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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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import OAI.Computability.DegreeRigidity.OrdinalCodes.LimitOmegaCertificates
import OAI.Computability.DegreeRigidity.SetModels.FinitePresentation
namespace OAI
namespace TuringRigidity.OrdinalArithmetic
open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible OrdinalCoding
universe u
noncomputable def boundedOmegaNextPairSentence : SentenceForm :=
.conj (.ex (.conj (.member 0 2) (.ex (fromBounded (.orderedPair 2 1 0))))) omegaNextPairSentence
theorem boundedOmegaNextPairSentence_spec (A : ZFSet.{u}) (hA : Transitive A)
(hω : ZFSet.omega.{u} ∈ A) (a : Ordinal.{u}) (ha : Order.IsSuccLimit a)
(haA : a.toZFSet ∈ A)
(hpow : ∀ c < a, (Ordinal.omega0 ^ c).toZFSet ∈ A ∧ OmegaPowerCertificates A c.toZFSet)
(z : ZFSet.{u}) (hz : z ∈ A) :
boundedOmegaNextPairSentence.Sat (A : Set ZFSet) (cons z (fun _ => a.toZFSet)) ↔
∃ x ∈ a.toZFSet, z = ZFSet.pair x (omegaNext x) := by
let e := cons z (fun _ => a.toZFSet)
have he (i : ℕ) : e i ∈ A := by cases i; exact hz; exact haA
have hbound : (SentenceForm.ex (.conj (.member 0 2) (.ex (fromBounded (.orderedPair 2 1 0))))).Sat
(A : Set ZFSet) e ↔ ∃ x ∈ a.toZFSet, ∃ y ∈ A, z = ZFSet.pair x y := by
change (∃ x ∈ A, x ∈ a.toZFSet ∧ ∃ y ∈ A,
(fromBounded (.orderedPair 2 1 0)).Sat (A : Set ZFSet) (cons y (cons x e))) ↔ _
have hp (x y : ZFSet.{u}) (hx : x ∈ A) (hy : y ∈ A) :
(fromBounded (.orderedPair 2 1 0)).Sat (A : Set ZFSet) (cons y (cons x e)) ↔
z = ZFSet.pair x y := by
rw [bounded_sat,Formula.absolute _ A hA _ (by
intro i; rcases i with _|_|i; exact hy; exact hx; exact he i)]
exact Formula.eval_orderedPair _ _ _ _
constructor
· rintro ⟨x,hx,hxa,y,hy,h⟩
exact ⟨x,hxa,y,hy,(hp x y hx hy).mp h⟩
· rintro ⟨x,hx,y,hy,h⟩
have hxA := hA _ haA _ hx
exact ⟨x,hxA,hx,y,hy,(hp x y hxA hy).mpr h⟩
change (_ ∧ omegaNextPairSentence.Sat (A : Set ZFSet) e) ↔ _
rw [hbound,omegaNextPairSentence_semantics A hA e he]
constructor
· rintro ⟨⟨x,hx,y,hy,hzxy⟩,x',hx',y',hy',k,hk,⟨ho,hks,hzp⟩,hp⟩
have heq : ZFSet.pair x y = ZFSet.pair x' y' := hzxy.symm.trans hzp
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp heq
have hv := (codePowerAt_sound A hA hω 1 0 _ (by
intro i; rcases i with _|_|_|i
exact hk; exact hy'; exact hx'; exact he i) 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,hks]
rw [hsucc,Ordinal.rank_toZFSet] at hv
exact ⟨x,hx,hzxy.trans (congrArg (ZFSet.pair x) hv)⟩
· rintro ⟨x,hx,hzdef⟩
obtain ⟨c,hc,rfl⟩ := Ordinal.mem_toZFSet_iff.mp hx
have hc : c < a := by simpa only [Ordinal.toZFSet_mem_toZFSet_iff] using hc
have hs : c+1 < a := ha.succ_lt hc
have hxA := hA _ haA _ (Ordinal.toZFSet_mem_toZFSet_iff.mpr hc)
have hk := hA _ haA _ (Ordinal.toZFSet_mem_toZFSet_iff.mpr hs)
obtain ⟨hy,hcert⟩ := hpow (c+1) hs
have hzdef' : z = ZFSet.pair c.toZFSet (Ordinal.omega0 ^ (c+1)).toZFSet := by
simpa only [omegaNext,Ordinal.rank_toZFSet] using hzdef
refine ⟨⟨c.toZFSet,Ordinal.toZFSet_mem_toZFSet_iff.mpr hc,_,hy,hzdef'⟩,
