File size: 3,302 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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaPowerDefinability
namespace OAI
namespace TuringRigidity.OrdinalArithmetic
open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible
universe u
theorem omegaStageStep_domain (M x r f : ZFSet.{u}) (hg : OmegaGraph M x r f) :
StageStep (1 : Ordinal.{u}).toZFSet f r x (Ordinal.omega0 ^ x.rank).toZFSet := by
refine ⟨?_,?_,?_⟩
· intro z hz; exact (omegaPower_members x.rank z).mpr (Or.inl hz)
· intro z hz
rcases (omegaPower_members x.rank z).mp hz with hz|⟨t,ht,hz⟩
· exact Or.inl hz
· rw [hg.1.toZFSet_rank_eq] at ht
obtain ⟨v,hv,hfv,_⟩ := hg.2.1.2 t ht
rw [← hg.correct t ht v hv hfv] at hz
exact Or.inr ⟨t,ht,v,hv,hfv,hz⟩
· intro t ht v hv hfv z hz
rw [hg.correct t ht v hv hfv] at hz
exact (omegaPower_members x.rank z).mpr
(Or.inr ⟨t,hg.1.toZFSet_rank_eq.symm ▸ ht,hz⟩)
theorem omegaStageStep_domain_exact {M x r f A : ZFSet.{u}} (hg : OmegaGraph M x r f)
(hs : StageStep (1 : Ordinal.{u}).toZFSet f r x A) :
A = (Ordinal.omega0 ^ x.rank).toZFSet := by
have ht := omegaStageStep_domain M x r f hg
apply ZFSet.ext; intro z
constructor
· intro hz
rcases hs.2.1 z hz with hz|⟨t,ht',v,hv,hfv,hz⟩
· exact ht.1 hz
· exact ht.2.2 t ht' v hv hfv hz
· intro hz
rcases ht.2.1 z hz with hz|⟨t,ht',v,hv,hfv,hz⟩
· exact hs.1 hz
· exact hs.2.2 t ht' v hv hfv hz
theorem internal_omegaGraph_below (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M)
(x : ZFSet.{u}) (hx : x ∈ M) (ho : x.IsOrdinal) :
∃ f ∈ M, OmegaGraph M x (iterUnion 2 f) f := by
obtain ⟨f,hf,hg⟩ := internal_omegaGraph M hM hT x hx ho
let p : Formula := .existsMem 1 (.existsMem 3 (.orderedPair 2 1 0))
let e : ℕ → ZFSet.{u} := cons x (fun _ => iterUnion 2 f)
let g := ZFSet.sep (fun z => p.Eval (cons z e)) f
have hgM : g ∈ M := sep_mem M hM hT.separation.finitePrefix.bounded p e
(by intro i; cases i; exact hx; exact iterUnion_mem M hM hT.union hf 2) hf
have hm (z : ZFSet.{u}) : z ∈ g ↔ ∃ t ∈ x, z = ZFSet.pair t (omegaNext t) := by
change z ∈ ZFSet.sep _ f ↔ _
simp only [ZFSet.mem_sep,p,Formula.Eval,Formula.eval_orderedPair,cons_zero,cons_succ]
change (z ∈ f ∧ ∃ t ∈ x, ∃ v ∈ iterUnion 2 f, z = ZFSet.pair t v) ↔ _
constructor
· rintro ⟨hz,t,ht,v,hv,rfl⟩
exact ⟨t,ht,by rw [hg.correct t (ZFSet.mem_insert_of_mem x ht) v hv hz]⟩
· rintro ⟨t,ht,rfl⟩
have hp := (hg.mem_iff _).mpr ⟨t,ZFSet.mem_insert_of_mem x ht,rfl⟩
exact ⟨hp,t,ht,_,second_mem_doubleUnion hp,rfl⟩
exact ⟨g,hgM,omegaGraph_of_mem_iff M x g hM hT hx hgM ho hm⟩
theorem OmegaGraph.value_subset_power {M r f : ZFSet.{u}} {a : Ordinal.{u}}
(hg : OmegaGraph M a.toZFSet r f) (t v : ZFSet.{u})
(ht : t ∈ a.toZFSet) (hv : v ∈ r) (hfv : ZFSet.pair t v ∈ f) :
v ⊆ (Ordinal.omega0 ^ a).toZFSet := by
rw [hg.correct t ht v hv hfv,omegaNext]
obtain ⟨b,hb,rfl⟩ := Ordinal.mem_toZFSet_iff.mp ht
rw [Ordinal.rank_toZFSet]
exact Ordinal.toZFSet_monotone
(Ordinal.opow_le_opow_right Ordinal.omega0_pos (Order.succ_le_of_lt hb))
end TuringRigidity.OrdinalArithmetic
end OAI
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