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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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaStageSyntax
namespace OAI
namespace TuringRigidity.OrdinalArithmetic
open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible
universe u
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)))))))))
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
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⟩
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
end TuringRigidity.OrdinalArithmetic
end OAI
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