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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 | import OAI.Computability.FourierTransform.LayerWords
namespace OAI
/-! Disjoint interval effects of layer words. In particular a composition of two rectangular
maps can be added by a commutator of adds on an arbitrary-content intermediate. -/
namespace PowerSaving.Slant
open Reel Stroke Spectral
noncomputable section
structure Zone where
lo : ℕ
wide : ℕ
namespace Zone
def has (u : Zone) (i : ℕ) : Prop := u.lo ≤ i ∧ i < u.lo+u.wide
instance instDecidableHas (u : Zone) (i : ℕ) : Decidable (u.has i) := by
unfold has
infer_instance
def Fits (u : Zone) (v : ℕ) : Prop := ∀ i, u.has i → i<v
lemma fits_of (u : Zone) (n : ℕ) (h : u.lo+u.wide ≤ n) : u.Fits n :=
fun _ hi => hi.2.trans_le h
lemma fits_empty (i n : ℕ) : Fits ⟨i,0⟩ n := by
simp [Fits,has]
def Sep (u v : Zone) : Prop := ∀ i, u.has i → ¬ v.has i
lemma sep_of (u v : Zone) (h : u.lo+u.wide ≤ v.lo) : u.Sep v :=
fun _ hi hj => Nat.not_lt.mpr (h.trans hj.1) hi.2
lemma Sep.symm {u v : Zone} (h : u.Sep v) : v.Sep u := fun i hi g => h i g hi
lemma inside (u : Zone) {i : ℕ} (hi : i<u.wide) : u.has (u.lo+i) :=
⟨by omega,by omega⟩
lemma enter (u : Zone) {i : ℕ} (hi : u.has i) :
i-u.lo < u.wide := by
unfold has at hi; omega
lemma identify (u : Zone) {i} (hi : u.has i) : u.lo+(i-u.lo)=i := by
unfold has at hi; omega
lemma boundary (u : Zone) {i : ℕ} (hi : ¬ u.has i) : i<u.lo ∨ u.lo+u.wide ≤ i := by
unfold has at hi; omega
/-- Only read entries within 'wide'; other values are ignored. -/
def peek (u : Zone) (x : Line) : Line := drift u.lo x
def slim (u : Zone) (n : ℕ) : Zone := ⟨u.lo,n⟩
lemma slim_has (u : Zone) {n i : ℕ} (hn : n ≤ u.wide)
(hi : (u.slim n).has i) : u.has i := ⟨hi.1,hi.2.trans_le (by change u.lo+n ≤ _; omega)⟩
lemma Fits.slim {u : Zone} {v n : ℕ} (h : u.Fits v) (hn : n ≤ u.wide) :
(u.slim n).Fits v := fun i hi => h i (slim_has u hn hi)
lemma Sep.slim_left {u v : Zone} (h : u.Sep v) {n : ℕ} (hn : n ≤ u.wide) :
(u.slim n).Sep v := fun i hi => h i (slim_has u hn hi)
lemma Sep.slim_right {u v : Zone} (h : u.Sep v) {n : ℕ} (hn : n ≤ v.wide) :
u.Sep (v.slim n) := (h.symm.slim_left hn).symm
end Zone
/-- f read only first p entries for the first q output entries -/
def Reads (p q : ℕ) (f : Line → Line) : Prop :=
∀ x y, (∀ i<p, x i=y i) → ∀ i<q, f x i=f y i
def AdditiveOn (q : ℕ) (f : Line → Line) : Prop :=
∀ x y, ∀ i<q, f (x+y) i=f x i+f y i
def minusFn (f : Line→Line) : Line → Line := fun x i => -f x i
structure Spill (r : Reel) (u v : Zone) (f : Line→Line) : Prop where
adds : ∀ (x : Line) (i), i<v.wide → v.peek (r.go x) i=v.peek x i + f (u.peek x) i
keeps : ∀ (x : Line) (i), ¬ v.has i → r.go x i=x i
namespace Spill
variable {r : Reel} {u v z : Zone} {f g : Line→Line}
lemma preserves (h : Spill r u v f) (hz : z.Sep v) (x : Line) :
∀ i<z.wide, z.peek (r.go x) i=z.peek x i :=
fun _ hi => h.keeps x _ (hz _ (z.inside hi))
lemma congr (h : Spill r u v f) (hg : ∀ (x : Line), ∀ i<v.wide, f x i = g x i) :
Spill r u v g :=
⟨fun x i hi => (h.adds x i hi).trans (by rw [hg _ i hi]),h.keeps⟩
end Spill
open Zone
/-- The commutator construction. Disjoint x->z, z->y adds. -/
def cameo (a b c d : Reel) : Reel := ((a.stack b).stack c).stack d
lemma sound_cameo (a b c d : Reel) (n) (h : a.Sound n) (g : b.Sound n)
(p : c.Sound n) (q : d.Sound n) :
(cameo a b c d).Sound n :=
((h.stack g).stack p).stack q
theorem spill_cameo {a b c d : Reel} {u v z : Zone}
{f g : Line→Line}
(ha : Spill a u z f) (hb : Spill b z v g)
(hc : Spill c u z (minusFn f)) (hd : Spill d z v (minusFn g))
(rf : Reads u.wide z.wide f) (rg : Reads z.wide v.wide g)
(hh : AdditiveOn v.wide g)
(huv : u.Sep v) (huz : u.Sep z) (hzv : z.Sep v) :
Spill (cameo a b c d) u v (g ∘ f) := by
have middle (x : Line) :
∀ i < z.wide, z.peek (c.go (b.go (a.go x))) i = z.peek x i := by
intro i hi
have H (j : ℕ) (hj : j < u.wide) :
u.peek (b.go (a.go x)) j = u.peek x j :=
(hb.preserves huv _ j hj).trans (ha.preserves huz _ j hj)
rw [hc.adds _ i hi, hb.preserves hzv _ i hi, ha.adds _ i hi, minusFn, rf _ _ H i hi]
ring
constructor
· intro x i hi
have H := rg _ _ (middle x) i hi
have mid : g (z.peek (a.go x)) i=g (z.peek x + f (u.peek x)) i := by
apply rg
· intro j hj
exact ha.adds x j hj
· exact hi
simp only [cameo,Reel.go_stack]
rw [hd.adds _ i hi,minusFn,H, hc.preserves hzv.symm _ i hi, hb.adds _ i hi,
ha.preserves hzv.symm _ i hi,mid, hh _ _ i hi,Function.comp_apply]
ring
· intro x i hi
simp only [cameo,Reel.go_stack]
rw [hd.keeps _ i hi]
by_cases h : z.has i
· have G := middle x (i-z.lo) (z.enter h)
simpa [Zone.peek, Stroke.drift,z.identify h] using G
· rw [hc.keeps _ i h,hb.keeps _ i hi,ha.keeps _ i h]
end
end PowerSaving.Slant
end OAI
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