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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 | import OAI.Computability.FourierTransform.BinaryFFT
namespace OAI
/-! Circulant addition between disjoint dirty intervals by two transformed words and inverses. -/
namespace PowerSaving.Slant
open Reel Zone Stroke Spectral Finset
noncomputable section
def eddy (d : ℕ) (r : ℂ) (u v : Zone) (h : Line) : Reel :=
let p := pulse d r; let b := backPulse d r
let f := flash v u (freq (2^d) r h)
((((atBase u.lo p).stack (atBase v.lo p)).stack f).stack (atBase u.lo b)).stack
(atBase v.lo b)
lemma beat_mounted (p : ℕ) (r : Reel) : (atBase p r).beats=r.beats := rfl
lemma beats_eddy (d : ℕ) (r : ℂ) (u v : Zone) (h : Line) :
(eddy d r u v h).beats=20*d+28 := by
unfold eddy; dsimp only
simp only [stack,beat_mounted,beats_pulse,beats_flash,beats_back]
omega
lemma Zone.Fits.end_le {v : Zone} {n : ℕ} (hv : v.Fits n) (hp : 0<v.wide) :
v.lo+v.wide ≤ n := by
have := hv _ (v.inside (by omega : v.wide-1<v.wide))
omega
lemma Zone.Fits.mount {v : Zone} {n : ℕ} (hv : v.Fits n) (hp : 0<v.wide)
{r : Reel} (h : r.Sound v.wide) :
(atBase v.lo r).Sound n :=
(h.atBase _).wider (hv.end_le hp)
section
variable {u v : Zone} {d n : ℕ} (r : ℂ) (h : Line) (hu : u.wide=2^d)
(hv : v.wide=2^d) (hs : u.Sep v)
include hu hv hs
theorem sound_eddy (pu : u.Fits n) (pv : v.Fits n) :
(eddy d r u v h).Sound n := by
have HU {a : Reel} (h : a.Sound (2^d)) :=
pu.mount (by rw [hu]; positivity) (show a.Sound u.wide by rwa [hu])
have HV {a : Reel} (h : a.Sound (2^d)) :=
pv.mount (by rw [hv]; positivity) (show a.Sound v.wide by rwa [hv])
have hq : v.wide=u.wide := hv.trans hu.symm
exact ((((HU (sound_pulse d r)).stack (HV (sound_pulse d r))).stack
(sound_flash pv pu hq hs.symm _)).stack (HU (sound_back d r))).stack
(HV (sound_back d r))
theorem spill_eddy (hr : IsPrimitiveRoot r (2^d)) :
Spill (eddy d r u v h) u v (cyclic (2^d) h) := by
let q := 2^d
let f := freq q r
let g := fun (x : Line) (i : ℕ) => (q:ℂ)⁻¹ * freq q r⁻¹ x i
let p := pulse d r; let b := backPulse d r
let pa := atBase u.lo p; let pb := atBase v.lo p
let ba := atBase u.lo b; let bb := atBase v.lo b
let fl := flash v u (f h)
have hq : v.wide=u.wide := hv.trans hu.symm
have h1 : Retrace pa u f := atBase_retrace (by rw [hu]; exact sound_pulse d _)
(by rw [hu]; exact flows_pulse d _ hr)
have h2 : Retrace pb v f := atBase_retrace (by rw [hv]; exact sound_pulse d _)
(by rw [hv]; exact flows_pulse d _ hr)
have h4 : Retrace ba u g := atBase_retrace (by rw [hu]; exact sound_back d _)
(by rw [hu]; exact flows_back d _ hr)
have h5 : Retrace bb v g := atBase_retrace (by rw [hv]; exact sound_back d _)
(by rw [hv]; exact flows_back d _ hr)
have h3 : Spill fl u v (fun x i => f h i*x i) := spill_flash hq hs.symm (f h)
have gf (x y : Line) (i : ℕ) (hi : i<q) (H : ∀ i<q, x i=f y i) :
g x i=y i := by
unfold g
rw [freq_congr H]
exact freq_inv hr y i hi
change u.wide=q at hu; change v.wide=q at hv
have clean (x : Line) (i : ℕ) (hi : i<u.wide) :
u.peek ((eddy d r u v h).go x) i=u.peek x i := by
change u.peek (((((pa.stack pb).stack fl).stack ba).stack bb).go x) i=_
simp only [go_stack]
rw [retrace_preserve h5 hs _ i hi,h4.draws _ i hi]
refine gf _ _ i (hu ▸ hi) ?_
intro j hj
have hk : j<u.wide := hu.symm ▸ hj
rw [h3.preserves hs _ j hk,retrace_preserve h2 hs _ j hk,h1.draws _ j hk]
constructor
· intro x i hi
change v.peek (((((pa.stack pb).stack fl).stack ba).stack bb).go x) i=_
simp only [go_stack]
rw [h5.draws _ i hi]
refine gf _ (v.peek x + cyclic q h (u.peek x)) i (hv ▸ hi) ?_
intro j hj
have hk : j<v.wide := hv.symm ▸ hj
have hp : j<u.wide := hq ▸ hk
rw [retrace_preserve h4 hs.symm _ j hk,h3.adds _ j hk,h2.draws _ j hk]
rw [retrace_preserve h2 hs _ j hp, h1.draws _ j hp]
have hx : freq q r (v.peek (pa.go x)) = f (v.peek x) := freq_congr
(fun j hj => retrace_preserve h1 hs.symm _ j (hv.symm ▸ hj)) r
unfold f
rw [hx, freq_add]
exact congr_arg (freq q r (v.peek x) j + ·) (freq_cyclic hr.pow_eq_one _ _ _).symm
· intro x i hi
by_cases hh : u.has i
· have H := clean x (i-u.lo) (u.enter hh)
simpa [Zone.peek,Stroke.drift,u.identify hh] using H
· change ((((pa.stack pb).stack fl).stack ba).stack bb).go x i=_
simp only [go_stack]
rw [h5.waits _ i hi,h4.waits _ i hh,h3.keeps _ i hi,h2.waits _ i hi,h1.waits _ i hh]
end
lemma cyclic_reads (v : ℕ) (h : Line) : Reads v v (cyclic v h) := by
intro x y hxy i hi
unfold cyclic
exact sum_congr rfl fun _ hj => congr_arg (_*·) (hxy _ (mem_range.mp hj))
lemma cyclic_add (v : ℕ) (h : Line) : AdditiveOn v (cyclic v h) := by
intro x y i _
simp [cyclic,mul_add,sum_add_distrib]
lemma cyclic_neg (v : ℕ) (h : Line) :
cyclic v (fun i => -h i) = minusFn (cyclic v h) := by
ext x i
simp [cyclic,minusFn]
end
end PowerSaving.Slant
end OAI
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