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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 | import OAI.Computability.DegreeRigidity.SetModels.DegreeSetFormula
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open EncodedForcing
universe u
namespace Formula
def columnBit (o Q n H k : ℕ) : Formula := .existsMem o
(.conj (naturalPair (o+1) (Q+1) (n+1) (k+1) 0) (.member 0 (H+1)))
theorem columnBit_spec (o Q n H k : ℕ) (e : ℕ → ZFSet.{u}) (ho : e o = ZFSet.omega)
(hQ : ∀ f : ℕ → ℕ, ∀ i, finiteNaturalGraph f i ∈ e Q)
(A : Oracle) (hA : e H = realCode A) (N K : ℕ) (hn : e n = natSet N) (hk : e k = natSet K) :
(columnBit o Q n H k).Eval e ↔ columns A N K = true := by
simp only [columnBit,Formula.Eval,cons_zero,cons_succ]
constructor
· rintro ⟨v,_,hp,hm⟩
have hv : v = natSet (Nat.pair N K) :=
(naturalPair_spec (o+1) (Q+1) (n+1) (k+1) 0 (cons v e) ho hQ N K hn hk).mp hp
rw [hv,hA,natSet_mem_realCode] at hm
exact hm
· intro h
refine ⟨natSet (Nat.pair N K),?_,?_,?_⟩
· rw [ho]; exact (mem_omega _).mpr ⟨Nat.pair N K,rfl⟩
· exact (naturalPair_spec (o+1) (Q+1) (n+1) (k+1) 0 (cons (natSet (Nat.pair N K)) e)
ho hQ N K hn hk).mpr rfl
· rw [hA,natSet_mem_realCode]; exact h
def columnSet (o Q n H B : ℕ) : Formula := .conj (.subset B o)
(allMem o (iff (.member 0 (B+1)) (columnBit (o+1) (Q+1) (n+1) (H+1) 0)))
theorem columnSet_spec (o Q n H B : ℕ) (e : ℕ → ZFSet.{u}) (ho : e o = ZFSet.omega)
(hQ : ∀ f : ℕ → ℕ, ∀ i, finiteNaturalGraph f i ∈ e Q)
(A : Oracle) (hA : e H = realCode A) (N : ℕ) (hn : e n = natSet N) :
(columnSet o Q n H B).Eval e ↔ e B = realCode (columns A N) := by
simp only [columnSet,Formula.Eval,eval_subset,eval_allMem,eval_iff,cons_zero,cons_succ]
rw [ho]
constructor
· rintro ⟨hsub,h⟩
apply ZFSet.ext; intro w
by_cases hw : w ∈ ZFSet.omega.{u}
· obtain ⟨k,rfl⟩ := (mem_omega w).mp hw
rw [natSet_mem_realCode]
exact (h _ ((mem_omega _).mpr ⟨k,rfl⟩)).trans
(columnBit_spec (o+1) (Q+1) (n+1) (H+1) 0 (cons (natSet k) e) ho hQ A hA N k hn rfl)
· exact ⟨fun h => False.elim (hw (hsub h)),fun h => False.elim (hw (realCode_subset _ h))⟩
· intro hb
refine ⟨hb.symm ▸ realCode_subset (columns A N),?_⟩
intro w hw
obtain ⟨k,rfl⟩ := (mem_omega w).mp hw
rw [hb,natSet_mem_realCode]
exact (columnBit_spec (o+1) (Q+1) (n+1) (H+1) 0 (cons (natSet k) e) ho hQ A hA N k hn rfl).symm
end Formula
end TuringRigidity.BoundedSetTheory
end OAI
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