openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /BindCanonicalArithmetic.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.CanonicalAdditionSyntax | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SetModelArithmetic | |
| universe u | |
| def squareFormula (w d : ℕ) : Formula := | |
| .conj (.allMem d (.existsMem (w+1) (.existsMem (w+2) (.orderedPair 2 1 0)))) | |
| (.allMem w (.allMem (w+1) (.existsMem (d+2) (.orderedPair 0 2 1)))) | |
| theorem squareFormula_spec (w d : ℕ) (e : ℕ → ZFSet.{u}) : | |
| (squareFormula w d).Eval e ↔ e d = ZFSet.prod (e w) (e w) := by | |
| simp only [squareFormula,Formula.Eval,Formula.eval_allMem,Formula.eval_orderedPair,cons_zero,cons_succ] | |
| constructor | |
| · rintro ⟨h1,h2⟩ | |
| apply ZFSet.ext; intro x | |
| constructor | |
| · intro hx | |
| exact ZFSet.mem_prod.mpr (h1 x hx) | |
| · intro hx | |
| obtain ⟨a,ha,b,hb,rfl⟩ := ZFSet.mem_prod.mp hx | |
| obtain ⟨z,hz,rfl⟩ := h2 a ha b hb | |
| exact hz | |
| · intro h | |
| rw [h] | |
| exact ⟨fun _ hx => ZFSet.mem_prod.mp hx, | |
| fun a ha b hb => ⟨ZFSet.pair a b,ZFSet.pair_mem_prod.mpr ⟨ha,hb⟩,rfl⟩⟩ | |
| noncomputable def bindArithmetic (p : SentenceForm) : SentenceForm := | |
| .ex (.conj (canonicalOmega 0) | |
| (.ex (.conj (fromBounded (squareFormula 1 0)) | |
| (.ex (.conj (fromBounded (additionSystemFormula 2 1 0)) p))))) | |
| theorem bindArithmetic_spec (M : ZFSet.{u}) (hM : Transitive M) | |
| (hw : ZFSet.omega.{u} ∈ M) (hd : pairNumbers.{u} ∈ M) (ha : additionSet.{u} ∈ M) | |
| (p : SentenceForm) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) : | |
| (bindArithmetic p).Sat (M : Set ZFSet) e ↔ | |
| p.Sat (M : Set ZFSet) (cons additionSet (cons pairNumbers (cons ZFSet.omega e))) := by | |
| have hc (x : ZFSet.{u}) (hx : x ∈ M) : ∀ i, cons x e i ∈ M := by | |
| intro i; cases i; exact hx; exact he _ | |
| have hwf (x : ZFSet.{u}) (hx : x ∈ M) := canonicalOmega_spec M hM hw 0 (cons x e) (hc x hx) | |
| have hdf (d : ZFSet.{u}) (hdM : d ∈ M) : | |
| (fromBounded (squareFormula 1 0)).Sat (M : Set ZFSet) (cons d (cons ZFSet.omega e)) ↔ d = pairNumbers := by | |
| rw [bounded_sat,Formula.absolute _ M hM _ | |
| (by intro i; rcases i with _|_|i; exact hdM; exact hw; exact he i),squareFormula_spec] | |
| rfl | |
| have haf (a : ZFSet.{u}) (haM : a ∈ M) : | |
| (fromBounded (additionSystemFormula 2 1 0)).Sat (M : Set ZFSet) | |
| (cons a (cons pairNumbers (cons ZFSet.omega e))) ↔ a = additionSet := by | |
| rw [bounded_sat,Formula.absolute _ M hM _ | |
| (by intro i; rcases i with _|_|_|i; exact haM; exact hd; exact hw; exact he i)] | |
| exact additionSystemFormula_unique 2 1 0 _ rfl rfl | |
| change (∃ w ∈ M, (canonicalOmega 0).Sat (M : Set ZFSet) (cons w e) ∧ | |
| ∃ d ∈ M, (fromBounded (squareFormula 1 0)).Sat (M : Set ZFSet) (cons d (cons w e)) ∧ | |
| ∃ a ∈ M, (fromBounded (additionSystemFormula 2 1 0)).Sat (M : Set ZFSet) (cons a (cons d (cons w e))) ∧ | |
| p.Sat (M : Set ZFSet) (cons a (cons d (cons w e)))) ↔ _ | |
| constructor | |
| · rintro ⟨w,hwM,hwS,d,hdM,hdS,a,haM,haS,hp⟩ | |
| obtain rfl := (hwf w hwM).mp hwS | |
| obtain rfl := (hdf d hdM).mp hdS | |
| obtain rfl := (haf a haM).mp haS | |
| exact hp | |
| · intro hp | |
| exact ⟨_,hw,(hwf _ hw).mpr rfl,_,hd,(hdf _ hd).mpr rfl,_,ha,(haf _ ha).mpr rfl,hp⟩ | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |