File size: 3,902 Bytes
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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.CodeOperationCertificates
import OAI.Computability.DegreeRigidity.OrdinalCodes.OrdinalCodeBranch
namespace OAI
namespace TuringRigidity.OrdinalArithmetic
open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible
open OrdinalCoding
universe u
noncomputable def omegaNextPairSentence : SentenceForm :=
.ex (.ex (.ex (.conj
(fromBounded (.conj (ordinalFormula 2) (.conj (.successor 0 2) (.orderedPair 3 2 1))))
(codePowerAt 1 0))))
theorem omegaNextPairSentence_bound : omegaNextPairSentence.bound = 1 := by
have hbounded :
(fromBounded (.conj (ordinalFormula 2) (.conj (.successor 0 2) (.orderedPair 3 2 1)))).bound = 4 := by
rfl
have hpower : (codePowerAt 1 0).bound ≤ 4 :=
code_rename_bound omegaPowerSentence _ 4 (by intro i _; split <;> omega)
have finish (p q : SentenceForm) (hp : p.bound = 4) (hq : q.bound ≤ 4) :
(SentenceForm.ex (.ex (.ex (.conj p q)))).bound = 1 := by
change max p.bound q.bound - 1 - 1 - 1 = 1
rw [hp, max_eq_left hq]
exact finish _ _ hbounded hpower
theorem omegaNextPairSentence_semantics (M : ZFSet.{u}) (hM : Transitive M)
(e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) :
omegaNextPairSentence.Sat (M : Set ZFSet) e ↔
∃ x ∈ M, ∃ y ∈ M, ∃ k ∈ M,
(x.IsOrdinal ∧ k = insert x x ∧ e 0 = ZFSet.pair x y) ∧
(codePowerAt 1 0).Sat (M : Set ZFSet) (cons k (cons y (cons x e))) := by
change (∃ x ∈ M, ∃ y ∈ M, ∃ k ∈ M, _) ↔ _
apply exists_congr; intro x
apply and_congr_right; intro hx
apply exists_congr; intro y
apply and_congr_right; intro hy
apply exists_congr; intro k
apply and_congr_right; intro hk
have hE : ∀ i, cons k (cons y (cons x e)) i ∈ M := by
intro i; rcases i with _|_|_|i
exact hk; exact hy; exact hx; exact he i
rw [SentenceForm.Sat,bounded_sat,Formula.absolute _ M hM _ hE]
simp only [Formula.Eval,eval_ordinalFormula,Formula.eval_successor,
Formula.eval_orderedPair,cons_zero,cons_succ]
theorem omegaNextPairSentence_at_cut (M : ZFSet.{u}) (hM : Transitive M)
(hω : ZFSet.omega.{u} ∈ M) (a : Ordinal.{u}) (ha : Order.IsSuccLimit a)
(hcut : ∀ c : Ordinal.{u}, c.toZFSet ∈ M ↔ c < a)
(hpow : ∀ c < a, (Ordinal.omega0 ^ c).toZFSet ∈ M ∧ OmegaPowerCertificates M c.toZFSet)
(e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) :
omegaNextPairSentence.Sat (M : Set ZFSet) e ↔
∃ c < a, e 0 = ZFSet.pair c.toZFSet (omegaNext c.toZFSet) := by
rw [omegaNextPairSentence_semantics M hM e he]
constructor
· rintro ⟨x,hx,y,hy,k,hk,⟨ho,hkdef,hz⟩,hp⟩
have hE : ∀ i, cons k (cons y (cons x e)) i ∈ M := by
intro i; rcases i with _|_|_|i
exact hk; exact hy; exact hx; exact he i
have hv := (codePowerAt_sound M hM hω 1 0 _ hE hp).2
change y = (Ordinal.omega0 ^ k.rank).toZFSet at hv
have hsucc : k = (x.rank+1).toZFSet := by
rw [Ordinal.toZFSet_add_one,ho.toZFSet_rank_eq,hkdef]
rw [hsucc,Ordinal.rank_toZFSet] at hv
refine ⟨x.rank,(hcut _).mp (ho.toZFSet_rank_eq.symm ▸ hx),?_⟩
simpa only [ho.toZFSet_rank_eq,omegaNext] using hz.trans (congrArg (ZFSet.pair x) hv)
· rintro ⟨c,hc,hz⟩
have hs : c+1 < a := ha.succ_lt hc
have hx := (hcut c).mpr hc
have hk := (hcut (c+1)).mpr hs
obtain ⟨hy,hcert⟩ := hpow (c+1) hs
refine ⟨c.toZFSet,hx,(Ordinal.omega0 ^ (c+1)).toZFSet,hy,(c+1).toZFSet,hk,
⟨ZFSet.isOrdinal_toZFSet _,Ordinal.toZFSet_add_one c,?_⟩,?_⟩
· simpa only [omegaNext,Ordinal.rank_toZFSet] using hz
· apply codePowerAt_of_certificates M hM hω 1 0 _ (fun i => by
rcases i with _|_|_|i
exact hk; exact hy; exact hx; exact he i) hcert
simp only [cons_zero,cons_succ,Ordinal.rank_toZFSet]
end TuringRigidity.OrdinalArithmetic
end OAI
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