openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /NaturalDefinitions.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.NaturalPairingFormula | |
| namespace OAI | |
| namespace TuringRigidity.BoundedSetTheory | |
| universe u | |
| def DefinesNatural (φ : Formula) (f : ℕ → ℕ) : Prop := | |
| ∀ e : ℕ → ZFSet.{u}, e 2 = ZFSet.omega → | |
| (∀ g : ℕ → ℕ, ∀ k, finiteNaturalGraph g k ∈ e 3) → ∀ n, e 1 = natSet n → | |
| (φ.Eval e ↔ e 0 = natSet (f n)) | |
| namespace Formula | |
| def unaryVars (o Q x y : ℕ) : ℕ → ℕ | |
| | 0 => y | |
| | 1 => x | |
| | 2 => o | |
| | _+3 => Q | |
| def unary (φ : Formula) (o Q x y : ℕ) : Formula := φ.rename (unaryVars o Q x y) | |
| theorem unary_spec {φ : Formula} {f : ℕ → ℕ} (hφ : DefinesNatural.{u} φ f) | |
| (o Q x y : ℕ) (e : ℕ → ZFSet.{u}) (ho : e o = ZFSet.omega) | |
| (hQ : ∀ g : ℕ → ℕ, ∀ k, finiteNaturalGraph g k ∈ e Q) (n : ℕ) (hx : e x = natSet n) : | |
| (unary φ o Q x y).Eval e ↔ e y = natSet (f n) := by | |
| rw [unary,eval_rename] | |
| exact hφ (fun i => e (unaryVars o Q x y i)) ho hQ n hx | |
| def numericComposition (φ ψ : Formula) : Formula := .existsMem 2 | |
| (.conj (unary ψ 3 4 2 0) (unary φ 3 4 0 1)) | |
| def numericPair (φ ψ : Formula) : Formula := .existsMem 2 (.existsMem 3 | |
| (.conj (unary φ 4 5 3 1) (.conj (unary ψ 4 5 3 0) (naturalPair 4 5 1 0 2)))) | |
| def pairedUnary (φ : Formula) (o Q a b y : ℕ) : Formula := .existsMem o | |
| (.conj (naturalPair (o+1) (Q+1) (a+1) (b+1) 0) (unary φ (o+1) (Q+1) 0 (y+1))) | |
| theorem pairedUnary_spec {φ : Formula} {f : ℕ → ℕ} (hφ : DefinesNatural.{u} φ f) | |
| (o Q x n y : ℕ) (e : ℕ → ZFSet.{u}) (ho : e o = ZFSet.omega) | |
| (hQ : ∀ g : ℕ → ℕ, ∀ k, finiteNaturalGraph g k ∈ e Q) | |
| (a b : ℕ) (ha : e x = natSet a) (hb : e n = natSet b) : | |
| (pairedUnary φ o Q x n y).Eval e ↔ e y = natSet (f (Nat.pair a b)) := by | |
| have hp (w : ZFSet.{u}) := naturalPair_spec (o+1) (Q+1) (x+1) (n+1) 0 (cons w e) ho hQ a b ha hb | |
| have hu := unary_spec hφ (o+1) (Q+1) 0 (y+1) (cons (natSet (Nat.pair a b)) e) ho hQ (Nat.pair a b) rfl | |
| simp only [pairedUnary,Formula.Eval] | |
| constructor | |
| · rintro ⟨w,_,hw,h⟩ | |
| obtain rfl := (hp w).mp hw | |
| exact hu.mp h | |
| · intro h | |
| refine ⟨natSet (Nat.pair a b),?_,(hp _).mpr rfl,hu.mpr h⟩ | |
| rw [ho]; exact (mem_omega _).mpr ⟨Nat.pair a b,rfl⟩ | |
| end Formula | |
| theorem definesNatural_zero : DefinesNatural.{u} (.empty 0) (fun _ => 0) := by | |
| intro e _ _ n _ | |
| exact Formula.eval_empty 0 e | |
| theorem definesNatural_succ : DefinesNatural.{u} (.successor 0 1) Nat.succ := by | |
| intro e _ _ n hn | |
| rw [Formula.eval_successor,hn] | |
| rfl | |
| theorem DefinesNatural.comp {φ ψ : Formula} {f g : ℕ → ℕ} | |
| (hf : DefinesNatural.{u} φ f) (hg : DefinesNatural.{u} ψ g) : | |
| DefinesNatural.{u} (Formula.numericComposition φ ψ) (fun n => f (g n)) := by | |
| intro e ho hQ n hn | |
| have hv (w : ZFSet.{u}) := Formula.unary_spec hg 3 4 2 0 (cons w e) ho hQ n hn | |
| have hu := Formula.unary_spec hf 3 4 0 1 (cons (natSet (g n)) e) ho hQ (g n) rfl | |
| simp only [Formula.numericComposition,Formula.Eval] | |
| constructor | |
| · rintro ⟨w,_,hw,h⟩ | |
| obtain rfl := (hv w).mp hw | |
| exact hu.mp h | |
| · intro h | |
| refine ⟨natSet (g n),?_,(hv _).mpr rfl,hu.mpr h⟩ | |
| rw [ho]; exact (mem_omega _).mpr ⟨g n,rfl⟩ | |
| theorem DefinesNatural.pair {φ ψ : Formula} {f g : ℕ → ℕ} | |
| (hf : DefinesNatural.{u} φ f) (hg : DefinesNatural.{u} ψ g) : | |
| DefinesNatural.{u} (Formula.numericPair φ ψ) (fun n => Nat.pair (f n) (g n)) := by | |
| intro e ho hQ n hn | |
| have hv (w z : ZFSet.{u}) := Formula.unary_spec hf 4 5 3 1 (cons z (cons w e)) ho hQ n hn | |
| have hw (w z : ZFSet.{u}) := Formula.unary_spec hg 4 5 3 0 (cons z (cons w e)) ho hQ n hn | |
| have hp := Formula.naturalPair_spec 4 5 1 0 2 (cons (natSet (g n)) (cons (natSet (f n)) e)) | |
| ho hQ (f n) (g n) rfl rfl | |
| have hmem (k : ℕ) : natSet.{u} k ∈ e 2 := by rw [ho]; exact (mem_omega _).mpr ⟨k,rfl⟩ | |
| simp only [Formula.numericPair,Formula.Eval] | |
| constructor | |
| · rintro ⟨w,_,z,_,hvw,hwz,h⟩ | |
| obtain rfl := (hv w z).mp hvw | |
| obtain rfl := (hw _ z).mp hwz | |
| exact hp.mp h | |
| · intro h | |
| exact ⟨natSet (f n),hmem _,natSet (g n),hmem _,(hv _ _).mpr rfl,(hw _ _).mpr rfl,hp.mpr h⟩ | |
| end TuringRigidity.BoundedSetTheory | |
| end OAI | |