| import OAI.Computability.FourierTransform.TensorIndexing |
|
|
| namespace OAI |
|
|
| /-! Prefix computation and tensor gather. -/ |
| namespace PowerSaving.Grove |
| open Slant Finset |
| noncomputable section |
|
|
| /-- Run state updates in order, with natural prefix addressing. |
| For step: parameters (axis, old prefix, extension digit, previous state). -/ |
| def climb {α : Type*} (r : Axes) (x : α) |
| (step : ℕ → ℕ → ℕ → α → α) : ℕ → ℕ → α |
| | 0 => fun _ => x |
| | l+1 => fun i => step l (i/r l) (i%r l) (climb r x step l (i/r l)) |
|
|
| def survey (r : Axes) (u : ℕ → Stroke) : ℕ → ℕ → ℕ × ℂ := |
| climb r (0,1) fun i _ a s => (s.1*r i+(u i).ix a,s.2*(u i).dye a) |
|
|
| lemma survey_lt (r : Axes) (l : ℕ) (u : ℕ → Stroke) (h : Splits r l) |
| (g : ∀ i<l, (u i).Sound (r i)) : |
| ∀ j<extent r l, (survey r u l j).1<extent r l := by |
| induction l with |
| | zero => intro i hi; change 0<1; omega |
| | succ l ih => |
| intro j hj |
| exact h.join_lt (ih (h.cut (by omega)) (fun i hi => g i (by omega)) _ |
| (h.div_lt hj)) ((g l (by omega)).bdix _ (h.mod_lt j)) |
|
|
| lemma canopy_survey (r : Axes) (l : ℕ) (u : ℕ → Stroke) (h : Splits r l) |
| (g : ∀ i<l, (u i).Sound (r i)) : |
| ∀ i<extent r l, ∀ j<extent r l, |
| canopy r (fun t => leafM (u t)) l i j = |
| if j=(survey r u l i).1 then (survey r u l i).2 else 0 := by |
| induction l with |
| | zero => |
| intro i hi j hj |
| change j<1 at hj |
| have h : j=0 := by omega |
| simp [survey,climb,h,canopy] |
| | succ l ih => |
| intro i hi j hj |
| let v := survey r u l (i/r l) |
| let b := (u l).ix (i%r l) |
| have hb : b < r l := (g l (by omega)).bdix _ (h.mod_lt i) |
| have H : j=v.1*r l+b ↔ j/r l=v.1 ∧ j%r l=b := by |
| constructor |
| · intro h; rw [h,chops _ _ _ hb,chips _ _ _ hb]; exact ⟨rfl,rfl⟩ |
| · rintro ⟨A,B⟩; rw [← A,← B]; exact (joined ..).symm |
| change canopy r _ l _ _ * leafM (u l) _ _ = |
| if j=v.1*r l+b then v.2 * (u l).dye (i%r l) else 0 |
| rw [ih (h.cut (by omega)) (fun i hi => g i (by omega)) _ (h.div_lt hi) _ |
| (h.div_lt hj)] |
| change (if j/r l=v.1 then v.2 else 0) * |
| (if j%r l=b then _ else 0) = if j=v.1*r l+b then _ else _ |
| by_cases A : j/r l=v.1 <;> by_cases B : j%r l=b <;> simp [H,A,B] |
|
|
| theorem graft_survey (r : Axes) (l : ℕ) (u : ℕ → Stroke) (h : Splits r l) |
| (g : ∀ i<l, (u i).Sound (r i)) (x : Line) (j : ℕ) (hj : j<extent r l) : |
| graft (extent r l) (canopy r (fun t => leafM (u t)) l) x j = |
| (survey r u l j).2 * x (survey r u l j).1 := by |
| let t := survey r u l j |
| have ht := survey_lt r l u h g j hj |
| change ∑ i ∈ range _, _ = t.2*x t.1 |
| rw [sum_eq_single t.1] |
| · rw [canopy_survey r l u h g j hj _ ht, ite_eq_left rfl] |
| · intro b hb hg |
| rw [canopy_survey r l u h g j hj _ (mem_range.mp hb), ite_eq_right hg, zero_mul] |
| · intro hh; exact absurd (mem_range.mpr ht) hh |
|
|
| end |
| end PowerSaving.Grove |
|
|
| end OAI |
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|
