| import Mathlib |
| import OAI.Computability.VertexCover.PCP.Preprocessing |
|
|
| namespace OAI |
|
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| |
|
|
| namespace UniqueGames.Foundations.PCP.FinalConstants |
|
|
| def alphabet : Nat := 64 |
| def compositionLoss : Nat := 12288 |
| def denominator : Nat := 16 * alphabet ^ 4 * 66 * compositionLoss |
| def windowHalf : Nat := denominator * Preprocessing.sizeFactor |
| def windowSize : Nat := 2 * windowHalf + 1 |
| def smoothingScale : Nat := 4 * alphabet * windowHalf |
| def endpointLength : Nat := smoothingScale ^ 2 |
| def walkLength : Nat := 2 * endpointLength + 1 |
|
|
| noncomputable def gain : ℝ := (windowSize : ℝ) / denominator |
| noncomputable def cap : ℝ := 1 / (walkLength : ℝ) |
|
|
| theorem denominator_positive : 0 < denominator := by |
| norm_num [denominator, alphabet, compositionLoss] |
|
|
| theorem windowHalf_positive : 0 < windowHalf := |
| Nat.mul_pos denominator_positive Preprocessing.sizeFactor_positive |
|
|
| theorem walkLength_positive : 0 < walkLength := by |
| simp [walkLength] |
|
|
| theorem gain_large : 2 * (Preprocessing.sizeFactor : ℝ) ≤ gain := by |
| have hd : (0 : ℝ) < denominator := by exact_mod_cast denominator_positive |
| apply (le_div_iff₀ hd).2 |
| simp only [windowSize, windowHalf, Nat.cast_add, Nat.cast_mul, |
| Nat.cast_ofNat] |
| nlinarith |
|
|
| theorem cap_positive : 0 < cap := by |
| exact one_div_pos.mpr (by exact_mod_cast walkLength_positive) |
|
|
| theorem cap_le_one : cap ≤ 1 := by |
| apply (div_le_one (by exact_mod_cast walkLength_positive)).2 |
| exact_mod_cast walkLength_positive |
|
|
| theorem composed_gap (epsilon : ℝ) (he : 0 ≤ epsilon) : |
| min (2 * epsilon) cap ≤ |
| gain * min (epsilon / (Preprocessing.sizeFactor : ℝ)) |
| (1 / (walkLength : ℝ)) := by |
| have hs : (0 : ℝ) < Preprocessing.sizeFactor := by |
| exact_mod_cast Preprocessing.sizeFactor_positive |
| have hs1 : (1 : ℝ) ≤ Preprocessing.sizeFactor := by |
| exact_mod_cast Preprocessing.sizeFactor_positive |
| have hg : 0 ≤ gain := le_trans (by positivity) gain_large |
| rw [mul_min_of_nonneg _ _ hg] |
| apply min_le_min |
| · calc |
| 2 * epsilon = (2 * (Preprocessing.sizeFactor : ℝ)) * |
| (epsilon / (Preprocessing.sizeFactor : ℝ)) := by field_simp |
| _ ≤ gain * (epsilon / (Preprocessing.sizeFactor : ℝ)) := |
| mul_le_mul_of_nonneg_right gain_large (div_nonneg he hs.le) |
| · change cap ≤ gain * cap |
| have hgain : 1 ≤ gain := by linarith [gain_large] |
| simpa only [one_mul] using mul_le_mul_of_nonneg_right hgain cap_positive.le |
|
|
| end UniqueGames.Foundations.PCP.FinalConstants |
|
|
| end OAI |
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|
