openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Refinement /ComputedKernelWork.lean
| 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 | |