openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Coarse /CoarseCompletePolynomial.lean
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseCompleteArrayWork | |
| import OAI.Combinatorics.EditApproximation.Geometry.TreeParameters | |
| namespace OAI | |
| namespace EditApproximation | |
| namespace BinaryFraction | |
| theorem coarseCompleteNodes_polynomial (N B n : ℕ) (hB : 2 ≤ B) | |
| (hn : n ≤ N) (hBP : B ≤ (N + 2) ^ 4) : | |
| coarseCompleteNodes B (intervalTreeDepth B n) ≤ (N + 2) ^ 6 := by | |
| have hc := coarseCompleteNodes_le B (intervalTreeDepth B n) hB | |
| have hp := intervalTreeDepth_power_le B n hB | |
| have hm : B * max 1 n ≤ (N + 2) ^ 4 * (N + 2) := | |
| Nat.mul_le_mul hBP (max_le (by omega) (by omega)) | |
| calc | |
| _ ≤ 2 * (B * max 1 n) := by omega | |
| _ ≤ 2 * ((N + 2) ^ 4 * (N + 2)) := Nat.mul_le_mul_left 2 hm | |
| _ = 2 * (N + 2) ^ 5 := by ring | |
| _ ≤ (N + 2) ^ 6 := by | |
| have h := Nat.mul_le_mul_right ((N + 2) ^ 5) (show 2 ≤ N + 2 by omega) | |
| nlinarith only [h] | |
| theorem coarseRootWidth_polynomial (N B k : ℕ) (hBP : B ≤ (N + 2) ^ 4) | |
| (hk : k ≤ 2 * (N + 2)) : B * (2 * k + 1) + 1 ≤ (N + 2) ^ 8 := by | |
| let X := N + 2 | |
| have hx : 2 ≤ X := by dsimp only [X]; omega | |
| have hw : 2 * k + 1 ≤ 5 * X := by dsimp only [X]; omega | |
| have hm := Nat.mul_le_mul hBP hw | |
| change B * (2 * k + 1) ≤ X ^ 4 * (5 * X) at hm | |
| have hp : 1 ≤ X ^ 5 := one_le_pow₀ (by omega) | |
| have h8 : 6 * X ^ 5 ≤ X ^ 8 := by | |
| have hX3 : 6 ≤ X ^ 3 := by | |
| have h := Nat.pow_le_pow_left hx 3 | |
| norm_num at h | |
| omega | |
| calc | |
| _ ≤ X ^ 3 * X ^ 5 := Nat.mul_le_mul_right _ hX3 | |
| _ = _ := by rw [← pow_add] | |
| change B * (2 * k + 1) + 1 ≤ X ^ 8 | |
| nlinarith only [hm, hp, h8] | |
| theorem coarseActiveArrayWork_polynomial {α : Type*} (N B n k : ℕ) (Q lam allowance : ℚ) | |
| (hB : 2 ≤ B) (hn : n ≤ N) (hBP : B ≤ (N + 2) ^ 4) (hk : k ≤ 2 * (N + 2)) | |
| (tree : CoarseSourceTree B α (intervalTreeDepth B n)) | |
| (draws : CoarseThresholdDraws B (intervalTreeDepth B n)) : | |
| coarseActiveArrayWork k Q lam tree draws allowance ≤ (N + 2) ^ 14 := by | |
| have hc := coarseActiveArrayWork_le_complete k Q lam tree draws allowance | |
| have hn' := coarseCompleteNodes_polynomial N B n hB hn hBP | |
| have hw : B * (2 * k + 1) ≤ (N + 2) ^ 8 := by | |
| have h := coarseRootWidth_polynomial N B k hBP hk | |
| omega | |
| calc | |
| _ ≤ _ := hc | |
| _ ≤ (N + 2) ^ 8 * (N + 2) ^ 6 := Nat.mul_le_mul hw hn' | |
| _ = _ := by rw [← pow_add] | |
| end BinaryFraction | |
| end EditApproximation | |
| end OAI | |