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92416a2 | 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 | import OAI.Computability.WLIdentification.PresentationArithmetic
namespace OAI
namespace WLIdentification
namespace FinitePresentation
open WordExpressions NumericFinite
variable {C B : Word → Type} [∀ w, DecidableEq (C w)]
structure Locals (CP : Presented C) (BP : Presented B) where
coordinates : ∀ w, B w → Finset (C w)
size : ∀ w b, (coordinates w b).card ≤ 5
member : Test (BP.prod CP) (fun w p => decide (p.2 ∈ coordinates w p.1))
namespace Locals
variable {CP : Presented C} {BP : Presented B} (D : Locals CP BP)
abbrev Value (w : Word) (b : B w) := (c : D.coordinates w b) → Bit
def good (w : Word) (b c : ℕ) : Bool := CP.good w c && D.member.eval w (Nat.pair b c)
theorem good_iff (w : Word) (b : B w) (i : ℕ) :
D.good w (BP.code w b) i=true ↔ ∃ x : D.coordinates w b, CP.code w x.val=i := by
simp only [good, Bool.and_eq_true]
constructor
· rintro ⟨hc,hm⟩
obtain ⟨c,rfl⟩ := (CP.good_iff w i).mp hc
have h := D.member.correct w (b,c)
change D.member.eval w (Nat.pair (BP.code w b) (CP.code w c))=decide (c∈D.coordinates w b) at h
exact ⟨⟨c,of_decide_eq_true (h ▸ hm)⟩,rfl⟩
· rintro ⟨c,rfl⟩
refine ⟨CP.good_code w c.val,?_⟩
have h := D.member.correct w (b,c.val)
change D.member.eval w (Nat.pair (BP.code w b) (CP.code w c.val))=_ at h
rw [h,decide_eq_true_eq]
exact c.property
noncomputable def localP (w : Word) (b : B w) : Presented (fun _ => D.coordinates w b) :=
Presented.ofCode _ (NaturalCode.subtype (CP.code w) _) (CP.bound w)
(D.good w (BP.code w b)) (fun c => CP.code_lt w c.val) (D.good_iff w b)
def count (w : Word) (b : ℕ) : ℕ := NumericFinite.count (CP.bound w) (D.good w b)
theorem count_eq (w : Word) (b : B w) : D.count w (BP.code w b)=(D.coordinates w b).card := by
change (D.localP w b).card [] = _
rw [Presented.card_eq, Fintype.card_coe]
def rank (w : Word) (b c : ℕ) : ℕ := NumericFinite.count c (D.good w b)
noncomputable def valueEquiv (w : Word) (b : B w) :
D.Value w b ≃ Fin (2^(D.count w (BP.code w b))) :=
(Equiv.arrowCongr ((D.localP w b).rankEquiv []) (ZMod.finEquiv 2).toEquiv.symm).trans finFunctionFinEquiv
noncomputable def valueCode (w : Word) (b : B w) : D.Value w b ↪ ℕ :=
(D.valueEquiv w b).toEmbedding.trans (NaturalCode.fin _)
def valueGood (w : Word) (b v : ℕ) : Bool := decide (v < min (2^(D.count w b)) 32)
theorem valueGood_iff (w : Word) (b : B w) (i : ℕ) :
D.valueGood w (BP.code w b) i=true ↔ ∃ v, D.valueCode w b v=i := by
have h : 2^(D.count w (BP.code w b)) ≤ 32 := by
rw [D.count_eq]
exact Nat.pow_le_pow_right (by decide) (D.size w b)
simp only [valueGood, min_eq_left h, decide_eq_true_eq]
exact ⟨fun hi => ⟨(D.valueEquiv w b).symm ⟨i,hi⟩, congrArg Fin.val ((D.valueEquiv w b).apply_symm_apply _)⟩,
fun ⟨v,hv⟩ => hv ▸ (D.valueEquiv w b v).isLt⟩
theorem good_poly {n} {w : (Fin n → Word) → Word} {b c : (Fin n → Word) → ℕ}
(hw : Poly w) (hb : NatPoly b) (hc : NatPoly c) : BoolPoly (fun S => D.good (w S) (b S) (c S)) :=
BoolPoly.and (CP.good_poly_of hw hc) (D.member.eval_poly hw (NatPoly.pair hb hc))
theorem count_poly {n} {w : (Fin n → Word) → Word} {b : (Fin n → Word) → ℕ}
(hw : Poly w) (hb : NatPoly b) : NatPoly (fun S => D.count (w S) (b S)) := by
apply (NatPoly.filterCount (CP.bound_poly_of hw)
(D.good_poly (hw.rename Fin.castSucc) (hb.rename Fin.castSucc) (NatPoly.var (Fin.last n)))).congr
intro S
simp only [Fin.snoc_comp_castSucc, Fin.snoc_last, unary, List.length_replicate]
rfl
theorem rank_poly {n} {w : (Fin n → Word) → Word} {b c : (Fin n → Word) → ℕ}
(hw : Poly w) (hb : NatPoly b) (hc : NatPoly c) : NatPoly (fun S => D.rank (w S) (b S) (c S)) := by
apply (NatPoly.filterCount hc
(D.good_poly (hw.rename Fin.castSucc) (hb.rename Fin.castSucc) (NatPoly.var (Fin.last n)))).congr
intro S
simp only [Fin.snoc_comp_castSucc, Fin.snoc_last, unary, List.length_replicate]
rfl
noncomputable def values : Presented (fun w => Σ b : B w, D.Value w b) :=
BP.sigma (D.Value) D.valueCode (fun _ => 32) D.valueGood
(by intro w b v; have := (D.valueEquiv w b v).isLt
have h := D.size w b
rw [← D.count_eq w b] at h
exact this.trans_le (Nat.pow_le_pow_right (by decide) h))
D.valueGood_iff (NatPoly.const _)
(NatPoly.lt (NatPoly.var 2) (NatPoly.cappedPow (NatPoly.const 2)
(D.count_poly (Poly.var 0) (NatPoly.var 1)) (NatPoly.const 32)))
def read (w : Word) (b v c : ℕ) : ℕ := NumericKernel.digit 2 v (D.rank w b c)
theorem read_value (w : Word) (b : B w) (v : D.Value w b) (c : D.coordinates w b) :
D.read w (BP.code w b) (D.valueCode w b v) (CP.code w c.val)=(v c).val := by
have h := (D.valueEquiv w b).symm_apply_apply v
have h' := congrArg (fun f => (f c).val) h
exact h'
theorem read_poly {n} {w : (Fin n → Word) → Word} {b v c : (Fin n → Word) → ℕ}
(hw : Poly w) (hb : NatPoly b) (hv : NatPoly v) (hc : NatPoly c) :
NatPoly (fun S => D.read (w S) (b S) (v S) (c S)) :=
NatPoly.digit (NatPoly.const 2) hv (D.rank_poly hw hb hc)
end Locals
end FinitePresentation
end WLIdentification
end OAI
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