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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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseRootHistory
import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseArrayStorage
namespace OAI
namespace EditApproximation
namespace BinaryFraction
open Filter
theorem computedCoarseRootHistory_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 →
∀ (hB : 0 < B) (hq : 0 < inputHeight N ^ 4) (hL : 0 < 64 * inputHeight N)
(i : CoarseGuessIndex (coarseEntryWidth x y)) (draws : CoarseThresholdDraws B d),
coarseRootArrayHistory (coarseGuess (coarseEntryWidth x y) i)
(inputHeight N ^ 4) (64 * inputHeight N) hB hq hL
(coarseTargetFrame (coarseGuess (coarseEntryWidth x y) i)
(coarsePadTo (coarseEntryWidth x y) y)) (coarseEntryTree B d x y) draws 0 ≤
(32 * (inputHeight N ^ 3 + 1)) *
((B * (2 * coarseGuess (coarseEntryWidth x y) i + 1) + 1) * coarseCompleteNodes B d) := by
filter_upwards [computedCoarse_bitParameters, eventually_ge_atTop 1,
inputSmallLog_tendsto.eventually (eventually_ge_atTop 200)] with N hbits hN hell
intro B d x y hBdef hddef hsize hB hq hL i draws
let H := inputHeight N
let k := coarseGuess (coarseEntryWidth x y) i
have hk : k ≤ 2 * (N + 2) := (coarseGuess_le _ i).trans (by omega)
obtain ⟨hden, hbound, hBbits, hqbits, hLbits, hNbits, _hD⟩ :=
hbits B d (coarseEntryWidth x y) k hBdef hddef hsize hk
have hcap : coarseEntryWidth x y ≤ B ^ d := by
rw [hBdef, hddef]
exact (computed_coarse_depth N _ (by omega) hsize hell).1
have htree : 0 + (coarseEntryTree B d x y).word.length ≤ N := by
rw [coarseEntryTree_word B d x y hcap, coarsePadTo_length _ _ (le_max_left _ _)]
simpa only [Nat.zero_add] using hsize
exact coarseRootArrayHistory_bound k (H ^ 4) (64 * H) N (10 * H ^ 2) (H ^ 3)
hB hq hL _ _ draws 0 hBbits hqbits hLbits hNbits hden hbound htree
theorem computedCoarseRootHistory_polynomial (N B d k n : ℕ)
(hN : 0 < N) (hell : 200 ≤ inputSmallLog N)
(hBdef : B = (integerParameters N).B) (hddef : d = computedCoarseDepth N n)
(hn : n ≤ N) (hk : k ≤ 2 * (N + 2)) :
(32 * (inputHeight N ^ 3 + 1)) *
((B * (2 * k + 1) + 1) * coarseCompleteNodes B d) ≤ (N + 2) ^ 32 := by
let X := N + 2
have hX : 2 ≤ X := by dsimp only [X]; omega
have hH := Nat.pow_le_pow_left (computed_height_polynomial N) 3
rw [← pow_mul] at hH
change inputHeight N ^ 3 ≤ X ^ 12 at hH
have hone : 1 ≤ X ^ 12 := one_le_pow₀ (by omega)
have hc : 64 ≤ X ^ 6 := (by decide : 64 ≤ (2 : ℕ) ^ 6).trans (Nat.pow_le_pow_left hX 6)
have hu : 32 * (inputHeight N ^ 3 + 1) ≤ X ^ 18 := by
calc
_ ≤ 64 * X ^ 12 := by omega
_ ≤ X ^ 6 * X ^ 12 := Nat.mul_le_mul_right _ hc
_ = _ := by rw [← pow_add]
have hB : 2 ≤ B := by
have h := computed_coarse_branching N hN hell
omega
have hBP : B ≤ (N + 2) ^ 4 := by
rw [hBdef]
exact computedCoarseBranching_polynomial N hN hell
have hd : d = intervalTreeDepth B n := by
simp only [hddef, computedCoarseDepth, intervalTreeDepth, hBdef]
have hnodes := coarseCompleteNodes_polynomial N B n hB hn hBP
rw [← hd] at hnodes
have hw := coarseRootWidth_polynomial N B k hBP hk
calc
_ ≤ X ^ 18 * (X ^ 8 * X ^ 6) := Nat.mul_le_mul hu (Nat.mul_le_mul hw hnodes)
_ = _ := by ring
/-- Total stored bits of all recursive array results from all guesses.
This bound holds for every draw, without a successful-approximation premise. -/
theorem computedCoarseAllHistories_polynomial :
∀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 →
∀ (hB : 0 < B) (hq : 0 < inputHeight N ^ 4) (hL : 0 < 64 * inputHeight N)
(draws : CoarseGuessIndex (coarseEntryWidth x y) → CoarseThresholdDraws B d),
(∑ i : CoarseGuessIndex (coarseEntryWidth x y),
coarseRootArrayHistory (coarseGuess (coarseEntryWidth x y) i)
(inputHeight N ^ 4) (64 * inputHeight N) hB hq hL
(coarseTargetFrame (coarseGuess (coarseEntryWidth x y) i)
(coarsePadTo (coarseEntryWidth x y) y)) (coarseEntryTree B d x y) (draws i) 0) ≤
(N + 2) ^ 37 := by
filter_upwards [computedCoarseRootHistory_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 hB hq hL draws
have hclog : Nat.clog 2 (coarseEntryWidth x y) ≤ inputHeight N :=
(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 hcount : Nat.clog 2 (coarseEntryWidth x y) + 1 ≤ (N + 2) ^ 5 := by
have hh := computed_height_polynomial N
have hp : 1 ≤ (N + 2) ^ 4 := one_le_pow₀ (by omega)
have hm := Nat.mul_le_mul_right ((N + 2) ^ 4) (show 2 ≤ N + 2 by omega)
nlinarith only [hclog, hh, hp, hm]
calc
_ ≤ ∑ _i : CoarseGuessIndex (coarseEntryWidth x y), (N + 2) ^ 32 := by
apply Finset.sum_le_sum
intro i _
exact (hroot B d x y hBdef hddef hsize hB hq hL i (draws i)).trans
(computedCoarseRootHistory_polynomial N B d _ (coarseEntryWidth x y)
(by omega) hell hBdef hddef hsize ((coarseGuess_le _ i).trans (by omega)))
_ = (Nat.clog 2 (coarseEntryWidth x y) + 1) * (N + 2) ^ 32 := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_eq_mul]
_ ≤ (N + 2) ^ 5 * (N + 2) ^ 32 := Nat.mul_le_mul_right _ hcount
_ = _ := by rw [← pow_add]
end BinaryFraction
end EditApproximation
end OAI
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