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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 | import OAI.Combinatorics.EditApproximation.Geometry.RankedMemoKernelWork
import OAI.Combinatorics.EditApproximation.Refinement.ComputedLocalKernelWork
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation
open Filter
/-- Local arithmetic annotations follow the actual memo run, hence are
incurred once per cache miss, with the original program and draw values. -/
theorem computedFamilyKernelWork_le (source target : List ℕ) (N : ℕ)
(input : ComputedFamilyInput source target N)
(cost : ComputedFamilyMemoKey source target N → ℕ) (C : ℕ)
(hcost : ∀ key, cost key ≤ C) :
computedFamilyKernelWork source target N input cost ≤
C * (computedFamilyResult source target N input).requests :=
rankedMemoKernelWork_le _ _ _ _ cost C hcost _ _
theorem computedFamilyKernelWork_near_linear (D : ℕ) (epsilon : ℝ) (hepsilon : 0 < epsilon) :
∀ᶠ N in atTop, ∀ source target : List ℕ, source.length + target.length ≤ N →
∀ (input : ComputedFamilyInput source target N)
(cost : ComputedFamilyMemoKey source target N → ℕ),
(∀ key, cost key ≤ computedLocalKernelBudget D N) →
(computedFamilyKernelWork source target N input cost : ℝ) ≤
(N + 2 : ℝ) ^ (1 + epsilon) := by
filter_upwards [computedLocalKernelBudget_requests_near_linear D epsilon hepsilon] with N hN
intro source target hsize input cost hcost
have h := computedFamilyKernelWork_le source target N input cost
(computedLocalKernelBudget D N) hcost
have hr : (computedFamilyKernelWork source target N input cost : ℝ) ≤
(computedLocalKernelBudget D N : ℝ) * (computedFamilyResult source target N input).requests := by
exact_mod_cast h
exact hr.trans (by simpa only [mul_comm] using hN source target hsize input)
end EditApproximation
end OAI
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