| import OAI.Computability.DegreeRigidity.SetModels.DegreeSetFormula |
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| namespace OAI |
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| namespace TuringRigidity.BoundedSetTheory |
| open EncodedForcing |
| universe u |
| namespace Formula |
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| 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))) |
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| 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 |
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| 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))) |
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| 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 |
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| end TuringRigidity.BoundedSetTheory |
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| end OAI |
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