c.toZFSet,hxA,_,hy,(c+1).toZFSet,hk,
⟨ZFSet.isOrdinal_toZFSet _,Ordinal.toZFSet_add_one c,hzdef'⟩,?_⟩
apply codePowerAt_of_certificates A hA hω 1 0 _ (fun i => by
rcases i with _|_|_|i; exact hk; exact hy; exact hxA; exact he i) hcert
simp only [cons_zero,cons_succ,Ordinal.rank_toZFSet]
theorem omega_limit_certificates_at_add_five (R : ZFSet.{u}) (γ a : Ordinal.{u})
(ha : Order.IsSuccLimit a) (haA : a.toZFSet ∈ level R γ)
(hω : ZFSet.omega.{u} ∈ level R γ)
(hpow : ∀ c < a, (Ordinal.omega0 ^ c).toZFSet ∈ level R γ ∧
OmegaPowerCertificates (level R γ) c.toZFSet) :
OmegaPowerCertificates (level R (γ+5)) a.toZFSet := by
let A := level R (γ+2)
let f := A.sep (fun z => boundedOmegaNextPairSentence.Sat (A : Set ZFSet)
(cons z (fun _ => a.toZFSet)))
have h02 : γ ≤ γ+2 := le_self_add
have h25 : γ+2 ≤ γ+5 := add_le_add le_rfl (by exact_mod_cast (show (2 : ℕ) ≤ 5 by decide))
have h05 := h02.trans h25
have h35 : γ+3 ≤ γ+5 := add_le_add le_rfl (by exact_mod_cast (show (3 : ℕ) ≤ 5 by decide))
have ha2 := level_mono R h02 haA
have hpow2 (c : Ordinal.{u}) (hc : c < a) :
(Ordinal.omega0 ^ c).toZFSet ∈ A ∧ OmegaPowerCertificates A c.toZFSet :=
⟨level_mono R h02 (hpow c hc).1,(hpow c hc).2.mono (level_mono R h02)⟩
have hspec := boundedOmegaNextPairSentence_spec A (level_transitive R (γ+2))
(level_mono R h02 hω) a ha ha2 hpow2
have hf : f ∈ level R (γ+3) := by
have h := separation_mem_definablePower A boundedOmegaNextPairSentence (fun _ => a.toZFSet)
(fun _ _ => ha2)
change f ∈ definablePower A at h
have h' : f ∈ level R (γ+2+1) := by rw [level_succ]; exact h
simpa only [add_assoc,show (2 : Ordinal.{u})+1=3 by norm_num] using h'
have hexact (z : ZFSet.{u}) : z ∈ f ↔ ∃ x ∈ a.toZFSet, z = ZFSet.pair x (omegaNext x) := by
change z ∈ A.sep _ ↔ _
rw [ZFSet.mem_sep]
constructor
· rintro ⟨hz,h⟩; exact (hspec z hz).mp h
· rintro ⟨x,hx,rfl⟩
obtain ⟨c,hc,rfl⟩ := Ordinal.mem_toZFSet_iff.mp hx
have hc : c < a := by simpa only [Ordinal.toZFSet_mem_toZFSet_iff] using hc
have hy : omegaNext c.toZFSet ∈ level R γ := by
simpa only [omegaNext,Ordinal.rank_toZFSet] using (hpow (c+1) (ha.succ_lt hc)).1
have hz := orderedPair_mem_level_add_two R γ
(level_transitive R γ _ haA _ (Ordinal.toZFSet_mem_toZFSet_iff.mpr hc)) hy
exact ⟨hz,(hspec _ hz).mpr ⟨_,Ordinal.toZFSet_mem_toZFSet_iff.mpr hc,rfl⟩⟩
have hr : iterUnion 2 f ∈ level R (γ+5) := by
simpa only [Nat.cast_ofNat,add_assoc,show (3 : Ordinal.{u})+2=5 by norm_num] using
iterUnion_mem_level_add R (γ+3) hf 2
refine ⟨level_mono R h05 haA,f,level_mono R h35 hf,_,hr,
omegaGraph_of_exact_pairs _ _ f (ZFSet.isOrdinal_toZFSet a) hexact ?_,?_⟩
· intro x hx
obtain ⟨c,hc,rfl⟩ := Ordinal.mem_toZFSet_iff.mp hx
rw [Ordinal.rank_toZFSet]
exact level_mono R h05 (hpow c hc).1
· intro x hx
obtain ⟨c,hc,rfl⟩ := Ordinal.mem_toZFSet_iff.mp hx
obtain ⟨_,g,hg,r,hr,hgraph,hprod⟩ := (hpow (c+1) (ha.succ_lt hc)).2
exact (hprod c.toZFSet (Ordinal.toZFSet_mem_toZFSet_iff.mpr (lt_add_one c))).mono
(level_mono R h05)
end TuringRigidity.OrdinalArithmetic
end OAI
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