| import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaStageSyntax |
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| namespace OAI |
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| namespace TuringRigidity.OrdinalArithmetic |
| open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible |
| universe u |
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| noncomputable def omegaGraphSentence (s w d r f : ℕ) : SentenceForm := |
| .conj (fromBounded (ordinalFormula d)) |
| (.conj (fromBounded (.functionGraph f d r)) |
| (ordinalAllMem d (ordinalAllMem (r+1) |
| (.imp (fromBounded (.pairMem 1 0 (f+2))) |
| (.ex (.conj (fromBounded (omegaStageMatrix (s+3) (f+3) (r+3) 2 0)) |
| (.conj (fromBounded (ordinalFormula 0)) (ordinalProductAt 1 0 (w+3))))))))) |
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| theorem omegaGraphSentence_semantics (M : ZFSet.{u}) (hM : Transitive M) |
| (s w d r f : ℕ) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) : |
| (omegaGraphSentence s w d r f).Sat (M : Set ZFSet) e ↔ |
| (e d).IsOrdinal ∧ FunctionGraph (e d) (e r) (e f) ∧ |
| ∀ x ∈ e d, ∀ v ∈ e r, ZFSet.pair x v ∈ e f → |
| ∃ A ∈ M, StageStep (e s) (e f) (e r) x A ∧ A.IsOrdinal ∧ |
| (ordinalProductAt 1 0 (w+3)).Sat (M : Set ZFSet) (cons A (cons v (cons x e))) := by |
| have hb (p : Formula) : (fromBounded p).Sat (M : Set ZFSet) e ↔ p.Eval e := by |
| rw [bounded_sat,Formula.absolute _ M hM e he] |
| change ((fromBounded _).Sat _ e ∧ (fromBounded _).Sat _ e ∧ |
| (ordinalAllMem d _).Sat _ e) ↔ _ |
| rw [hb,hb,eval_ordinalFormula,Formula.eval_functionGraph, |
| ordinalAllMem_sat M hM d _ e (he d)] |
| apply and_congr_right; intro _ |
| apply and_congr_right; intro _ |
| apply forall_congr'; intro x |
| apply forall_congr'; intro hx |
| have hxM := hM _ (he d) x hx |
| rw [ordinalAllMem_sat M hM (r+1) _ (cons x e) (he r)] |
| apply forall_congr'; intro v |
| apply forall_congr'; intro hv |
| have hvM := hM _ (he r) v hv |
| rw [SentenceForm.sat_imp,bounded_sat,Formula.absolute _ M hM _ (by |
| intro i; rcases i with _|_|i; exact hvM; exact hxM; exact he i),Formula.eval_pairMem] |
| apply forall_congr'; intro _ |
| change (∃ A ∈ M, (fromBounded _).Sat _ _ ∧ (fromBounded _).Sat _ _ ∧ _) ↔ _ |
| apply exists_congr; intro A |
| apply and_congr_right; intro hA |
| have heA : ∀ i, cons A (cons v (cons x e)) i ∈ M := by |
| intro i; rcases i with _|_|_|i; exact hA; exact hvM; exact hxM; exact he i |
| rw [bounded_sat,Formula.absolute _ M hM _ heA,omegaStageMatrix_eval, |
| bounded_sat,Formula.absolute _ M hM _ heA,eval_ordinalFormula] |
| rfl |
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| theorem omegaGraphSentence_sound (M : ZFSet.{u}) (hM : Transitive M) |
| (s w d r f : ℕ) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) |
| (hs : e s = (1 : Ordinal.{u}).toZFSet) (hw : e w = ZFSet.omega) : |
| (omegaGraphSentence s w d r f).Sat (M : Set ZFSet) e → |
| OmegaGraph M (e d) (e r) (e f) := by |
| rw [omegaGraphSentence_semantics M hM s w d r f e he] |
| rintro ⟨hd,hf,hstep⟩ |
| refine ⟨hd,hf,?_⟩ |
| intro x hx v hv hfv |
| obtain ⟨A,hA,hst,ho,hp⟩ := hstep x hx v hv hfv |
| have hwOrd : ZFSet.omega.{u}.IsOrdinal := toZFSet_omega ▸ ZFSet.isOrdinal_toZFSet Ordinal.omega0 |
| have hm := ordinalProductAt_sound M hM 1 0 (w+3) _ (by |
| intro i; rcases i with _|_|_|i; exact hA; exact hM _ (he r) v hv; |
| exact hM _ (he d) x hx; exact he i) ho (by simpa [hw] using hwOrd) hp |
| simp only [cons_zero,cons_succ,hw,← toZFSet_omega,Ordinal.rank_toZFSet] at hm |
| exact ⟨A,hA,hs ▸ hst,ho,hm⟩ |
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| theorem omegaGraphSentence_sourceT (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M) |
| (s w d r f : ℕ) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) |
| (hs : e s = (1 : Ordinal.{u}).toZFSet) (hw : e w = ZFSet.omega) : |
| (omegaGraphSentence s w d r f).Sat (M : Set ZFSet) e ↔ |
| OmegaGraph M (e d) (e r) (e f) := by |
| refine ⟨omegaGraphSentence_sound M hM s w d r f e he hs hw,?_⟩ |
| intro hg |
| apply (omegaGraphSentence_semantics M hM s w d r f e he).mpr |
| refine ⟨hg.1,hg.2.1,?_⟩ |
| intro x hx v hv hfv |
| obtain ⟨A,hA,hst,ho,hp⟩ := hg.2.2 x hx v hv hfv |
| refine ⟨A,hA,hs.symm ▸ hst,ho,?_⟩ |
| have hwOrd : ZFSet.omega.{u}.IsOrdinal := toZFSet_omega ▸ ZFSet.isOrdinal_toZFSet Ordinal.omega0 |
| apply (ordinalProductAt_sourceT M hM hT 1 0 (w+3) _ (by |
| intro i; rcases i with _|_|_|i; exact hA; exact hM _ (he r) v hv; |
| exact hM _ (he d) x hx; exact he i) ho (by simpa [hw] using hwOrd)).mpr |
| simpa only [cons_zero,cons_succ,hw,← toZFSet_omega,Ordinal.rank_toZFSet] using hp |
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| end TuringRigidity.OrdinalArithmetic |
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| end OAI |
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