openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /NaturalTermFormula.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.NaturalMultiplicationFormula | |
| namespace OAI | |
| namespace TuringRigidity.BoundedSetTheory | |
| universe u | |
| inductive NaturalTerm where | |
| | var (i : ℕ) | |
| | binary (mul : Bool) (a b : NaturalTerm) | |
| namespace NaturalTerm | |
| def eval (v : ℕ → ℕ) : NaturalTerm → ℕ | |
| | .var i => v i | |
| | .binary m a b => if m then eval v a * eval v b else eval v a + eval v b | |
| def operation (m : Bool) (o Q a b y : ℕ) : Formula := | |
| if m then .multiplication o Q a b y else .addition o Q a b y | |
| theorem operation_spec (m : Bool) (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) : | |
| (operation m o Q x n y).Eval e ↔ e y = natSet (if m then a*b else a+b) := by | |
| cases m with | |
| | false => exact Formula.addition_spec o Q x n y e ho hQ a b ha hb | |
| | true => exact Formula.multiplication_spec o Q x n y e ho hQ a b ha hb | |
| def graph (o Q y : ℕ) (v : ℕ → ℕ) : NaturalTerm → Formula | |
| | .var i => .equal y (v i) | |
| | .binary m a b => .existsMem o (.existsMem (o+1) | |
| (.conj (graph (o+2) (Q+2) 1 (fun i => v i+2) a) | |
| (.conj (graph (o+2) (Q+2) 0 (fun i => v i+2) b) (operation m (o+2) (Q+2) 1 0 (y+2))))) | |
| theorem graph_spec (t : NaturalTerm) (o Q y : ℕ) (v : ℕ → ℕ) (e : ℕ → ZFSet.{u}) | |
| (ho : e o = ZFSet.omega) (hQ : ∀ f : ℕ → ℕ, ∀ k, finiteNaturalGraph f k ∈ e Q) | |
| (a : ℕ → ℕ) (ha : ∀ i, e (v i) = natSet (a i)) : | |
| (graph o Q y v t).Eval e ↔ e y = natSet (t.eval a) := by | |
| induction t generalizing o Q y v e with | |
| | var i => exact congrArg (fun z => e y = z) (ha i) |>.to_iff | |
| | binary m t s iht ihs => | |
| have hωmem (k : ℕ) : natSet.{u} k ∈ e o := by | |
| rw [ho]; exact (mem_omega _).mpr ⟨k,rfl⟩ | |
| have ht (w z : ZFSet.{u}) : | |
| (graph (o+2) (Q+2) 1 (fun i => v i+2) t).Eval (cons z (cons w e)) ↔ w = natSet (t.eval a) := | |
| iht (o+2) (Q+2) 1 (fun i => v i+2) (cons z (cons w e)) ho hQ ha | |
| have hs (w z : ZFSet.{u}) : | |
| (graph (o+2) (Q+2) 0 (fun i => v i+2) s).Eval (cons z (cons w e)) ↔ z = natSet (s.eval a) := | |
| ihs (o+2) (Q+2) 0 (fun i => v i+2) (cons z (cons w e)) ho hQ ha | |
| have hop := operation_spec m (o+2) (Q+2) 1 0 (y+2) | |
| (cons (natSet (s.eval a)) (cons (natSet (t.eval a)) e)) ho hQ (t.eval a) (s.eval a) rfl rfl | |
| simp only [graph,Formula.Eval] | |
| constructor | |
| · rintro ⟨w,_,z,_,hw,hz,h⟩ | |
| obtain rfl := (ht w z).mp hw | |
| obtain rfl := (hs _ z).mp hz | |
| exact hop.mp h | |
| · intro h | |
| refine ⟨natSet (t.eval a),hωmem (t.eval a), | |
| natSet (s.eval a),hωmem (s.eval a), | |
| (ht _ _).mpr rfl,(hs _ _).mpr rfl,hop.mpr h⟩ | |
| end NaturalTerm | |
| end TuringRigidity.BoundedSetTheory | |
| end OAI | |