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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 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseCandidateWork
import OAI.Combinatorics.EditApproximation.Coarse.CoarseInitialScalar
namespace OAI
namespace EditApproximation
open Filter
namespace BinaryFraction
/-- All actual initial-estimator branches have the same source-width charging
form. In the long-target branch no target scan or root is executed. -/
theorem computedCoarseInitialWithWork_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),
(coarseInitialWithWork (inputHeight N ^ 4) (64 * inputHeight N) N hB hq hL x y draws).2 ≤
inputHeight N ^ 36 * (coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N)
x y draws + x.length + 1) := by
filter_upwards [computedCoarseCandidate_bounds, eventually_ge_atTop 1,
inputSmallLog_tendsto.eventually (eventually_ge_atTop 200)] with N hcandidate hN hell
intro B d x y hBdef hddef hsize hsymbols hB hq hL draws
let H := inputHeight N
change (coarseInitialWithWork (H ^ 4) (64 * H) N hB hq hL x y draws).2 ≤
H ^ 36 * (coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draws + x.length + 1)
have hH : 100 ≤ H := (by decide : 100 ≤ 256).trans
(computed_coarse_height_large N (by omega) hell)
have hlength : Nat.size N ≤ H + 1 :=
(Nat.size_le_size (Nat.le_add_right N 2)).trans (inputLength_register_word N)
have hxlen : x.length ≤ N := (le_max_left _ _).trans hsize
have hylen : y.length ≤ N := (le_max_right _ _).trans hsize
have hx : Nat.size x.length ≤ H + 1 := (Nat.size_le_size hxlen).trans hlength
have hy : Nat.size y.length ≤ H + 1 := (Nat.size_le_size hylen).trans hlength
have hlongcost : (binaryNaturalCompareWithWork (2 * x.length) y.length).2 ≤ 8 * H + 17 := by
have hc := binaryNaturalCompareWithWork_cost (2 * x.length) y.length
have hp := natural_product_bit_length 2 x.length
rw [show Nat.size 2 = 2 by decide] at hp
omega
have hpow36 : 100 * H ^ 3 ≤ H ^ 36 := by
have hh := Nat.mul_le_mul_right (H ^ 3) hH
have hp := Nat.pow_le_pow_right (by omega : 0 < H) (by decide : 4 ≤ 36)
nlinarith only [hh, hp]
have hH3 : H ≤ H ^ 3 := Nat.le_pow (by decide)
have hmul := Nat.mul_le_mul_left (H ^ 36)
(show 1 ≤ coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draws +
x.length + 1 by omega)
by_cases hlong : 2 * x.length < y.length
· have hb := (binaryNaturalCompareWithWork_lt (2 * x.length) y.length).mpr hlong
change (coarseInitialWithWork _ _ _ _ _ _ _ _ _).2 ≤ _
simp only [coarseInitialWithWork, hb, ite_true]
nlinarith only [hlongcost, hlength, hy, hH, hpow36, hmul, hH3]
have hb : ¬ (binaryNaturalCompareWithWork (2 * x.length) y.length).1 = Ordering.lt :=
fun h => hlong ((binaryNaturalCompareWithWork_lt _ _).mp h)
by_cases he : x = y
· have heq := (naturalListEqualWithWork_value x y).mpr he
have hc := naturalListEqualWithWork_cost x y (H ^ 3) hsymbols
have hsource : y.length = x.length := congrArg List.length he.symm
have hf : (2 * x.length + 1) * (8 * H ^ 3 + 4) + (8 * H + 17) + 3 ≤
H ^ 36 * (x.length + 1) := by
have ha := Nat.mul_le_mul_right (x.length + 1) hpow36
have h3 : 1 ≤ H ^ 3 := one_le_pow₀ (by omega)
nlinarith only [ha, hH, h3, hH3]
simp only [coarseInitialWithWork, ite_eq_right hb, heq, ite_true]
rw [hsource] at hc
have hz : coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draws = 0 := by
simp only [coarseInitialArrayWork, ite_eq_right hlong, ite_eq_left he]
rw [hz, zero_add]
nlinarith only [hc, hf, hlongcost]
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 hN4 : Nat.size N ≤ H ^ 4 := by
have hp2 : H + 1 ≤ H ^ 2 := by nlinarith
have hp : H + 1 ≤ H ^ 4 := hp2.trans
(Nat.pow_le_pow_right (by omega : 0 < H) (by decide : 2 ≤ 4))
exact hlength.trans hp
have hw := coarseInitialWithWork_cost (inputHeight N ^ 4) (64 * inputHeight N) N
(H ^ 4) (H ^ 3) hB hq hL x y draws (by
have hp : H ≤ H ^ 4 := Nat.le_pow (by decide)
omega) hN4
((Nat.size_le_size hxlen).trans hN4) ((Nat.size_le_size hylen).trans hN4) hsymbols
(fun i => (hcandidate B d x y hBdef hddef hsize hB hq hL i (draws i)).1)
let f := fun i : CoarseGuessIndex (coarseEntryWidth x y) =>
(coarseCandidateWithWork (H ^ 4) (64 * H) hB hq hL x y i (draws i)).2
have hsum : ((List.finRange (Nat.clog 2 (coarseEntryWidth x y) + 1)).map f).sum =
∑ i, f i := by
rw [← List.ofFn_eq_map, List.sum_ofFn]
have hr : ((List.finRange (Nat.clog 2 (coarseEntryWidth x y) + 1)).map f).sum ≤
H ^ 32 * (coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draws +
(Nat.clog 2 (coarseEntryWidth x y) + 1)) := by
rw [hsum]
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 _ => (hcandidate B d x y hBdef hddef hsize hB hq hL i (draws i)).2)
_ = _ := by
simp only [coarseInitialArrayWork, ite_eq_right hlong, ite_eq_right he,
← Finset.mul_sum, Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, smul_eq_mul, mul_one]
have hscan : x.length + y.length + 1 ≤ 3 * x.length + 1 := by omega
apply coarseInitial_aggregate_scalar H x.length _ (Nat.clog 2 (coarseEntryWidth x y) + 1)
_ _ hH (by omega) hr
exact hw.trans (Nat.add_le_add_right (Nat.add_le_add_left
(Nat.mul_le_mul_right (8 * H ^ 3 + 4) hscan) _) _)
end BinaryFraction
end EditApproximation
end OAI
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