| import OAI.Computability.DegreeRigidity.OrdinalCodes.TrimmedSumGraph |
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| namespace OAI |
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| namespace TuringRigidity.OrdinalArithmetic |
| open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible OrdinalCoding |
| universe u |
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| 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 |
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| end TuringRigidity.OrdinalArithmetic |
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| end OAI |
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