File size: 5,818 Bytes
e63eb7f | 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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseRootBounds
import OAI.Combinatorics.EditApproximation.Coarse.CoarseIndexedInitial
namespace OAI
namespace EditApproximation
namespace BinaryFraction
/-- The stored numerator bits, denominator bits, sign and list links. -/
def coarseArrayStorage : CoarseBinaryArray → ℕ
| none => 1
| some cells => 1 + (cells.map fun a => a.numerator.bits.length + a.denominator.length + 3).sum
theorem coarseArrayStorage_le (two : BinaryFraction) (n V D A : ℕ)
(array : CoarseBinaryArray) (h : CoarseBinaryBox two n V D A array) :
coarseArrayStorage array ≤ 1 + n * (2 * A + 3) := by
cases array with
| none => simp only [coarseArrayStorage]; omega
| some cells =>
have hw (a : BinaryFraction) (ha : a ∈ cells) :
a.numerator.bits.length + a.denominator.length + 3 ≤ 2 * A + 3 := by
have hab := ((h cells rfl).2 a ha).2.2.2.2
unfold width at hab
omega
have hsum (values : List BinaryFraction)
(hv : ∀ a ∈ values, a.numerator.bits.length + a.denominator.length + 3 ≤ 2 * A + 3) :
(values.map fun a => a.numerator.bits.length + a.denominator.length + 3).sum ≤
values.length * (2 * A + 3) := by
induction values with
| nil => simp
| cons a values ih =>
have ha := hv a List.mem_cons_self
have ht := ih (fun b hb => hv b (List.mem_cons_of_mem _ hb))
simp only [List.map_cons, List.sum_cons, List.length_cons, Nat.add_mul, Nat.one_mul]
omega
have hs := hsum cells hw
rw [(h cells rfl).1] at hs
exact Nat.add_le_add_left hs 1
/-- The actual array immediately before the root lookup. -/
def coarseRootArray {B d : ℕ} {α : Type*} [DecidableEq α]
(k q L : ℕ) (hB : 0 < B) (hq : 0 < q) (hL : 0 < L)
(frame : List (Option α)) (tree : CoarseSourceTree B α d)
(draws : CoarseThresholdDraws B d) (start : ℕ) : CoarseBinaryArray :=
let den := coarseRootDenominator B q L d
let denPos := coarseRootDenominator_pos B q L d hB hq hL
let two := (coarseGridNatWithWork den.1 denPos 2).1
let allowance := (coarseGridNatWithWork den.1 denPos k).1
(coarseBinaryEval k q L frame two tree draws allowance start).1
theorem coarseRootArray_storage {B d : ℕ} {α : Type*} [DecidableEq α]
(k q L N D A : ℕ) (hB : 0 < B) (hq : 0 < q) (hL : 0 < L)
(frame : List (Option α)) (tree : CoarseSourceTree B α d)
(draws : CoarseThresholdDraws B d) (start : ℕ)
(hBbits : Nat.size B + 2 ≤ A) (hqbits : Nat.size q + 2 ≤ A)
(hLbits : Nat.size L + 2 ≤ A) (hNbits : Nat.size N + 2 ≤ A)
(hd : Nat.size (coarseCommonDenominator 1 q L B d) ≤ D)
(hbound : Nat.size (coarseLevelMagnitude N k B d + 2) + 2 * D + 1 ≤ A)
(hstart : start + tree.word.length ≤ N) :
coarseArrayStorage (coarseRootArray k q L hB hq hL frame tree draws start) ≤
1 + (2 * k + 1) * (2 * A + 3) := by
let den := coarseRootDenominator B q L d
let denPos := coarseRootDenominator_pos B q L d hB hq hL
let two := (coarseGridNatWithWork den.1 denPos 2).1
let allowance := (coarseGridNatWithWork den.1 denPos k).1
have hden := coarseRootDenominator_value B q L d
have hdenLen : den.1.length ≤ D := (coarseRootDenominator_length B q L d).trans_le hd
have htwo : two.value = 2 := coarseGridNatWithWork_value _ _ 2
have htwoDen : two.representation.den = coarseCommonDenominator 1 q L B d :=
(coarseGridNatWithWork_den _ _ 2).trans hden
have hkwidth : Nat.size k + 2 * D + 1 ≤ A :=
(Nat.add_le_add_right (Nat.add_le_add_right (Nat.size_le_size
((coarseLevelMagnitude_ge N k B d).trans (by omega))) _) _).trans hbound
have h2width : Nat.size 2 + 2 * D + 1 ≤ A :=
(Nat.add_le_add_right (Nat.add_le_add_right (Nat.size_le_size
(by omega : 2 ≤ coarseLevelMagnitude N k B d + 2)) _) _).trans hbound
have htwocell : CoarseGridCell two two 2 D A :=
coarseGridNatWithWork_bounds two den.1 denPos 2 2 D A
(coarseGridNatWithWork_den _ _ 2).symm hdenLen le_rfl h2width
have hacell : CoarseGridCell two allowance k D A :=
coarseGridNatWithWork_bounds two den.1 denPos k k D A
(coarseGridNatWithWork_den _ _ 2).symm hdenLen le_rfl hkwidth
have hav : allowance.value = k := coarseGridNatWithWork_value _ _ k
have hpath : allowance.value.den ∣ (L * B * q) ^ 0 := by
simp only [hav, Rat.den_natCast, one_dvd]
have hc := (coarseBinaryEval_bitWork k q L N d D A frame two tree draws hB hq hL htwo
htwocell.2.2.2.2 (by rw [htwoDen]; exact hd) htwoDen hBbits hqbits hLbits hNbits hbound
0 (by omega) allowance start hacell hpath hstart).1
exact coarseArrayStorage_le two _ _ D A _ hc
def coarseIndexedRootArray {B d : ℕ} (k q L : ℕ)
(hB : 0 < B) (hq : 0 < q) (hL : 0 < L) (x y : List ℕ)
(draws : CoarseThresholdDraws B d) : CoarseBinaryArray :=
let den := coarseRootDenominator B q L d
let denPos := coarseRootDenominator_pos B q L d hB hq hL
let two := (coarseGridNatWithWork den.1 denPos 2).1
let allowance := (coarseGridNatWithWork den.1 denPos k).1
(coarseIndexedWithWork k q L (coarseEntryWidth x y) x y two d 0 draws allowance 0).1
theorem coarseIndexedRootArray_eq {B d : ℕ} (k q L : ℕ)
(hB : 0 < B) (hq : 0 < q) (hL : 0 < L) (x y : List ℕ)
(draws : CoarseThresholdDraws B d) :
coarseIndexedRootArray k q L hB hq hL x y draws =
coarseRootArray k q L hB hq hL
(coarseTargetFrame k (coarsePadTo (coarseEntryWidth x y) y))
(coarseEntryTree B d x y) draws 0 := by
unfold coarseIndexedRootArray coarseRootArray
dsimp only
rw [coarseIndexedWithWork_value, coarseReadCore_eq k q L (coarseEntryWidth x y) x y
(le_max_left _ _) (le_max_right _ _)]
simp only [Nat.zero_add, coarseReadTree_entry]
end BinaryFraction
end EditApproximation
end OAI
|