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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 | import OAI.Combinatorics.EditApproximation.Geometry.RankedMemoAddressWork
import OAI.Combinatorics.EditApproximation.Refinement.ComputedExecutionWork
import OAI.Combinatorics.EditApproximation.Geometry.HeightPowerWork
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation
open Filter
theorem computedFamilyAddressWork_le (source target : List ℕ) (N : ℕ)
(input : ComputedFamilyInput source target N) :
computedFamilyAddressWork source target N input ≤
2 * (7 * intervalTreeDepth (integerParameters N).M source.length + 60) *
(computedFamilyResult source target N input).requests :=
rankedMemoAddressWork_le _ _ _ _ _ _
(computedFamilyMemoCodeWithWork_cost source target N (inputHeight N ^ 2)) _ _
private theorem address_factor_bound (H J : ℕ) (hH : 100 ≤ H) (hJ : J ≤ H) :
2 * (7 * J + 60) ≤ H ^ 2 := by nlinarith only [hH, hJ]
theorem computedFamilyAddressWork_near_linear (epsilon : ℝ) (hepsilon : 0 < epsilon) :
∀ᶠ N in atTop, ∀ source target : List ℕ, source.length + target.length ≤ N →
∀ input : ComputedFamilyInput source target N,
(computedFamilyAddressWork source target N input : ℝ) ≤ (N + 2 : ℝ) ^ (1 + epsilon) := by
have hh : 0 < epsilon / 2 := half_pos hepsilon
filter_upwards [computedFamilyResult_requests_near_linear (epsilon / 2) hh,
fixed_height_power_subpolynomial 2 (epsilon / 2) hh,
eventually_ge_atTop 1, inputSmallLog_tendsto.eventually (eventually_ge_atTop 200)]
with N hr hp hN hell
intro source target hsize input
have hdepth := computed_tree_depth_bound N source.length (by omega) (by omega) hell
have hJ : intervalTreeDepth (integerParameters N).M source.length ≤ inputHeight N := by
nlinarith only [hdepth, hell]
have hH : 100 ≤ inputHeight N := (by omega : 100 ≤ inputSmallLog N).trans
(computed_smallLog_le_height N)
have hfactor := address_factor_bound (inputHeight N) _ hH hJ
have ha := computedFamilyAddressWork_le source target N input
have haR : (computedFamilyAddressWork source target N input : ℝ) ≤
(inputHeight N : ℝ) ^ 2 * (computedFamilyResult source target N input).requests := by
exact_mod_cast ha.trans (Nat.mul_le_mul_right _ hfactor)
calc
_ ≤ (inputHeight N : ℝ) ^ 2 * (computedFamilyResult source target N input).requests := haR
_ ≤ (N + 2 : ℝ) ^ (epsilon / 2) * (N + 2 : ℝ) ^ (1 + epsilon / 2) :=
mul_le_mul hp (hr source target hsize input) (Nat.cast_nonneg _) (Real.rpow_nonneg (by positivity) _)
_ = _ := by rw [← Real.rpow_add (by positivity)]; congr 1; ring
end EditApproximation
end OAI
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