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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | import OAI.Computability.FourierTransform.ToeplitzCompiler
import OAI.Computability.FourierTransform.FourierWords
namespace OAI
namespace PowerSaving.RAM
open Ty Slant Spectral Complex
noncomputable section
universe U
namespace Bench
variable {m : Bool} {ι : Type U} {b : Bench m ι}
namespace Hue
lemma either {P : ι→Prop} {f : ι→ℂ} (h : b.Study P)
(a : (b.scrutin P).Hue (fun i => f i.val))
(q : (b.scrutin (fun i => ¬P i)).Hue (fun i => f i.val)) :
b.Hue f := ⟨chooseM h a.bright q.bright,
chooseM (t:=sc) (y:=fun i => starRingEnd ℂ (f i)) h a.dull q.dull⟩
lemma gapV {u : ι→ℂ} (h : b.Hue u) {n : ι→ℕ} (hh : b.Digit n) :
b.Hue (fun i => Spectral.gap (u i) (n i)) :=
Hue.prod hh _ (Hue.one.sub
((h.transfer (Feeds.scan b n)).exponent ((slot hh).add (Digit.wc 1))))
lemma roofV {u : ι→ℂ} (h : b.Hue u) {n : ι→ℕ} (hh : b.Digit n) :
b.Hue (fun i => Spectral.roof (u i) (n i)) :=
Hue.prod hh _ (((h.transfer (Feeds.scan b n)).exponent (slot hh)).neg)
end Hue
private abbrev swab := Ty.a (p sc sc)
private def drum (h : Line) (n : ℕ) : T swab :=
Tape.tab n fun i => (recoil h i, starRingEnd ℂ (recoil h i))
namespace Fabric
variable {n : ι→ℕ} {h : ι→Line} (hh : b.Fabric n h) (hg : ∀ i,h i 0≠0)
include hh hg
private lemma firstCoil (hp : ∀ i,0<n i) :
b.Hue (fun i => recoil (h i) 0) :=
((hh.pick (Makes.wc 0) hp).inv hg).congr (fun i => (recoil_zero (h i)).symm)
private lemma refill {k : ι→ℕ} (hk : b.Digit k) (hy : ∀ i,k i+1 < n i)
(hf : b.Fabric (fun i => k i+1) (fun i => recoil (h i))) :
b.Hue (fun i => recoil (h i) (k i+1)) := by
have hx := hk.add (Digit.wc 1)
let U i := k i+1
have F := Feeds.scan b U
have hu (i : During U) : k i.i-i.j<k i.i+1 := by omega
have hv (i : During U) : i.j+1<n i.i := (Nat.add_one_le_iff.mpr i.hj).trans_lt (hy i.i)
have hs := (slot hx)
have hq := ((hf.transfer F).pick ((draw F hk).sub hs).run hu).mul
((hh.transfer F).pick (hs.add (Digit.wc 1)).run hv)
have hsum := Hue.sum hx (fun i j => recoil (h i) (k i-j)*h i (j+1)) hq
have hy' i : 0<n i := lt_trans (show 0<k i+1 by omega) (hy i)
have h0 := hh.pick (Makes.wc 0) hy'
exact (hsum.neg.div h0 hg).congr (fun i => (recoil_next (h i) (k i)).symm)
private lemma oneLonger {k : ι→ℕ} (hk : b.Digit k) (hy : ∀ i,k i+1<n i)
(hf : b.Fabric (fun i => k i+1) (fun i => recoil (h i))) :
b.Fabric (fun i => k i+1+1) (fun i => recoil (h i)) := by
let U i := k i+1
let W i := k i+1+1
have hu := hk.add (Digit.wc 1)
have F := Feeds.scan b W
let T (j : During W) := j.j<U j.i
have hp := slot (hu.add (Digit.wc 1))
have H : (b.scan W).Study T := Study.lt hp (draw F hu)
unfold Bench.Fabric
refine Hue.either H ?_ ?_
· have G := restricted (b.scan W) T
exact ((hf.transfer F).transfer G).pick (draw G hp).run (fun i => i.property)
