| import Mathlib |
|
|
| namespace OAI |
|
|
| /-! |
| # Predicted-band geometry |
|
|
| The prefix-cost predictor is 1-Lipschitz, while its velocity has a strict |
| margin below one. This gives narrow bands without a convexity assumption |
| on the set of allowed endpoints (Sections 5 and 6 of the manuscript). |
| -/ |
|
|
| namespace EditApproximation |
|
|
| def inPredictedBand (prefixCost : ℝ → ℝ) (origin velocity margin width point : ℝ) : Prop := |
| |point - origin - velocity * prefixCost point| ≤ margin * width |
|
|
| theorem predicted_band_diameter {prefixCost : ℝ → ℝ} |
| {origin velocity margin width p q : ℝ} |
| (hmargin : 0 < margin) (hvelocity : |velocity| ≤ 1 - margin) |
| (hLipschitz : |prefixCost p - prefixCost q| ≤ |p - q|) |
| (hp : inPredictedBand prefixCost origin velocity margin width p) |
| (hq : inPredictedBand prefixCost origin velocity margin width q) : |
| |p - q| ≤ 2 * width := by |
| have hid : p - q = |
| (p - origin - velocity * prefixCost p) - (q - origin - velocity * prefixCost q) + |
| velocity * (prefixCost p - prefixCost q) := by ring |
| have htri := (abs_add_le |
| ((p - origin - velocity * prefixCost p) - (q - origin - velocity * prefixCost q)) |
| (velocity * (prefixCost p - prefixCost q))).trans |
| (add_le_add (abs_sub _ _) le_rfl) |
| rw [← hid, abs_mul] at htri |
| have hprod := mul_le_mul hvelocity hLipschitz (abs_nonneg _) |
| ((abs_nonneg velocity).trans hvelocity) |
| change |p - origin - velocity * prefixCost p| ≤ margin * width at hp |
| change |q - origin - velocity * prefixCost q| ≤ margin * width at hq |
| have hscaled : margin * |p - q| ≤ margin * (2 * width) := by nlinarith |
| exact le_of_mul_le_mul_left hscaled hmargin |
|
|
| /-- The comparison prefix error is the weighted deviation plus two |
| warmup-mass discrepancy charges. -/ |
| theorem comparison_prefix_prediction_error |
| {point origin velocity predictedVelocity prefixMass prefixCost totalMass discrepancy deviation : ℝ} |
| (hprefix : 0 ≤ prefixMass) (hprefixTotal : prefixMass ≤ totalMass) |
| (hprediction : |velocity - predictedVelocity| * totalMass ≤ discrepancy) |
| (hvelocity : |predictedVelocity| ≤ 1) |
| (hcostLower : prefixMass - discrepancy ≤ prefixCost) |
| (hcostUpper : prefixCost ≤ prefixMass) |
| (hdeviation : |point - origin - velocity * prefixMass| ≤ deviation) : |
| |point - origin - predictedVelocity * prefixCost| ≤ deviation + 2 * discrepancy := by |
| have hcostDiff : |prefixMass - prefixCost| ≤ discrepancy := by |
| rw [abs_of_nonneg (by linarith : 0 ≤ prefixMass - prefixCost)] |
| linarith |
| have hfirst : |(velocity - predictedVelocity) * prefixMass| ≤ discrepancy := by |
| rw [abs_mul, abs_of_nonneg hprefix] |
| exact (mul_le_mul_of_nonneg_left hprefixTotal (abs_nonneg _)).trans hprediction |
| have hsecond : |predictedVelocity * (prefixMass - prefixCost)| ≤ discrepancy := by |
| rw [abs_mul] |
| have hprod := mul_le_mul hvelocity hcostDiff (abs_nonneg _) |
| (by norm_num : (0 : ℝ) ≤ 1) |
| simpa only [one_mul] using hprod |
| have hid : point - origin - predictedVelocity * prefixCost = |
| (point - origin - velocity * prefixMass) + |
| (velocity - predictedVelocity) * prefixMass + |
| predictedVelocity * (prefixMass - prefixCost) := by ring |
| rw [hid] |
| exact ((abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)).trans |
| (by linarith) |
|
|
| end EditApproximation |
|
|
| end OAI |
|
|
