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9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | 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
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