File size: 3,360 Bytes
e63eb7f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | 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
|