openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /NumericParameters.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.NaturalPrimitiveRecursion | |
| import OAI.Computability.DegreeRigidity.Syntax.SentenceCodeDefinability | |
| namespace OAI | |
| namespace TuringRigidity.NumericSyntax | |
| open BoundedSetTheory TransitiveNameModel BoundedDefinability SentenceCoding | |
| universe u | |
| def params (Q : ZFSet.{u}) : ℕ → ZFSet.{u} := | |
| cons ZFSet.omega (cons Q (fun _ => natSet 0)) | |
| def Covers (Q : ZFSet.{u}) : Prop := ∀ f : ℕ → ℕ, ∀ k, finiteNaturalGraph f k ∈ Q | |
| def Numeric (P : (ℕ → ZFSet.{u}) → Prop) : Prop := | |
| ∃ p : Formula, ∀ Q, Covers Q → ∀ e, p.Eval (mix e (params Q)) ↔ P e | |
| variable {P Q : (ℕ → ZFSet.{u}) → Prop} | |
| theorem Numeric.congr (h : Numeric P) (he : ∀ e, P e ↔ Q e) : Numeric Q := by | |
| obtain ⟨p,hp⟩ := h | |
| exact ⟨p,fun B hB e => (hp B hB e).trans (he e)⟩ | |
| theorem Numeric.subst (h : Numeric P) (r : ℕ → ℕ) : Numeric (fun e => P (e ∘ r)) := by | |
| obtain ⟨p,hp⟩ := h | |
| refine ⟨p.rename (slotMap r id),?_⟩ | |
| intro B hB e | |
| simpa only [Formula.eval_rename_comp,mix_slotMap,Function.comp_id] using hp B hB (e ∘ r) | |
| theorem Numeric.neg (h : Numeric P) : Numeric (fun e => ¬ P e) := by | |
| obtain ⟨p,hp⟩ := h | |
| exact ⟨.neg p,fun B hB e => not_congr (hp B hB e)⟩ | |
| theorem Numeric.and (h : Numeric P) (k : Numeric Q) : Numeric (fun e => P e ∧ Q e) := by | |
| obtain ⟨p,hp⟩ := h; obtain ⟨q,hq⟩ := k | |
| exact ⟨.conj p q,fun B hB e => and_congr (hp B hB e) (hq B hB e)⟩ | |
| theorem Numeric.existsMem (h : Numeric P) (i : ℕ) : | |
| Numeric (fun e => ∃ x ∈ e i, P (cons x e)) := by | |
| obtain ⟨p,hp⟩ := h | |
| refine ⟨.existsMem (2*i) (p.rename bindSlots),?_⟩ | |
| intro B hB e | |
| simp only [Formula.Eval,mix_even,Formula.eval_rename_comp,bind_mix,hp B hB] | |
| theorem Numeric.existsNat (h : Numeric P) : | |
| Numeric (fun e => ∃ x ∈ ZFSet.omega, P (cons x e)) := by | |
| obtain ⟨p,hp⟩ := h | |
| refine ⟨.existsMem 1 (p.rename bindSlots),?_⟩ | |
| intro B hB e | |
| simp only [Formula.Eval,Formula.eval_rename_comp,bind_mix,hp B hB] | |
| rfl | |
| theorem defOr (h : Numeric P) (k : Numeric Q) : Numeric (fun e => P e ∨ Q e) := | |
| (h.neg.and k.neg).neg.congr (fun _ => by tauto) | |
| theorem defImp (h : Numeric P) (k : Numeric Q) : Numeric (fun e => P e → Q e) := | |
| (h.and k.neg).neg.congr (fun _ => by tauto) | |
| theorem defIff (h : Numeric P) (k : Numeric Q) : Numeric (fun e => P e ↔ Q e) := | |
| ((defImp h k).and (defImp k h)).congr (fun _ => iff_def.symm) | |
| theorem defAllMem (h : Numeric P) (i : ℕ) : Numeric (fun e => ∀ x ∈ e i, P (cons x e)) := | |
| (h.neg.existsMem i).neg.congr (fun _ => by simp) | |
| theorem defAllNat (h : Numeric P) : Numeric (fun e => ∀ x ∈ ZFSet.omega, P (cons x e)) := | |
| h.neg.existsNat.neg.congr (fun _ => by simp) | |
| theorem equal_definable (i j : ℕ) : Numeric (fun e => e i = e j) := | |
| ⟨.equal (2*i) (2*j),fun _ _ e => by simp [Formula.Eval]⟩ | |
| theorem member_definable (i j : ℕ) : Numeric (fun e => e i ∈ e j) := | |
