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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseReadBudget
import OAI.Combinatorics.EditApproximation.Coarse.CoarseIndexedRootBound
import OAI.Combinatorics.EditApproximation.Coarse.CoarseIndexedInitialCost
import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseInitialWork
namespace OAI
namespace EditApproximation
open Filter
namespace BinaryFraction
attribute [local irreducible] coarseRootReadExtra coarseInitialReadExtra
coarseIndexedInitialWithWork coarseInitialWithWork
theorem computedCoarseRootReadExtra_bound :
∀ᶠ N : ℕ in atTop, ∀ B d : ℕ, ∀ x y : List ℕ,
B = (integerParameters N).B → d = computedCoarseDepth N (coarseEntryWidth x y) →
coarseEntryWidth x y ≤ N → (∀ a ∈ x ++ y, Nat.size a ≤ inputHeight N ^ 3) →
∀ (hB : 0 < B) (hq : 0 < inputHeight N ^ 4) (hL : 0 < 64 * inputHeight N)
(i : CoarseGuessIndex (coarseEntryWidth x y)) (draws : CoarseThresholdDraws B d),
coarseRootReadExtra (coarseGuess (coarseEntryWidth x y) i)
(inputHeight N ^ 4) (64 * inputHeight N) hB hq hL x y draws ≤
inputHeight N ^ 32 * (coarseActiveArrayWork (coarseGuess (coarseEntryWidth x y) i)
(initialThresholdQ N) (initialLambda N) (coarseEntryTree B d x y) draws
(coarseGuess (coarseEntryWidth x y) i) + 1) := by
filter_upwards [computedCoarseReadBudget, eventually_ge_atTop 1,
inputSmallLog_tendsto.eventually (eventually_ge_atTop 200)] with N hbudget hN hell
intro B d x y hBdef hddef hsize hsymbol hB hq hL i draws
let H := inputHeight N
have hH : 100 ≤ H := (by decide : 100 ≤ 256).trans
(computed_coarse_height_large N (by omega) hell)
have hc := coarseRootReadExtra_bound (coarseGuess (coarseEntryWidth x y) i)
(H ^ 4) (64 * H) (H ^ 30) hB hq hL (one_le_pow₀ (by omega)) x y draws
(hbudget B d x y hBdef hddef hsize hsymbol i)
have hscale : 64 * H ^ 30 ≤ H ^ 32 := by
calc
_ ≤ H ^ 2 * H ^ 30 := Nat.mul_le_mul_right _ (by nlinarith only [hH])
_ = _ := by rw [← pow_add]
have hs := Nat.mul_le_mul_right
(coarseActiveArrayWork (coarseGuess (coarseEntryWidth x y) i) (initialThresholdQ N)
(initialLambda N) (coarseEntryTree B d x y) draws (coarseGuess (coarseEntryWidth x y) i) + 1) hscale
simp only [H, initialThresholdQ, initialLambda, Nat.cast_pow, Nat.cast_mul, Nat.cast_ofNat] at hc hs ⊢
nlinarith only [hc, hs]
theorem computedCoarseInitialReadExtra_bound :
∀ᶠ N : ℕ in atTop, ∀ B d : ℕ, ∀ x y : List ℕ,
B = (integerParameters N).B → d = computedCoarseDepth N (coarseEntryWidth x y) →
coarseEntryWidth x y ≤ N → (∀ a ∈ x ++ y, Nat.size a ≤ inputHeight N ^ 3) →
∀ (hB : 0 < B) (hq : 0 < inputHeight N ^ 4) (hL : 0 < 64 * inputHeight N)
(draws : CoarseGuessIndex (coarseEntryWidth x y) → CoarseThresholdDraws B d),
coarseInitialReadExtra (inputHeight N ^ 4) (64 * inputHeight N) hB hq hL x y draws ≤
inputHeight N ^ 34 * (coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N)
x y draws + 1) := by
filter_upwards [computedCoarseRootReadExtra_bound, eventually_ge_atTop 1,
inputSmallLog_tendsto.eventually (eventually_ge_atTop 200)] with N hroot hN hell
intro B d x y hBdef hddef hsize hsymbol hB hq hL draws
let H := inputHeight N
have hH : 100 ≤ H := (by decide : 100 ≤ 256).trans
(computed_coarse_height_large N (by omega) hell)
unfold coarseInitialReadExtra
split_ifs with hlong hequal
· exact Nat.zero_le _
· exact Nat.zero_le _
· have hlong' : ¬ 2 * x.length < y.length :=
fun h => hlong ((binaryNaturalCompareWithWork_lt _ _).mpr h)
have hequal' : x ≠ y := fun h => hequal ((naturalListEqualWithWork_value x y).mpr h)
have hclog : Nat.clog 2 (coarseEntryWidth x y) ≤ H :=
