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namespace OAI
namespace MatroidProphet
open Finset
noncomputable def hazardBandBound (l : ℕ) : ℝ :=
(((l : ℝ) + 1) * Real.log 2 + 1) / 2 ^ l
noncomputable def hazardTailPotential (l : ℕ) : ℝ :=
2 * (((l : ℝ) + 2) * Real.log 2 + 1) / 2 ^ l
lemma hazardTailPotential_nonneg (l : ℕ) : 0 ≤ hazardTailPotential l := by
have hlog : 0 ≤ Real.log (2 : ℝ) := Real.log_nonneg (by norm_num)
have hl : 0 ≤ (l : ℝ) + 2 := by positivity
exact div_nonneg (mul_nonneg (by norm_num)
(add_nonneg (mul_nonneg hl hlog) (by norm_num))) (pow_nonneg (by norm_num) _)
lemma hazardTailPotential_step (l : ℕ) :
hazardTailPotential l = hazardBandBound l + hazardTailPotential (l + 1) := by
unfold hazardTailPotential hazardBandBound
push_cast
rw [pow_succ]
field_simp
ring
lemma hazardBandBound_sum (start length : ℕ) :
(∑ k ∈ range length, hazardBandBound (start + k)) +
hazardTailPotential (start + length) = hazardTailPotential start := by
induction length with
| zero => simp
| succ length ih =>
rw [sum_range_succ]
have h := hazardTailPotential_step (start + length)
rw [Nat.add_succ]
linarith
lemma hazardTailPotential_twelve : hazardTailPotential 12 < (1 : ℝ) / 2 := by
have hlog : Real.log (2 : ℝ) ≤ 1 := by
have h := Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2)
linarith
unfold hazardTailPotential
norm_num
linarith
theorem hazardBandBound_sum_lt_half (length : ℕ) :
(∑ k ∈ range length, hazardBandBound (12 + k)) < (1 : ℝ) / 2 := by
have h := hazardBandBound_sum 12 length
have hn := hazardTailPotential_nonneg (12 + length)
have hu := hazardTailPotential_twelve
linarith
theorem squared_exit_mass {ι : Type*} [Fintype ι]
(mass hazard : ι → ℝ) (eta : ℝ)
(hmass : ∀ i, 0 ≤ mass i) (hhazard : ∀ i, 0 ≤ hazard i) (heta : 0 ≤ eta)
(htotal : ∑ i, mass i * hazard i = 1)
(hlow : (∑ i, if hazard i < eta then mass i * hazard i else 0) ≤ (1 : ℝ) / 2) :
eta / 2 ≤ ∑ i, mass i * (hazard i) ^ 2 := by
classical
have hpoint (i : ι) : eta * (mass i * hazard i) ≤
mass i * (hazard i)^2 + eta * (if hazard i < eta then mass i * hazard i else 0) := by
by_cases h : hazard i < eta
· rw [ite_eq_left h]
have hn := mul_nonneg (hmass i) (sq_nonneg (hazard i))
linarith
· rw [ite_eq_right h]
have hh : eta ≤ hazard i := le_of_not_gt h
have hm := mul_le_mul_of_nonneg_left hh (mul_nonneg (hmass i) (hhazard i))
nlinarith
have hsum := sum_le_sum (fun i (_ : i ∈ (univ : Finset ι)) => hpoint i)
rw [sum_add_distrib, ← mul_sum, ← mul_sum, htotal] at hsum
have hl := mul_le_mul_of_nonneg_left hlow heta
nlinarith
theorem exit_band_bound {ι : Type*} [Fintype ι]
(mass hazard : ι → ℝ) (upper budget : ℝ)
(hmass : ∀ i, 0 ≤ mass i) (hupper : 0 ≤ upper)
(hhazard : ∀ i, hazard i ≤ upper)
(hsuccess : (∑ i, mass i * (1 - hazard i)) ≤ budget)
(hexit : (∑ i, mass i * hazard i) ≤ 1) :
(∑ i, mass i * hazard i) ≤ upper * (budget + 1) := by
have htotal : (∑ i, mass i) =
(∑ i, mass i * (1 - hazard i)) + ∑ i, mass i * hazard i := by
rw [← sum_add_distrib]
apply sum_congr rfl
intro i hi
ring
have ht : (∑ i, mass i) ≤ budget + 1 := by linarith
have hb : (∑ i, mass i * hazard i) ≤ upper * ∑ i, mass i := by
rw [mul_sum]
apply sum_le_sum
intro i hi
simpa [mul_comm] using mul_le_mul_of_nonneg_left (hhazard i) (hmass i)
exact hb.trans (mul_le_mul_of_nonneg_left ht hupper)
lemma hazardBandBound_eq_certificate_bound (l : ℕ) :
2 * (((2 : ℝ) ^ (l + 1))⁻¹) *
(Real.log (1 / (((2 : ℝ) ^ (l + 1))⁻¹)) + 1) = hazardBandBound l := by
rw [one_div, inv_inv, Real.log_pow]
unfold hazardBandBound
push_cast
rw [pow_succ]
field_simp
end MatroidProphet
end OAI
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