| ⟨.member (2*i) (2*j),fun _ _ e => by simp [Formula.Eval]⟩ | |
| theorem memberNat (i : ℕ) : Numeric (fun e => e i ∈ ZFSet.omega) := | |
| ⟨.member (2*i) 1,fun _ _ e => by simp [Formula.Eval,mix,params]⟩ | |
| theorem equalZero (i : ℕ) : Numeric (fun e => e i = natSet 0) := | |
| ⟨.empty (2*i),fun _ _ e => by simp only [Formula.eval_empty,mix_even]; rfl⟩ | |
| theorem defSubset (i j : ℕ) : Numeric (fun e => e i ⊆ e j) := | |
| defAllMem (member_definable 0 (j+1)) i | |
| theorem defPairMem (i j g : ℕ) : Numeric (fun e => ZFSet.pair (e i) (e j) ∈ e g) := | |
| ⟨.pairMem (2*i) (2*j) (2*g),fun _ _ e => by simp only [Formula.eval_pairMem,mix_even]⟩ | |
| theorem defOrderedPair (z i j : ℕ) : Numeric (fun e => e z = ZFSet.pair (e i) (e j)) := | |
| ⟨.orderedPair (2*z) (2*i) (2*j),fun _ _ e => by simp only [Formula.eval_orderedPair,mix_even]⟩ | |
| theorem unaryGraph_definable (f : ℕ → ℕ) (hf : Primrec f) (i j : ℕ) : | |
| Numeric (fun e => UnaryGraph f (e i) (e j)) := by | |
| obtain ⟨p,hp⟩ := primitive_bounded_definition (Primrec.nat_iff.mp hf) | |
| refine ⟨.conj (.member (2*i) 1) (Formula.unary p 1 3 (2*i) (2*j)),?_⟩ | |
| intro B hB e | |
| simp only [Formula.Eval,mix_even] | |
| change (e i ∈ ZFSet.omega ∧ (Formula.unary p 1 3 (2*i) (2*j)).Eval (mix e (params B))) ↔ _ | |
| constructor | |
| · rintro ⟨hn,h⟩ | |
| obtain ⟨n,hn⟩ := (mem_omega _).mp hn | |
| refine ⟨n,hn,?_⟩ | |
| simpa only [mix_even] using (Formula.unary_spec hp 1 3 (2*i) (2*j) (mix e (params B)) rfl hB n | |
| (by simpa only [mix_even] using hn)).mp h | |
| · rintro ⟨n,hn,hj⟩ | |
| exact ⟨(mem_omega _).mpr ⟨n,hn⟩,(Formula.unary_spec hp 1 3 (2*i) (2*j) (mix e (params B)) rfl hB n | |
| (by simpa only [mix_even] using hn)).mpr (by simpa only [mix_even] using hj)⟩ | |
| theorem binaryGraph_definable (f : ℕ → ℕ → ℕ) (hf : Primrec₂ f) (i j k : ℕ) : | |
| Numeric (fun e => BinaryGraph f (e i) (e j) (e k)) := by | |
| have hu : Primrec (Nat.unpaired f) := hf.comp (Primrec.fst.comp Primrec.unpair) (Primrec.snd.comp Primrec.unpair) | |
| obtain ⟨p,hp⟩ := primitive_bounded_definition (Primrec.nat_iff.mp hu) | |
| refine ⟨.conj (.member (2*i) 1) (.conj (.member (2*j) 1) | |
| (Formula.pairedUnary p 1 3 (2*i) (2*j) (2*k))),?_⟩ | |
| intro B hB e | |
| simp only [Formula.Eval,mix_even] | |
| change (e i ∈ ZFSet.omega ∧ e j ∈ ZFSet.omega ∧ | |
| (Formula.pairedUnary p 1 3 (2*i) (2*j) (2*k)).Eval (mix e (params B))) ↔ _ | |
| have hs (a b : ℕ) (ha : e i = natSet a) (hb : e j = natSet b) := | |
| Formula.pairedUnary_spec hp 1 3 (2*i) (2*j) (2*k) (mix e (params B)) rfl hB a b | |
| (by simpa only [mix_even] using ha) (by simpa only [mix_even] using hb) | |
| simp only [mix_even,Nat.unpaired,Nat.unpair_pair] at hs | |
| constructor | |
| · rintro ⟨ha,hb,h⟩ | |
| obtain ⟨a,ha⟩ := (mem_omega _).mp ha | |
| obtain ⟨b,hb⟩ := (mem_omega _).mp hb | |
| exact ⟨a,b,ha,hb,(hs a b ha hb).mp h⟩ | |
| · rintro ⟨a,b,ha,hb,hc⟩ | |
| exact ⟨(mem_omega _).mpr ⟨a,ha⟩,(mem_omega _).mpr ⟨b,hb⟩,(hs a b ha hb).mpr hc⟩ | |
| end TuringRigidity.NumericSyntax | |
| end OAI | |