(Nat.clog_le_iff_le_pow (by decide : 1 < (2 : ℕ))).mpr
(hsize.trans ((Nat.le_add_right N 2).trans (inputHeight_covers_length N)))
have hsum : ((List.finRange (Nat.clog 2 (coarseEntryWidth x y) + 1)).map fun i =>
coarseRootReadExtra (coarseGuess (coarseEntryWidth x y) i) (inputHeight N ^ 4)
(64 * inputHeight N) hB hq hL x y (draws i)).sum ≤
H ^ 32 * (coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N)
x y draws + (Nat.clog 2 (coarseEntryWidth x y) + 1)) := by
rw [← List.ofFn_eq_map, List.sum_ofFn]
calc
_ ≤ ∑ i : CoarseGuessIndex (coarseEntryWidth x y), H ^ 32 *
(coarseActiveArrayWork (coarseGuess (coarseEntryWidth x y) i)
(initialThresholdQ N) (initialLambda N) (coarseEntryTree B d x y) (draws i)
(coarseGuess (coarseEntryWidth x y) i) + 1) :=
Finset.sum_le_sum (fun i _ => hroot B d x y hBdef hddef hsize hsymbol hB hq hL i (draws i))
_ = _ := by
simp only [coarseInitialArrayWork, ite_eq_right hlong', ite_eq_right hequal',
← Finset.mul_sum, Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, smul_eq_mul, mul_one]
have ha : coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draws +
(Nat.clog 2 (coarseEntryWidth x y) + 1) ≤
H ^ 2 * (coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draws + 1) := by
have hs : H + 1 ≤ H ^ 2 := by nlinarith only [hH]
have hm := Nat.mul_le_mul_right
(coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draws)
(show 1 ≤ H ^ 2 from one_le_pow₀ (by omega))
nlinarith only [hs, hm, hclog]
calc
_ ≤ _ := hsum
_ ≤ H ^ 32 * (H ^ 2 * (coarseInitialArrayWork B d (initialThresholdQ N)
(initialLambda N) x y draws + 1)) := Nat.mul_le_mul_left _ ha
_ = _ := by ring
theorem computedCoarseIndexedInitialWithWork_bound :
∀ᶠ N : ℕ in atTop, ∀ B d : ℕ, ∀ x y : List ℕ,
B = (integerParameters N).B → d = computedCoarseDepth N (coarseEntryWidth x y) →
coarseEntryWidth x y ≤ N → (∀ a ∈ x ++ y, Nat.size a ≤ inputHeight N ^ 3) →
∀ (hB : 0 < B) (hq : 0 < inputHeight N ^ 4) (hL : 0 < 64 * inputHeight N)
(draws : CoarseGuessIndex (coarseEntryWidth x y) → CoarseThresholdDraws B d),
(coarseIndexedInitialWithWork (inputHeight N ^ 4) (64 * inputHeight N) N hB hq hL x y draws).2 ≤
inputHeight N ^ 40 * (coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N)
x y draws + x.length + 1) := by
filter_upwards [computedCoarseInitialWithWork_bound, computedCoarseInitialReadExtra_bound,
eventually_ge_atTop 1, inputSmallLog_tendsto.eventually (eventually_ge_atTop 200)]
with N hold hextra hN hell
intro B d x y hBdef hddef hsize hsymbol hB hq hL draws
have h1 := hold B d x y hBdef hddef hsize hsymbol hB hq hL draws
have h2 := hextra B d x y hBdef hddef hsize hsymbol hB hq hL draws
have hc := coarseIndexedInitialWithWork_cost_compare (inputHeight N ^ 4) (64 * inputHeight N)
N hB hq hL x y draws
let H := inputHeight N
have hH : 100 ≤ H := (by decide : 100 ≤ 256).trans
(computed_coarse_height_large N (by omega) hell)
have hp : H ^ 34 ≤ H ^ 36 := Nat.pow_le_pow_right (by omega) (by decide)
have hs : 2 * H ^ 36 ≤ H ^ 40 := by
calc
_ ≤ H ^ 4 * H ^ 36 := Nat.mul_le_mul_right _
((by omega : 2 ≤ H).trans (Nat.le_pow (by decide)))
_ = _ := by rw [← pow_add]
have hm := Nat.mul_le_mul_right
(coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draws + x.length + 1) hs
have he := Nat.mul_le_mul hp
(show coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draws + 1 ≤
coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draws + x.length + 1 by omega)
dsimp only [H] at hm he
nlinarith only [h1, h2, hc, hm, he]
end BinaryFraction
end EditApproximation
end OAI
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