File size: 3,856 Bytes
6fe30b6 | 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 | import OAI.Computability.FourierTransform.CompilerEnvironment
namespace OAI
namespace PowerSaving.RAM
open Ty
noncomputable section
universe U V
namespace Bench
variable {m : Bool} {ι : Type U} {κ : Type V} {b : Bench m ι} {q : Bench m κ}
lemma restricted (b : Bench m ι) (p : ι→Prop) :
(b.scrutin p).Feeds b Subtype.val := Feeds.along b _
/-- Branches tested by integer registers. -/
def Study (b : Bench m ι) (P : ι→Prop) : Prop :=
∃ x : ι→ℕ, b.Digit x ∧ ∀ i, P i ↔ x i ≠ 0
namespace Study
variable {P Q : ι→Prop}
lemma transport {P : κ→Prop} {f : ι→κ} (h : b.Feeds q f) (k : q.Study P) :
b.Study (fun i => P (f i)) :=
let ⟨_,hx,g⟩ := k
⟨_,draw h hx,fun i => g (f i)⟩
lemma equiv (h : b.Study P) (g : ∀ i, P i ↔ Q i) :
b.Study Q :=
let ⟨x,hx,k⟩ := h
⟨x,hx,fun i => (g i).symm.trans (k i)⟩
lemma lt {x y} (h : b.Digit x) (g : b.Digit y) :
b.Study (fun i => x i < y i) :=
⟨_,h.isLess g, fun i => by split <;> simp [*]⟩
lemma neg (h : b.Study P) :
b.Study (fun i => ¬ P i) := by
obtain ⟨x,hx,k⟩ := h
refine (lt hx (Digit.wc 1)).equiv ?_
intro i; rw [k]; omega
lemma le {x y} (h : b.Digit x) (g : b.Digit y) :
b.Study (fun i => x i ≤ y i) :=
((lt g h).neg).equiv (fun i => by simp)
lemma both (h : b.Study P) (g : b.Study Q) :
b.Study (fun i => P i ∧ Q i) := by
obtain ⟨x,hx,k⟩ := h
obtain ⟨y,hy,j⟩ := g
refine ⟨_,hx.mul hy,fun i => ?_⟩
simp [k,j,not_or]
lemma eq {x y} (h : b.Digit x) (g : b.Digit y) :
b.Study (fun i => x i=y i) :=
((le h g).both (le g h)).equiv (fun i => by omega)
end Study
theorem chooseM {t : Ty} {y : ι→T t} {P}
(h : b.Study P)
(a : (b.scrutin P).Makes t (fun i => y i.val))
(c : (b.scrutin (fun i => ¬P i)).Makes t (fun i => y i.val)) :
b.Makes t y := by
obtain ⟨x,hx,g⟩ := h
refine PolyCan.testz hx.run ?_ ?_
· exact PolyCan.reindex c (fun i : {i // x i = 0} => ⟨i.val,by
rw [g]; exact not_not_intro i.property⟩)
· exact PolyCan.reindex a (fun i : {i // x i ≠ 0} => ⟨i.val,(g _).mpr i.property⟩)
theorem branchM {t : Ty} {P : ι→Prop} [DecidablePred P] {x y : ι→T t}
(h : b.Study P)
(a : (b.scrutin P).Makes t (fun i => x i.val))
(c : (b.scrutin (fun i => ¬ P i)).Makes t (fun i => y i.val)) :
b.Makes t (fun i => if P i then x i else y i) :=
chooseM h (a.congr (fun i => (ite_eq_left i.property).symm))
(c.congr (fun i => (ite_eq_right i.property).symm))
theorem splice {F : ι→ℕ} {P : ι→Prop}
(h : Small (b.scrutin P).size (fun i => F i.val))
(g : Small (b.scrutin (fun i => ¬ P i)).size (fun i => F i.val)) :
Small b.size F := by
obtain ⟨x,hx⟩ := h
obtain ⟨y,hy⟩ := g
refine ⟨x+y,fun i => ?_⟩
by_cases hi : P i
· exact (hx ⟨i,hi⟩).trans (Small.pow_mono_degree (b.size i) (by omega))
· exact (hy ⟨i,hi⟩).trans (Small.pow_mono_degree (b.size i) (by omega))
lemma chooseN {x : ι→ℕ} {P} (h : b.Study P)
(a : (b.scrutin P).Digit (fun i => x i.val))
(c : (b.scrutin (fun i => ¬P i)).Digit (fun i => x i.val)) :
b.Digit x := ⟨chooseM h a.run c.run,splice a.small c.small⟩
lemma branchN {P : ι→Prop} [DecidablePred P] {x y : ι→ℕ}
(h : b.Study P)
(a : (b.scrutin P).Digit (fun i => x i.val))
(c : (b.scrutin (fun i => ¬ P i)).Digit (fun i => y i.val)) :
b.Digit (fun i => if P i then x i else y i) :=
chooseN h (a.congr (fun i => (ite_eq_left i.property).symm)) (c.congr
(fun i => (ite_eq_right i.property).symm))
lemma iteN {P : ι→Prop} [DecidablePred P] {x y : ι→ℕ}
(h : b.Study P) (a : b.Digit x) (c : b.Digit y) :
b.Digit (fun i => if P i then x i else y i) :=
branchN h (draw (restricted ..) a) (draw (restricted ..) c)
end Bench
end
end PowerSaving.RAM
end OAI
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