| import OAI.Computability.FourierTransform.ToeplitzLayers |
|
|
| namespace OAI |
|
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| /-! Fourier layer word. Correct on each axis at exact width, even when coincident |
| intermediate wires hold arbitrary data. -/ |
| namespace PowerSaving.Slant |
| open Reel Stroke Spectral Zone Matrix Finset |
| noncomputable section |
|
|
| def Creates (p : Reel) (n : ℕ) (M : RootMat n) : Prop := |
| ∀ x, front n (p.go x)=M *ᵥ front n x |
|
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| lemma Creates.stack {p q : Reel} {n} {M N : RootMat n} |
| (h : Creates p n M) (g : Creates q n N) : |
| Creates (p.stack q) n (N*M) := |
| fun x => by rw [go_stack,g,h,Matrix.mulVec_mulVec] |
|
|
| lemma creates_swatch (n : ℕ) (f : Line) : |
| Creates (swatch n f) n (fade n f) := by |
| intro x; ext i |
| change (swatch n f).go x i.val = _ |
| rw [go_swatch n f x _ i.isLt] |
| change f i.val*x i.val=∑ j : Fin n, Matrix.diagonal (front n f) i j*front n x j |
| simp [ite_mul, diagonal_apply,front] |
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|
| lemma front_flows {p : Reel} {n : ℕ} {f : Line→Line} (h : Flows p n f) (x : Line) : |
| front n (p.go x)=front n (f x) := |
| funext fun i => h x _ i.isLt |
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| def echo (n : ℕ) : Reel := lace n fun i => n-1-i |
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| lemma sound_echo (n : ℕ) : (echo n).Sound n := |
| sound_lace _ _ (fun i hi => by omega) |
| lemma echo_go (n : ℕ) (i : ℕ) (hi : i<n) (x : Line) : |
| (echo n).go x i=x (n-1-i) := go_lace n _ x i hi |
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| def coda (n : ℕ) (u : Reel) : Reel := ((echo n).stack u).stack (echo n) |
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| lemma sound_coda (n : ℕ) (u : Reel) (h : u.Sound n) : (coda n u).Sound n := |
| ((sound_echo n).stack h).stack (sound_echo n) |
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| lemma creep_coda (n : ℕ) (u : Reel) (h : Line) (hh : Flows u n (stencil n h)) : |
| Flows (coda n u) n (creep n h) := by |
| intro x i hi |
| unfold coda |
| rw [go_stack,go_stack,echo_go n i hi,hh _ _ (by omega)] |
| exact (congrFun (stencil_congr n h (x:=(echo n).go x) (y:=fun j => x (n-1-j)) |
| (fun j hj => echo_go n j hj x)) _).trans (stencil_reversed n h x i hi) |
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| def voices (n : ℕ) (u : ℂ) (p : Reel) : Reel := |
| ((swatch n (roof u)).stack p).stack (swatch n (gap u)) |
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| def repeatV (n : ℕ) (u : ℂ) (p : Reel) : Reel := |
| ((swatch n (gap u)).stack (coda n p)).stack (swatch n (roof u)) |
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| def ode (n : ℕ) (u : ℂ) (p : Reel) : Reel := |
| ((repeatV n u p).stack (swatch n (fun i => (cog u i)⁻¹))).stack (voices n u p) |
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| def ballad (n : ℕ) (e : ℕ) (u : ℂ) (r : Line) : Reel := |
| let h := floorH u |
| ode n u (harp h (recoil h) r e n) |
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| lemma sound_ode (n : ℕ) (u : ℂ) (p : Reel) (h : p.Sound n) : (ode n u p).Sound n := |
| ((((sound_swatch n _).stack (sound_coda n p h)).stack (sound_swatch n _)).stack |
| (sound_swatch n _)).stack (((sound_swatch n _).stack h).stack (sound_swatch n _)) |
|
|
| lemma beats_ode (n : ℕ) (u : ℂ) (p : Reel) : |
| (ode n u p).beats=2*p.beats+7 := by |
| change 1+(1+p.beats+1)+1+1+(1+p.beats+1)=_ |
| omega |
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|
| lemma fade_flip (n : ℕ) (d : Line) : (fade n d)ᵀ = fade n d := by |
| ext i j |
| rcases eq_or_ne i j with rfl|h <;> simp [fade, eq_comm, *] |
|
|
| lemma creates_voices (n : ℕ) (u : ℂ) (p : Reel) (h : Flows p n (stencil n (floorH u))) : |
| Creates (voices n u p) n (steps n u) := by |
| have hp : Creates p n (stairs n (floorH u)) := |
| fun x => (front_flows h x).trans (front_stencil _ _ x) |
| have H := ((creates_swatch n (roof u)).stack hp).stack (creates_swatch n (gap u)) |
| rw [steps,Matrix.mul_assoc] |
| exact H |
|
|
| lemma creates_repeatV (n : ℕ) (u : ℂ) (p : Reel) (h : Flows p n (stencil n (floorH u))) : |
| Creates (repeatV n u p) n (steps n u)ᵀ := by |
| have hp : Creates (coda n p) n (stairs n (floorH u))ᵀ := |
| fun x => (front_flows (creep_coda n p _ h) x).trans (front_creep _ _ x) |
| have H := ((creates_swatch n (gap u)).stack hp).stack (creates_swatch n (roof u)) |
| rw [steps,transpose_mul,transpose_mul,fade_flip,fade_flip] |
| exact H |
|
|
| lemma creates_ode (n : ℕ) (u : ℂ) (p : Reel) (h : Flows p n (stencil n (floorH u))) |
| (hu : IsPrimitiveRoot u n) : |
| Creates (ode n u p) n (winds n u) := by |
| have H := ((creates_repeatV n u p h).stack (creates_swatch n (fun i => (cog u i)⁻¹))).stack |
| (creates_voices n u p h) |
| rw [winds_reduction n u hu, Matrix.mul_assoc] |
| exact H |
|
|
| lemma sound_ballad (n e : ℕ) (u : ℂ) (r : Line) (h : n ≤ 2^e) : |
| (ballad n e u r).Sound n := |
| sound_ode n u _ (sound_harp _ _ r e n h) |
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| lemma creates_ballad (n e : ℕ) (u : ℂ) (r : Line) (h : n ≤ 2^e) (hr : Springs e r) |
| (hu : IsPrimitiveRoot u n) : |
| Creates (ballad n e u r) n (winds n u) := |
| creates_ode _ _ _ (flows_harp _ _ _ _ h (by simp [floorH,gap]) hr) hu |
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| def score (e : ℕ) := 2*chorus e+7 |
|
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| lemma beats_ballad (n e : ℕ) (u : ℂ) (r : Line) : |
| (ballad n e u r).beats=score e := by |
| unfold ballad |
| dsimp only; rw [beats_ode,beats_harp]; rfl |
|
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| lemma score_le (e : ℕ) : score e ≤ 20000000*(e+1)^2+7 := by |
| have := chorus_le e |
| unfold score; omega |
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|
| end |
| end PowerSaving.Slant |
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| end OAI |
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