File size: 3,464 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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.TrimmedSumGraph
namespace OAI
namespace TuringRigidity.OrdinalArithmetic
open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible OrdinalCoding
universe u
theorem sum_graph_spec_with_internal_nodes (A : ZFSet.{u}) (hA : Transitive A) (a b : Ordinal.{u}) (hω : ZFSet.omega.{u} ∈ A)
(ha : a.toZFSet ∈ A) (hb : b.toZFSet ∈ A)
(h0 : (∅ : ZFSet.{u}) ∈ A) (h1 : ({∅} : ZFSet.{u}) ∈ A)
(hlocal : ∀ t < b, (a+t).toZFSet ∈ A ∧ SumCertificates (A) a t)
(hnodes : sumDomain a.toZFSet b.toZFSet ⊆ A)
(z : ZFSet.{u}) (hz : z ∈ A) :
trimmedSumGraphSentence.Sat (A : Set ZFSet)
(cons z (cons a.toZFSet (cons b.toZFSet (cons ∅ (fun _ => {∅}))))) ↔
∃ x ∈ sumDomain a.toZFSet b.toZFSet, z = ZFSet.pair x (sumValue a x) := by
let e := cons z (cons a.toZFSet (cons b.toZFSet (cons ∅ (fun _ => {∅}))))
have he (i : ℕ) : e i ∈ A := by
rcases i with _|_|_|_|i; exact hz; exact ha; exact hb; exact h0; exact h1
rw [trimmedSumGraphSentence_semantics A hA e he]
constructor
· rintro ⟨t,ht,x,hx,y,hy,hleft|⟨hright,hq⟩⟩
· obtain ⟨hta,hxdef,hzdef,hydef⟩ := hleft
change x = leftNode t at hxdef
refine ⟨x,(mem_sumDomain _ _ _).mpr (Or.inl ⟨t,hta,hxdef⟩),?_⟩
change z = ZFSet.pair x y at hzdef
rw [hzdef,hydef,hxdef,sumValue_left]
· obtain ⟨htb,hxdef,hzdef⟩ := hright
change t ∈ b.toZFSet at htb
obtain ⟨t,htb,rfl⟩ := Ordinal.mem_toZFSet_iff.mp htb
have htb := Ordinal.toZFSet_mem_toZFSet_iff.mp htb
have hE : ∀ i, cons y (cons x (cons t.toZFSet e)) i ∈ A := by
intro i; rcases i with _|_|_|i; exact hy; exact hx; exact ht; exact he i
have hydef := codeSumAt_sound A hA 0 4 2 _ hE a t rfl rfl hq
change x = rightNode t.toZFSet at hxdef
refine ⟨x,(mem_sumDomain _ _ _).mpr (Or.inr
⟨t.toZFSet,Ordinal.toZFSet_mem_toZFSet_iff.mpr htb,hxdef⟩),?_⟩
change z = ZFSet.pair x y at hzdef
change y = (a+t).toZFSet at hydef
rw [hzdef,hydef,hxdef,sumValue_right,Ordinal.rank_toZFSet]
· rintro ⟨x,hx,hzdef⟩
rcases (mem_sumDomain _ _ _).mp hx with ⟨t,ht,rfl⟩|⟨t,ht,rfl⟩
· have htA := hA _ ha _ ht
have hxA := hnodes ((mem_sumDomain _ _ _).mpr (Or.inl ⟨t,ht,rfl⟩))
refine ⟨t,htA,_,hxA,t,htA,Or.inl ⟨ht,rfl,?_,rfl⟩⟩
rw [sumValue_left] at hzdef
simpa only [e,cons_zero,leftNode] using hzdef
· obtain ⟨t,ht,rfl⟩ := Ordinal.mem_toZFSet_iff.mp ht
have ht := Ordinal.toZFSet_mem_toZFSet_iff.mp ht
have htA := hA _ hb _ (Ordinal.toZFSet_mem_toZFSet_iff.mpr ht)
have hxA := hnodes ((mem_sumDomain _ _ _).mpr (Or.inr ⟨t.toZFSet,Ordinal.toZFSet_mem_toZFSet_iff.mpr ht,rfl⟩))
obtain ⟨hy,hq⟩ := hlocal t ht
let E := cons (a+t).toZFSet (cons (rightNode t.toZFSet) (cons t.toZFSet e))
have hE (i : ℕ) : E i ∈ A := by
rcases i with _|_|_|i; exact hy; exact hxA; exact htA; exact he i
refine ⟨t.toZFSet,htA,_,hxA,(a+t).toZFSet,hy,Or.inr
⟨⟨Ordinal.toZFSet_mem_toZFSet_iff.mpr ht,rfl,?_⟩,?_⟩⟩
· rw [sumValue_right,Ordinal.rank_toZFSet] at hzdef
simpa only [e,cons_zero,rightNode] using hzdef
· exact codeSumAt_of_certificates A hA hω 0 4 2 E hE a t rfl rfl hq rfl
end TuringRigidity.OrdinalArithmetic
end OAI
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