openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Representation /RepresentationFamilyWork.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationFamilyPreparation | |
| namespace OAI | |
| /-! # The complete family draw fits the half-exponential work scale -/ | |
| namespace LowSpaceSubsetSum.RepresentationFamilyWork | |
| variable {n : ℕ} | |
| theorem bound (I : Instance n) (hn : 10000 ≤ n) : | |
| 60 * RepresentationFamilyPreparation.width n + 51 + | |
| 44 * 2 ^ RepresentationFamilyPreparation.width n * | |
| (RepresentationFamilyPreparation.width n * n ^ 3 + | |
| 32 * (RepresentationFamilyPreparation.width n + 1) ^ 3 + 4) + 22 ≤ | |
| inputScale I ^ 6 * 2 ^ ((n + 1) / 2) := by | |
| let g := RepresentationFamilyPreparation.width n | |
| let u := inputScale I | |
| let H := 2 ^ ((n + 1) / 2) | |
| have hu : n + 1 ≤ u := by dsimp [u, inputScale]; omega | |
| have hu0 : 10000 ≤ u := by omega | |
| have hg : g ≤ n := by | |
| dsimp [g, RepresentationFamilyPreparation.width, StoredTopCoordinates.width, | |
| RepresentationSetup.blockHalf] | |
| omega | |
| have hgh : g ≤ (n + 1) / 2 := by | |
| dsimp [g, RepresentationFamilyPreparation.width, StoredTopCoordinates.width, | |
| RepresentationSetup.blockHalf] | |
| omega | |
| have hH : 1 ≤ H := Nat.one_le_two_pow | |
| have hpow : 2 ^ g ≤ H := Nat.pow_le_pow_right (by decide) hgh | |
| have hu1 : 1 ≤ u ^ 4 := Nat.one_le_pow _ _ (by omega) | |
| have hu3 : u ^ 3 ≤ u ^ 4 := pow_le_pow_right' (by omega) (by decide) | |
| have hgu : g ≤ u ^ 4 := by | |
| exact (show g ≤ u by omega).trans (by | |
| simpa only [pow_one] using pow_le_pow_right' (by omega : 1 ≤ u) (by decide : 1 ≤ 4)) | |
| have hgn : g * n ^ 3 ≤ u ^ 4 := by | |
| have h := Nat.mul_le_mul (show g ≤ u by omega) (Nat.pow_le_pow_left (show n ≤ u by omega) 3) | |
| simpa only [← pow_succ'] using h | |
| have hg3 : (g + 1) ^ 3 ≤ u ^ 4 := | |
| (Nat.pow_le_pow_left (show g + 1 ≤ u by omega) 3).trans hu3 | |
| have hinner : g * n ^ 3 + 32 * (g + 1) ^ 3 + 4 ≤ 37 * u ^ 4 := by | |
| nlinarith only [hgn, hg3, hu1] | |
| have hmain := Nat.mul_le_mul (Nat.mul_le_mul_left 44 hpow) hinner | |
| have hlinear : 60 * g + 73 ≤ 133 * u ^ 4 := by nlinarith only [hgu, hu1] | |
| have hlinearH := hlinear.trans (Nat.le_mul_of_pos_right _ hH) | |
| have htotal : 60 * g + 51 + 44 * 2 ^ g * (g * n ^ 3 + 32 * (g + 1) ^ 3 + 4) + 22 ≤ | |
| 1761 * u ^ 4 * H := by nlinarith only [hmain, hlinearH] | |
| have hu2 : 1761 ≤ u ^ 2 := by nlinarith only [hu0] | |
| have hf : 1761 * u ^ 4 ≤ u ^ 6 := by | |
| calc | |
| _ ≤ u ^ 2 * u ^ 4 := Nat.mul_le_mul_right _ hu2 | |
| _ = _ := by ring | |
| exact htotal.trans (Nat.mul_le_mul_right H hf) | |
| end LowSpaceSubsetSum.RepresentationFamilyWork | |
| end OAI | |