| import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseInitialWork |
| import OAI.Combinatorics.EditApproximation.Refinement.PhysicalCoarseSeedLaw |
| import OAI.Combinatorics.EditApproximation.Probability.FreshCoordinateCost |
|
|
| namespace OAI |
|
|
| namespace EditApproximation |
|
|
| / |
| def coarseEntryBinaryCost (N B d : ℕ) [NeZero B] (x y : List ℕ) |
| (draws : CoarseGuessIndex (coarseEntryWidth x y) → CoarseThresholdDraws B d) : ℕ := |
| (BinaryFraction.coarseInitialWithWork (inputHeight N ^ 4) (64 * inputHeight N) N |
| (Nat.pos_of_neZero B) (pow_pos (Nat.two_pow_pos _) _) |
| (Nat.mul_pos (by decide) (Nat.two_pow_pos _)) x y draws).2 |
|
|
| theorem coarseEntryBinaryCost_computed_bound : |
| ∀ᶠ N : ℕ in Filter.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) → |
| ∀ (_hB : NeZero B) (draw : CoarseGuessIndex (coarseEntryWidth x y) → CoarseThresholdDraws B d), |
| (coarseEntryBinaryCost N B d x y draw : ℝ) ≤ (inputHeight N : ℝ) ^ 36 * |
| ((coarseInitialArrayWork B d (initialThresholdQ N) (initialLambda N) x y draw : ℝ) + |
| (x.length + 1 : ℕ)) := by |
| filter_upwards [BinaryFraction.computedCoarseInitialWithWork_bound] with N hN |
| intro B d x y hB hd hxy hsymbols hinstance draw |
| have hb := hN B d x y hB hd hxy hsymbols |
| (Nat.pos_of_neZero _) (pow_pos (Nat.two_pow_pos _) _) |
| (Nat.mul_pos (by decide) (Nat.two_pow_pos _)) draw |
| exact_mod_cast hb |
|
|
| noncomputable def coarseEntryCostSum {M J B : ℕ} [NeZero B] (source target : List ℕ) |
| (N : ℕ) (depth : PhysicalEntry M J target.length → ℕ) |
| (draw : PhysicalCoarseDraw source target B depth) |
| (selected : PhysicalEntry M J target.length → Prop) : ℝ := by |
| classical |
| exact ∑ entry : PhysicalEntry M J target.length, selectionIndicator (selected entry) * |
| (coarseEntryBinaryCost N B (depth entry) (physicalSource source entry.1) |
| (substring target entry.2.lo entry.2.hi) (draw entry) : ℝ) |
|
|
| theorem coarseEntryCostSum_bound {M J B : ℕ} [NeZero B] (source target : List ℕ) |
| (N : ℕ) (depth : PhysicalEntry M J target.length → ℕ) |
| (draw : PhysicalCoarseDraw source target B depth) |
| (selected : PhysicalEntry M J target.length → Prop) (C : ℝ) |
| (hc : ∀ entry : PhysicalEntry M J target.length, |
| (coarseEntryBinaryCost N B (depth entry) (physicalSource source entry.1) |
| (substring target entry.2.lo entry.2.hi) (draw entry) : ℝ) ≤ |
| C * ((coarseInitialArrayWork B (depth entry) (initialThresholdQ N) (initialLambda N) |
| (physicalSource source entry.1) (substring target entry.2.lo entry.2.hi) (draw entry) : ℝ) + |
| ((physicalSource source entry.1).length + 1 : ℕ))) : |
| coarseEntryCostSum source target N depth draw selected ≤ C * |
| ((∑ entry : PhysicalEntry M J target.length, selectionIndicator (selected entry) * |
| (coarseInitialArrayWork B (depth entry) (initialThresholdQ N) (initialLambda N) |
| (physicalSource source entry.1) (substring target entry.2.lo entry.2.hi) (draw entry) : ℝ)) + |
| ∑ entry : PhysicalEntry M J target.length, selectionIndicator (selected entry) * |
| ((physicalSource source entry.1).length + 1 : ℕ)) := by |
| classical |
| rw [← Finset.sum_add_distrib, Finset.mul_sum] |
| unfold coarseEntryCostSum |
| apply Finset.sum_le_sum |
| intro entry _ |
| have hs : 0 ≤ selectionIndicator (selected entry) := by |
| unfold selectionIndicator |
| split <;> norm_num |
| exact (mul_le_mul_of_nonneg_left (hc entry) hs).trans_eq (by ring) |
|
|
| end EditApproximation |
|
|
| end OAI |
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|
