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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.Computability.FourierTransform.WordCompilation
import OAI.Computability.FourierTransform.DisjointIntervals
namespace OAI
namespace PowerSaving.RAM
open Ty Slant Slant.Stroke Complex
noncomputable section
universe U V
namespace Bench
variable {m : Bool} {ι : Type U} {κ : Type V} {b : Bench m ι} {q : Bench m κ}
structure Spots (b : Bench m ι) (u : ι→Zone) : Prop where
low : b.Digit (fun i => (u i).lo)
wide : b.Digit (fun i => (u i).wide)
namespace Spots
variable {u v : ι→Zone} {n : ι→ℕ}
lemma transfer {f : ι→κ} {x : κ→Zone} (h : q.Spots x) (g : b.Feeds q f) :
b.Spots (fun i => x (f i)) := ⟨draw g h.low,draw g h.wide⟩
lemma has (h : b.Spots u) {x} (hh : b.Digit x) :
b.Study (fun i => (u i).has (x i)) :=
(Study.le h.low hh).both (Study.lt hh (h.low.add h.wide))
lemma overlay (h : b.Spots u) {x} (hh : b.Digit x) {l : ι→Line}
(g : b.Fabric (fun i => (u i).wide) l) {p} (hk : b.Hue p) :
b.Hue (fun i => if (u i).has (x i) then l i (x i-(u i).lo) else p i) := by
let P i := (u i).has (x i)
have H := h.has hh
refine Hue.fork H ?_ (hk.transfer (restricted b fun i => ¬P i))
have M := restricted b P
have hu (i : {i // P i}) := (u i.val).enter i.property
exact (g.transfer M).pick (draw M (hh.sub h.low)).run hu
lemma shade (h : b.Spots u) (k : b.Spots v) {l g : ι→Line}
(hf : b.Fabric (fun i => (u i).wide) l)
(hg : b.Fabric (fun i => (v i).wide) g) (hn : b.Digit n)
(ha : ∀ i,(u i).Fits (n i)) (hb : ∀ i,(v i).Fits (n i)) :
b.Engrave n (fun i => Slant.shade (u i) (v i) (l i) (g i)) := by
unfold Slant.shade
refine Engrave.single (Band.fresh hn ?_ ?_)
· intro i; exact sound_shade (f:=l i) (g:=g i) (ha i) (hb i) 0 (show 0<1 by omega)
· have H := Feeds.scan b n
have hx := slot hn
exact hx.run.pair ((((h.transfer H).overlay hx (hf.transfer H)
((k.transfer H).overlay hx (hg.transfer H) Hue.one)).bright).pair hx.run)
lemma finds (h : b.Spots u) (k : b.Spots v) {x} (hh : b.Digit x) :
b.Digit (fun i => Slant.jumps (u i) (v i) (x i)) :=
iteN (h.has hh) (k.low.add (hh.sub h.low))
(iteN (k.has hh) (h.low.add (hh.sub k.low)) hh)
lemma snap (h : b.Spots u) (k : b.Spots v) (hn : b.Digit n)
(ha : ∀ i,(u i).Fits (n i)) (hb : ∀ i,(v i).Fits (n i))
(hx : ∀ i,(u i).wide=(v i).wide) (hg : ∀ i,(u i).Sep (v i)) :
b.Engrave n (fun i => Slant.snap (u i) (v i)) := by
unfold Slant.snap
refine Engrave.single (Band.fresh hn ?_ ?_)
· intro i; exact sound_snap (ha i) (hb i) (hx i) (hg i) 0 (show 0<1 by omega)
· have H := Feeds.scan b n
exact (slot hn).run.pair (Makes.one.pair
(((h.transfer H).finds (k.transfer H) (slot hn)).run))
lemma slim (h : b.Spots u) (k : b.Digit n) :
b.Spots (fun i => (u i).slim (n i)) := ⟨h.low,k⟩
end Spots
/- Numeric convenience -/
lemma Hue.smallNumeral (b : Bench m ι) (n : ℕ) [Nat.AtLeastTwo n] :
b.Hue (fun _ => (OfNat.ofNat n:ℂ)) := (Hue.numeral n).congr fun _ => rfl
section
variable {n : ι→ℕ} {u v : ι→Zone} (h : b.Spots u) (k : b.Spots v)
(hn : b.Digit n)
(ha : ∀ i,(u i).Fits (n i)) (hb : ∀ i,(v i).Fits (n i))
(hx : ∀ i,(u i).wide=(v i).wide) (hg : ∀ i,(u i).Sep (v i))
(hy : b.Hue fun _ => I)
include h k hn ha hb hx hg hy
lemma Spots.hinge :
b.Engrave n (fun i => Slant.hinge (u i) (v i)) :=
((h.shade k Fabric.one (Fabric.flat hy) hn ha hb).stack (h.snap k hn ha hb hx hg)).stack
(h.shade k (Fabric.flat (Hue.one.sub hy)) (Fabric.flat (Hue.one.add hy)) hn ha hb)
lemma Spots.brick :
b.Engrave n (fun i => Slant.brick (u i) (v i)) :=
have H := h.hinge k hn ha hb hx hg hy
have p := h.shade k (Fabric.flat (Hue.smallNumeral b 3).neg) Fabric.one hn ha hb
have q := h.shade k (Fabric.flat (Hue.smallNumeral b 2)) Fabric.one hn ha hb
(((H.stack p).stack H).stack q).stack H
lemma Spots.spark {f : ι→Line}
(hh : b.Fabric (fun i => (u i).wide) f) (hf : ∀ i, ∀ j<(u i).wide, f i j ≠ 0) :
b.Engrave n (fun i => Slant.spark (u i) (v i) (f i)) := by
let F (i : ι) : Line := fun j => 4*f i j/5
let U i := (u i).wide
have H : b.Fabric U F :=
Hue.div ((Hue.smallNumeral (b.scan U) 4).mul hh) (Hue.smallNumeral _ 5)
(by intro i; norm_num)
have hx' : ∀ i, ∀ j<U i, F i j≠0 := fun i j hj => by simp [F,hf i j hj]
have hc6 : b.Hue (fun _ => -1/6) := Hue.one.neg.div (Hue.smallNumeral _ 6) (by
intro i; norm_num)
have hc8 : b.Hue (fun _ => -1/8) := Hue.one.neg.div (Hue.smallNumeral _ 8) (by
intro i; norm_num)
have hP := h.shade k (H.inv hx') Fabric.one hn ha hb
have hQ := h.shade k ((Fabric.flat hc8).mul H) (Fabric.flat hc6) hn ha hb
exact (hP.stack (h.brick k hn ha hb hx hg hy)).stack hQ
lemma Spots.flash {f : ι→Line} (hh : b.Fabric (fun i => (u i).wide) f) :
b.Engrave n (fun i => Slant.flash (u i) (v i) (f i)) := by
have hk : b.Fabric (fun i => (u i).wide) (fun i => blaze (f i)) :=
Fabric.one.add (hh.mul hh.flip)
exact (h.spark k hn ha hb hx hg hy hk (fun i j _ => blaze_ne _ _)).stack
(h.spark k hn ha hb hx hg hy (hh.sub hk) (fun i j _ => after_ne _ _))
end
lemma engrave_cameo {n : ι→ℕ} {p u v z : ι→Reel}
(hp : b.Engrave n p) (hu : b.Engrave n u)
(hv : b.Engrave n v) (hz : b.Engrave n z) :
b.Engrave n fun i => cameo (p i) (u i) (v i) (z i) :=
((hp.stack hu).stack hv).stack hz
end Bench
end
end PowerSaving.RAM
end OAI
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