File size: 3,250 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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseReadBudget
import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseCandidateWork
namespace OAI
namespace EditApproximation
open Filter
/-- Every descriptor, stored-symbol comparison and frame operation used by a
computed coarse root has a uniform fixed-power bit budget. -/
theorem computedCoarseReadBudget :
∀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) →
∀ i : CoarseGuessIndex (coarseEntryWidth x y),
CoarseReadBudget B d (coarseGuess (coarseEntryWidth x y) i) (coarseEntryWidth x y)
x y (inputHeight N ^ 30) := by
filter_upwards [eventually_ge_atTop 1,
inputSmallLog_tendsto.eventually (eventually_ge_atTop 200)] with N hN hell
intro B d x y hBdef hddef hsize hsymbol i
let H := inputHeight N
let k := coarseGuess (coarseEntryWidth x y) i
have hH : 256 ≤ H := computed_coarse_height_large N (by omega) hell
have hdepth : d ≤ H := hddef.symm ▸ (computed_coarse_depth N _ (by omega) hsize hell).2
have hBpos : 1 ≤ B := by
rw [hBdef]
change 1 ≤ inputHeight N ^ coarseExponent (inputSmallLog N)
exact one_le_pow₀ (Nat.succ_le_of_lt (Nat.two_pow_pos _))
have hb : Nat.size B ≤ 2 * H := by
have h := Nat.size_le_size (computed_coarse_B_upper N (by omega) hell)
rw [← hBdef] at h
simp only [Nat.size_pow] at h
exact h.trans (by dsimp only [H]; omega)
have hcube : 100 * H ^ 2 ≤ H ^ 3 := by
have h := Nat.mul_le_mul_right (H ^ 2) (show 100 ≤ H by omega)
nlinarith only [h]
have hp : Nat.size (B ^ d) ≤ H ^ 3 := by
have h := natural_power_bit_length B d
have hh := Nat.mul_le_mul hdepth hb
have hsq := Nat.mul_le_mul_right (H ^ 2) (show 3 ≤ H by omega)
nlinarith only [h, hh, hsq, hH]
have hn : Nat.size (coarseEntryWidth x y) ≤ H ^ 3 := by
have h := (Nat.size_le_size hsize).trans
((Nat.size_le_size (Nat.le_add_right N 2)).trans (inputLength_register_word N))
change Nat.size (coarseEntryWidth x y) ≤ H + 1 at h
have hpow : H ≤ H ^ 3 := Nat.le_pow (by decide)
have hsq := Nat.mul_le_mul_right (H ^ 2) (show 3 ≤ H by omega)
nlinarith only [h, hpow, hsq, hH, hcube]
have hk : Nat.size (2 * k + 1) ≤ H ^ 3 := by
have hkn : k ≤ 2 * (N + 2) := (BinaryFraction.coarseGuess_le _ i).trans (by omega)
have hks := (Nat.size_le_size hkn).trans (natural_product_bit_length 2 (N + 2))
have hn' := inputLength_register_word N
change Nat.size (N + 2) ≤ H + 1 at hn'
have hs := natural_sum_bit_length (2 * k) 1
have hm := natural_product_bit_length 2 k
norm_num only [show Nat.size 2 = 2 by decide, show Nat.size 1 = 1 by decide] at hks hs hm
have hsq := Nat.mul_le_mul_right (H ^ 2) (show 3 ≤ H by omega)
nlinarith only [hks, hn', hs, hm, hsq, hH, hcube]
exact coarseReadBudget_height B d k (coarseEntryWidth x y) H x y (by omega) hdepth hBpos
(by have hpow : H ≤ H ^ 2 := Nat.le_pow (by decide); nlinarith only [hb, hpow, hH, hcube]) hp hn hk hsymbol
end EditApproximation
end OAI
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