· have G := restricted (b.scan W) fun i => ¬T i
refine (((hh.refill hg hk hy hf).transfer F).transfer G).congr ?_
intro i
have hh0 : i.val.j < U i.val.i+1 := i.val.hj
have hg0 : ¬i.val.j<U i.val.i := i.property
rw [show i.val.j=U i.val.i by omega]
lemma uncoil (hn : b.Digit n) (hp : ∀ i,0<n i) :
b.Fabric n (fun i => recoil (h i)) := by
let U i k := drum (h i) (k+1)
let y i := n i-1
have hy : b.Digit y := hn.sub (Digit.wc 1)
have hf : b.Makes swab (fun i => drum (h i) (n i)) := by
have H : b.Makes swab (fun i => U i (y i)) := by
refine loop (t:=swab) hy U ?_ ?_
· have hx := hh.firstCoil hg hp
have H1 : b.Fabric (fun _ => 1) (fun i => recoil (h i)) := by
refine (hx.transfer (Feeds.scan b (fun _ => 1))).congr ?_
intro d; rw [show d.j=0 by have := d.hj; omega]
exact H1.supply (Digit.wc 1)
· have F := Feeds.climbs b y U
have hw := stepi hy U
have hq : ∀ j : During y, j.j+1<n j.i := by
intro j; have hx : j.j<n j.i-1 := j.hj; omega
have H1 := (hh.transfer F).oneLonger (fun i => hg _) hw hq (Fabric.recover
(stepsum b y U))
exact H1.supply ((hw.add (Digit.wc 1)).add (Digit.wc 1))
exact H.congr fun i => congrArg (drum (h i)) (show y i+1=n i by unfold y; have := hp i; omega)
exact Fabric.recover hf
end Fabric
lemma print_echo {n : ι→ℕ} (h : b.Digit n) :
b.Engrave n (fun i => echo (n i)) :=
Engrave.lace _ h (fun i _ _ => by omega)
(((draw (Feeds.scan b n) (h.sub (Digit.wc 1))).sub (slot h)).run)
theorem publish {n d : ι→ℕ} {u : ι→ℂ} {r : ι→Line}
(hd : b.Arch d) (hh : b.Digit n) (hn : ∀ i,0<n i) (hv : ∀ i,n i ≤ 2^d i)
(hr : b.Fabric (fun i => d i+1) r) (hy : ∀ i, Springs (d i) (r i))
(h : b.Hue u) (hu : ∀ i, IsPrimitiveRoot (u i) (n i))
(hI : b.Hue fun _ => I) :
b.Engrave n (fun i => ballad (n i) (d i) (u i) (r i)) := by
have F := Feeds.scan b n
have hX : b.Fabric n fun i => gap (u i) := (h.transfer F).gapV (slot hh)
have hx (i : ι) (j : ℕ) (hj : j<n i) : gap (u i) j≠0 :=
gap_ne_zero (u i) (hu i) j hj
let f i := floorH (u i)
have hF : b.Fabric n f := hX.inv hx
have hg := hF.uncoil (fun i => by simp [f,floorH,gap]) hh hn
have hL : b.Library n (fun i => d i+1) f (fun i => recoil (f i)) r :=
⟨hI,hF,hg,hr,fun i j hj => (hy i j (by omega)).ne_zero (by positivity)⟩
have ht := hL.buildHarp hd hh hn (fun _ => le_rfl) (fun i => by exact Nat.lt_succ_self _)
hv
have hY : b.Fabric n fun i => roof (u i) := (h.transfer F).roofV (slot hh)
have hhR : b.Fabric n fun i j => (cog (u i) j)⁻¹ :=
(Fabric.inv (x:=fun i => cog (u i)) (hX.mul hY) (fun i => cog_ne_zero (u i) (hu i)))
have He := print_echo hh
have Hg := Engrave.swatch hh _ hX
have Hh := Engrave.swatch hh _ hY
unfold ballad ode
exact (((Hg.stack ((He.stack ht).stack He)).stack Hh).stack
(Engrave.swatch hh _ hhR)).stack ((Hh.stack ht).stack Hg)
end Bench
end
end PowerSaving.RAM
end OAI
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