File size: 4,193 Bytes
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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.NaturalProjectionFormula
import Mathlib.Computability.Primrec.Basic
namespace OAI
namespace TuringRigidity.BoundedSetTheory
universe u
namespace Formula
def primitiveStep (ψ : Formula) : Formula := .existsMem 8
(.conj (naturalPair 9 10 3 2 0) (pairedUnary ψ 9 10 6 0 1))
def primitiveRecursion (φ ψ : Formula) : Formula := .existsMem 2 (.existsMem 3 (.existsMem 4
(.conj (naturalPair 5 6 2 1 4) (.conj (unary φ 5 6 2 0)
(iteration 5 6 1 0 3 (primitiveStep ψ))))))
theorem primitiveStep_spec {ψ : Formula} {g : ℕ → ℕ} (hg : DefinesNatural.{u} ψ g)
(e : ℕ → ZFSet.{u}) (ho : e 5 = ZFSet.omega)
(hQ : ∀ f : ℕ → ℕ, ∀ k, finiteNaturalGraph f k ∈ e 6) (z : ℕ) (hz : e 2 = natSet z)
(i v : ℕ) (w : ZFSet.{u}) :
(primitiveStep ψ).Eval (cons w (cons (natSet v) (cons (natSet i) e))) ↔
w = natSet (g (Nat.pair z (Nat.pair i v))) := by
let E := cons w (cons (natSet v) (cons (natSet i) e))
have hp (t : ZFSet.{u}) := naturalPair_spec 9 10 3 2 0 (cons t E) ho hQ i v rfl rfl
have hu := pairedUnary_spec hg 9 10 6 0 1 (cons (natSet (Nat.pair i v)) E)
ho hQ z (Nat.pair i v) hz rfl
simp only [primitiveStep,Formula.Eval]
constructor
· rintro ⟨t,_,ht,h⟩
obtain rfl := (hp t).mp ht
exact hu.mp h
· intro h
refine ⟨natSet (Nat.pair i v),?_,(hp _).mpr rfl,hu.mpr h⟩
change natSet (Nat.pair i v) ∈ e 5
rw [ho]; exact (mem_omega _).mpr ⟨Nat.pair i v,rfl⟩
end Formula
theorem DefinesNatural.prec {φ ψ : Formula} {f g : ℕ → ℕ}
(hf : DefinesNatural.{u} φ f) (hg : DefinesNatural.{u} ψ g) :
DefinesNatural.{u} (Formula.primitiveRecursion φ ψ)
(Nat.unpaired fun z n => Nat.rec (f z) (fun i v => g (Nat.pair z (Nat.pair i v))) n) := by
intro e ho hQ N hn
have hmem (k : ℕ) : natSet.{u} k ∈ e 2 := by rw [ho]; exact (mem_omega _).mpr ⟨k,rfl⟩
have decode (x : ZFSet.{u}) (hx : x ∈ e 2) : ∃ n, x = natSet n := by
rw [ho] at hx; exact (mem_omega x).mp hx
have hp (z n : ℕ) (b : ZFSet.{u}) := Formula.naturalPair_spec 5 6 2 1 4
(cons b (cons (natSet n) (cons (natSet z) e))) ho hQ z n rfl rfl
have hb (z n : ℕ) (b : ZFSet.{u}) := Formula.unary_spec hf 5 6 2 0
(cons b (cons (natSet n) (cons (natSet z) e))) ho hQ z rfl
have hi (z n : ℕ) := Formula.iteration_spec 5 6 1 0 3 (Formula.primitiveStep ψ)
(cons (natSet (f z)) (cons (natSet n) (cons (natSet z) e))) ho hQ n (f z) rfl rfl
(fun i v => g (Nat.pair z (Nat.pair i v)))
(Formula.primitiveStep_spec hg _ ho hQ z rfl)
simp only [Formula.primitiveRecursion,Formula.Eval]
constructor
· rintro ⟨z,hz,n,hnum,b,_,hp',hb',hi'⟩
obtain ⟨z,rfl⟩ := decode z hz
obtain ⟨n,rfl⟩ := decode n hnum
have hN : N = Nat.pair z n := natSet_injective (hn.symm.trans ((hp z n b).mp hp'))
obtain rfl := (hb z n b).mp hb'
have h := (hi z n).mp hi'
simpa only [cons_succ,hN,Nat.unpaired,Nat.unpair_pair] using h
· intro h
let z := (Nat.unpair N).1
let n := (Nat.unpair N).2
refine ⟨natSet z,hmem _,natSet n,hmem _,natSet (f z),hmem _,?_,(hb z n _).mpr rfl,?_⟩
· apply (hp z n _).mpr
simpa only [cons_succ,z,n,Nat.pair_unpair] using hn
· apply (hi z n).mpr
exact h
theorem primitive_bounded_definition {f : ℕ → ℕ} (hf : Nat.Primrec f) :
∃ φ : Formula, DefinesNatural.{u} φ f := by
induction hf with
| zero => exact ⟨.empty 0,definesNatural_zero⟩
| succ => exact ⟨.successor 0 1,definesNatural_succ⟩
| left => exact ⟨.naturalProjection false,Formula.naturalProjection_spec false⟩
| right => exact ⟨.naturalProjection true,Formula.naturalProjection_spec true⟩
| pair _ _ ihf ihg =>
obtain ⟨φ,hφ⟩ := ihf; obtain ⟨ψ,hψ⟩ := ihg
exact ⟨.numericPair φ ψ,hφ.pair hψ⟩
| comp _ _ ihf ihg =>
obtain ⟨φ,hφ⟩ := ihf; obtain ⟨ψ,hψ⟩ := ihg
exact ⟨.numericComposition φ ψ,hφ.comp hψ⟩
| prec _ _ ihf ihg =>
obtain ⟨φ,hφ⟩ := ihf; obtain ⟨ψ,hψ⟩ := ihg
exact ⟨.primitiveRecursion φ ψ,hφ.prec hψ⟩
end TuringRigidity.BoundedSetTheory
end OAI
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