| import OAI.Computability.DegreeRigidity.OrdinalCodes.NumericDomain |
| import OAI.Computability.DegreeRigidity.Constructibility.SeedBoundStageSyntax |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.NumericSyntax |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding RelativeConstructible |
| open BoundedDefinability SetModelFunctions SetModelReals |
| universe u |
| theorem stageStep_definable (s f r x A : ℕ) : |
| Numeric.{u} (fun e => StageStep (e s) (e f) (e r) (e x) (e A)) := by |
| have hn := defAllMem (defOr (member_definable 0 (s+1)) |
| ((((defPairMem 1 0 (f+3)).and (member_definable 2 0)).existsMem (r+2)).existsMem (x+1))) A |
| have hc := defAllMem (defAllMem (defImp (defPairMem 1 0 (f+2)) |
| (defSubset 0 (A+2))) (r+1)) x |
| exact (defSubset s A).and (hn.and hc) |
|
|
| theorem functionGraph_definable (d r f : ℕ) : |
| Numeric.{u} (fun e => FunctionGraph (e d) (e r) (e f)) := by |
| refine ⟨.functionGraph (2*f) (2*d) (2*r),?_⟩ |
| intro B hB e |
| simp only [Formula.eval_functionGraph,mix_even,FunctionGraph] |
|
|
| theorem insertedSelf_definable (s R : ℕ) : |
| Numeric.{u} (fun e => e s = insert (e R) (e R)) := by |
| have h := (member_definable R s).and ((defSubset R s).and |
| (defAllMem (defOr (equal_definable 0 (R+1)) (member_definable 0 (R+1))) s)) |
| apply h.congr |
| intro e |
| constructor |
| · rintro ⟨hR,hsub,hall⟩ |
| apply ZFSet.ext; intro z |
| rw [ZFSet.mem_insert_iff] |
| exact ⟨fun hz => hall z hz,fun h => h.elim (fun he => he ▸ hR) (fun hz => hsub hz)⟩ |
| · rintro he |
| rw [he] |
| exact ⟨ZFSet.mem_insert_iff.mpr (Or.inl rfl), |
| fun _ hz => ZFSet.mem_insert_iff.mpr (Or.inr hz),fun _ hz => ZFSet.mem_insert_iff.mp hz⟩ |
|
|
| theorem checkedDefSuccessor_domainDefinable : |
| Domain.{u} (fun N e => CheckedDefSuccessor N (e 2) (e 3) (e 1) (e 0)) := by |
| have hs := (defSystem_allBounds 5 4 7 3 2 1 0 6).toDomain.existsSet.existsSet.existsSet.existsSet |
| have hb := (powerBound_domain 4 1).and (powerBound_domain 5 0) |
| exact (hb.and hs).existsSet.existsSet |
|
|
| theorem checkedHierarchyGraph_domainDefinable |
| (w sq s d r f : ℕ) : |
| Domain.{u} (fun N e => CheckedHierarchyGraph N (e w) (e sq) (e s) (e d) (e r) (e f)) := by |
| have hd := (defAllMem (defSubset 0 (d+1)) d).toDomain |
| have hf := (functionGraph_definable d r f).toDomain |
| have hv := (checkedDefSuccessor_domainDefinable).subst |
| (fun i => match i with | 0 => 1 | 1 => 0 | 2 => w+3 | _ => sq+3) |
| have hs := (stageStep_definable (s+3) (f+3) (r+3) 2 0).toDomain.and hv |
| have hi := (defPairMem 1 0 (f+2)).toDomain.imp hs.existsSet |
| exact hd.and (hf.and ((hi.allMem (r+1)).allMem d)) |
|
|
| theorem seedBoundStage_domainDefinable : |
| Domain.{u} (fun N e => SeedBoundStage N (e 3) (e 4) (e 2) (e 1) (e 0)) := by |
| have h := (insertedSelf_definable 3 6).toDomain.and |
| ((member_definable 5 2).toDomain.and |
| ((checkedHierarchyGraph_domainDefinable 7 8 3 2 0 1).and |
| (stageStep_definable 3 1 0 5 4).toDomain)) |
| exact h.existsSet.existsSet.existsSet.existsSet |
|
|
| end TuringRigidity.NumericSyntax |
|
|
| end OAI |
|
|
