| import Mathlib.Basic.Real.Basic |
| import Mathlib.Tactic.NormNum |
| import Mathlib.Tactic.Linarith |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
|
|
| noncomputable section |
|
|
| def weightBase : ℝ := 2 ^ 32 |
| def densityThreshold : ℝ := 2 ^ 100 |
| def thinningRate : ℝ := ((2 : ℝ) ^ 140)⁻¹ |
| def exitThreshold : ℝ := ((2 : ℝ) ^ 12)⁻¹ |
| def retainedFraction : ℝ := ((2 : ℝ) ^ 16)⁻¹ |
|
|
| theorem constants_positive : 0 < weightBase ∧ 0 < densityThreshold ∧ |
| 0 < thinningRate ∧ 0 < exitThreshold ∧ 0 < retainedFraction := by |
| norm_num [weightBase, densityThreshold, thinningRate, exitThreshold, retainedFraction] |
|
|
| theorem error_constant : |
| ((2 : ℝ) ^ 23)⁻¹ + thinningRate * densityThreshold + |
| (1 + thinningRate * densityThreshold) / (weightBase - 1) < ((2 : ℝ) ^ 21)⁻¹ := by |
| norm_num [thinningRate, densityThreshold, weightBase] |
|
|
| theorem retained_fraction_slack : |
| retainedFraction / 8 ≤ retainedFraction / 4 - ((2 : ℝ) ^ 21)⁻¹ := by |
| norm_num [retainedFraction] |
|
|
| theorem hidden_fraction_exact : |
| thinningRate * retainedFraction / (32 * densityThreshold * weightBase) = |
| ((2 : ℝ) ^ 293)⁻¹ := by |
| rw [show 293 = 140 + 153 from rfl, pow_add] |
| norm_num [thinningRate, retainedFraction, densityThreshold, weightBase] |
|
|
| theorem public_fraction_slack : ((2 : ℝ) ^ 310)⁻¹ ≤ ((2 : ℝ) ^ 293)⁻¹ := by |
| rw [show 310 = 155 + 155 from rfl, show 293 = 140 + 153 from rfl, pow_add, pow_add] |
| norm_num |
|
|
| theorem branch_mixture_lower (W maximumWeight mainPayoff maximumPayoff : ℝ) |
| (hW : 0 ≤ W) (hmw : 0 ≤ maximumWeight) (hmain : 0 ≤ mainPayoff) |
| (hmax : maximumWeight / 4 ≤ maximumPayoff) |
| (hsmall : maximumWeight ≤ W / (8 * densityThreshold) → |
| thinningRate * retainedFraction / (16 * densityThreshold) * W ≤ mainPayoff) : |
| thinningRate * retainedFraction / (32 * densityThreshold) * W ≤ |
| (mainPayoff + maximumPayoff) / 2 := by |
| by_cases hm : maximumWeight ≤ W / (8 * densityThreshold) |
| · have hs := hsmall hm |
| have hz : 0 ≤ maximumPayoff := by linarith |
| norm_num [thinningRate, retainedFraction, densityThreshold] at hs ⊢ |
| linarith |
| · have hm' : W / (8 * densityThreshold) < maximumWeight := lt_of_not_ge hm |
| have hc : thinningRate * retainedFraction / (32 * densityThreshold) ≤ |
| 1 / (64 * densityThreshold) := by |
| norm_num [thinningRate, retainedFraction, densityThreshold] |
| have hcW := mul_le_mul_of_nonneg_right hc hW |
| norm_num [densityThreshold] at hm' hcW ⊢ |
| linarith |
|
|
| end |
|
|
| end MatroidProphet |
|
|
| end OAI |
|
|
