openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /NaturalIterationFormula.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.NaturalIterationTrace | |
| import OAI.Computability.DegreeRigidity.OrdinalCodes.NaturalAdditionTrace | |
| namespace OAI | |
| namespace TuringRigidity.BoundedSetTheory | |
| open TransitiveNameModel | |
| universe u | |
| namespace Formula | |
| theorem addition_spec (o Q x n y : ℕ) (e : ℕ → ZFSet.{u}) | |
| (ho : e o = ZFSet.omega) (hQ : ∀ f : ℕ → ℕ, ∀ k, finiteNaturalGraph f k ∈ e Q) | |
| (a b : ℕ) (ha : e x = natSet a) (hb : e n = natSet b) : | |
| (addition o Q x n y).Eval e ↔ e y = natSet (a+b) := by | |
| have h := additionFormula_correct (e Q) hQ a b (e y) | |
| rw [eval_addition] at h | |
| simp only [cons_zero,cons_succ] at h | |
| rw [eval_addition,ho,ha,hb] | |
| exact h | |
| def iterationOuterVars : ℕ → ℕ | |
| | 0 => 0 | |
| | 1 => 1 | |
| | 2 => 2 | |
| | i+3 => i+6 | |
| def iteration (o Q n b y : ℕ) (θ : Formula) : Formula := | |
| .existsMem o (.existsMem (o+1) (.existsMem (Q+2) | |
| (.conj (iterationTrace (o+3) 1 2 (n+3) (b+3) 0 (θ.rename iterationOuterVars)) | |
| (.pairMem (n+3) (y+3) 0)))) | |
| theorem eval_iteration (o Q n b y : ℕ) (θ : Formula) (e : ℕ → ZFSet.{u}) : | |
| (iteration o Q n b y θ).Eval e ↔ ∃ k ∈ e o, ∃ z ∈ e o, ∃ f ∈ e Q, | |
| IterationTrace (e o) z k (e n) (e b) f | |
| (fun i v w => θ.Eval (cons w (cons v (cons i e)))) ∧ ZFSet.pair (e n) (e y) ∈ f := by | |
| simp only [iteration,Formula.Eval,eval_iterationTrace,eval_pairMem,eval_rename,cons_zero,cons_succ] | |
| have eq (k z f i v w : ZFSet.{u}) : | |
| (fun n => cons w (cons v (cons i (cons f (cons z (cons k e))))) (iterationOuterVars n)) = | |
| cons w (cons v (cons i e)) := by | |
| funext n | |
| rcases n with _|n; rfl | |
| rcases n with _|n; rfl | |
| rcases n with _|n <;> rfl | |
| simp only [eq] | |
| theorem iteration_spec (o Q n b y : ℕ) (θ : Formula) (e : ℕ → ZFSet.{u}) | |
| (ho : e o = ZFSet.omega) (hQ : ∀ f : ℕ → ℕ, ∀ k, finiteNaturalGraph f k ∈ e Q) | |
| (N B : ℕ) (hn : e n = natSet N) (hb : e b = natSet B) (g : ℕ → ℕ → ℕ) | |
| (hθ : ∀ i v w, θ.Eval (cons w (cons (natSet v) (cons (natSet i) e))) ↔ w = natSet (g i v)) : | |
| (iteration o Q n b y θ).Eval e ↔ e y = natSet (Nat.rec B g N) := by | |
| rw [eval_iteration,ho,hn,hb] | |
| constructor | |
| · rintro ⟨k,_,z,_,f,_,hf,hy⟩ | |
| exact hf.correct B N g _ (fun i v w h => (hθ i v w).mp h) hy | |
| · intro hy | |
| refine ⟨natSet (N+1),(mem_omega _).mpr ⟨N+1,rfl⟩,natSet 0,(mem_omega _).mpr ⟨0,rfl⟩, | |
| finiteNaturalGraph (fun i => Nat.rec B g i) N,hQ _ _,?_,?_⟩ | |
| · exact iterationTrace_exists B g _ (fun i v => (hθ i v _).mpr rfl) N | |
| · exact (finiteNaturalGraph_pair _ N N _).mpr ⟨le_rfl,hy⟩ | |
| end Formula | |
| end TuringRigidity.BoundedSetTheory | |
| end OAI | |