| import OAI.Probability.MatroidProphet.ExitNumerics |
| import Mathlib.Algebra.Order.Archimedean.Basic |
| import Mathlib.Data.Nat.Find |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| open Finset |
|
|
| noncomputable def inHazardBand (p : ℝ) (l : ℕ) : Prop := |
| ((2 : ℝ) ^ (l+1))⁻¹ ≤ p ∧ p < ((2 : ℝ) ^ l)⁻¹ |
|
|
| lemma exists_hazard_band {p : ℝ} (hp : 0 < p) (hsmall : p < ((2 : ℝ)^12)⁻¹) : |
| ∃ l : ℕ, 12 ≤ l ∧ inHazardBand p l := by |
| have hex : ∃ n : ℕ, ((1/2 : ℝ)^n) ≤ p := by |
| obtain ⟨n, hn⟩ := exists_pow_lt_of_lt_one hp (by norm_num : (1/2 : ℝ) < 1) |
| exact ⟨n, hn.le⟩ |
| let n := Nat.find hex |
| have hn : (1/2 : ℝ)^n ≤ p := Nat.find_spec hex |
| have hn12 : 12 < n := by |
| by_contra h |
| have hle : n ≤ 12 := by omega |
| have hpow := pow_le_pow_of_le_one (by norm_num : (0 : ℝ) ≤ 1/2) |
| (by norm_num : (1/2 : ℝ) ≤ 1) hle |
| have hsmall' : p < (1/2 : ℝ)^12 := by simpa [one_div, inv_pow] using hsmall |
| linarith |
| refine ⟨n-1, by omega, ?_⟩ |
| have hpred : p < (1/2 : ℝ)^(n-1) := |
| lt_of_not_ge (Nat.find_min hex (by dsimp only [n]; omega)) |
| unfold inHazardBand |
| have hsucc : n-1+1 = n := by omega |
| rw [hsucc] |
| simpa only [one_div, inv_pow] using And.intro hn hpred |
|
|
| theorem finite_hazard_cover {ι : Type*} [Fintype ι] (p : ι → ℝ) : |
| ∃ length : ℕ, ∀ i, 0 < p i → p i < ((2 : ℝ)^12)⁻¹ → |
| ∃ k ∈ range length, inHazardBand (p i) (12+k) := by |
| classical |
| let band : ι → ℕ := fun i => if h : 0 < p i ∧ p i < ((2 : ℝ)^12)⁻¹ |
| then (exists_hazard_band h.1 h.2).choose else 12 |
| let length := (univ : Finset ι).sup band + 1 |
| refine ⟨length, ?_⟩ |
| intro i hi hs |
| have hband : 12 ≤ band i ∧ inHazardBand (p i) (band i) := by |
| dsimp only [band] |
| rw [dite_eq_left ⟨hi, hs⟩] |
| exact (exists_hazard_band hi hs).choose_spec |
| refine ⟨band i-12, mem_range.mpr ?_, ?_⟩ |
| · have hb := le_sup (f := band) (mem_univ i) |
| dsimp only [length] |
| omega |
| · have heq : 12 + (band i-12) = band i := by omega |
| rw [heq] |
| exact hband.2 |
|
|
| lemma hazard_band_disjoint {p : ℝ} {l k : ℕ} |
| (hl : inHazardBand p l) (hk : inHazardBand p k) : l = k := by |
| have impossible {a b : ℕ} (hab : a < b) |
| (ha : inHazardBand p a) (hb : inHazardBand p b) : False := by |
| have hpow : ((2 : ℝ)^b)⁻¹ ≤ ((2 : ℝ)^(a+1))⁻¹ := by |
| rw [← inv_pow, ← inv_pow] |
| exact pow_le_pow_of_le_one (by norm_num) (by norm_num) (by omega) |
| exact (not_lt_of_ge (hpow.trans ha.1)) hb.2 |
| exact le_antisymm (le_of_not_gt (fun h => impossible h hk hl)) |
| (le_of_not_gt (fun h => impossible h hl hk)) |
|
|
| end MatroidProphet |
|
|
| end OAI |
|
|
