| import OAI.Probability.GammaPassage.CheapPathCounting |
| import Mathlib.Analysis.SpecialFunctions.Log.Basic |
| import Mathlib.Tactic.FieldSimp |
| import Mathlib.Tactic.NormNum |
|
|
| namespace OAI |
|
|
| namespace GammaFPP |
| open MeasureTheory ProbabilityTheory PlanarFPP |
| open scoped ENNReal |
|
|
| noncomputable def cheapChernoff (shape rate : ℝ) : ℝ := |
| rate * (Real.exp (Real.log 16 / shape) - 1) |
|
|
| noncomputable def cheapSlope (shape rate : ℝ) : ℝ := |
| Real.log 2 / cheapChernoff shape rate |
|
|
| lemma cheapChernoff_pos {shape rate : ℝ} (hs : 0 < shape) (hr : 0 < rate) : |
| 0 < cheapChernoff shape rate := by |
| have hl : 0 < Real.log (16 : ℝ) := Real.log_pos (by norm_num) |
| exact mul_pos hr (sub_pos.mpr (Real.one_lt_exp_iff.mpr (div_pos hl hs))) |
|
|
| lemma cheapSlope_pos {shape rate : ℝ} (hs : 0 < shape) (hr : 0 < rate) : |
| 0 < cheapSlope shape rate := |
| div_pos (Real.log_pos (by norm_num)) (cheapChernoff_pos hs hr) |
|
|
| / |
| lemma cheapChernoff_factor {shape rate : ℝ} (hs : 0 < shape) (hr : 0 < rate) : |
| (rate / (rate + cheapChernoff shape rate)) ^ shape = (1 / 16 : ℝ) := by |
| have hd : rate + cheapChernoff shape rate = |
| rate * Real.exp (Real.log 16 / shape) := by |
| unfold cheapChernoff |
| ring |
| have hratio : rate / (rate + cheapChernoff shape rate) = |
| Real.exp (-(Real.log 16 / shape)) := by |
| rw [hd, Real.exp_neg] |
| field_simp [hr.ne', Real.exp_ne_zero] |
| rw [hratio, ← Real.exp_mul] |
| have he : -(Real.log 16 / shape) * shape = -Real.log 16 := by |
| field_simp [hs.ne'] |
| rw [he, Real.exp_neg, Real.exp_log (by norm_num : (0 : ℝ) < 16)] |
| norm_num |
|
|
| lemma cheapChernoff_mul_slope {shape rate : ℝ} (hs : 0 < shape) (hr : 0 < rate) : |
| cheapChernoff shape rate * cheapSlope shape rate = Real.log 2 := by |
| unfold cheapSlope |
| field_simp [(cheapChernoff_pos hs hr).ne'] |
| |
| /-- Exact cancellation of counting, Laplace factor and linear threshold. -/ |
| lemma cheap_count_bound_eq {shape rate : ℝ} (hs : 0 < shape) (hr : 0 < rate) (n : ℕ) : |
| (4 : ℝ)^n * (((rate / (rate + cheapChernoff shape rate)) ^ shape)^n / |
| Real.exp (-cheapChernoff shape rate * (cheapSlope shape rate * (n : ℝ)))) = |
| (1 / 2 : ℝ)^n := by |
| rw [cheapChernoff_factor hs hr] |
| have he : -cheapChernoff shape rate * (cheapSlope shape rate * (n : ℝ)) = |
| (n : ℝ) * (-Real.log 2) := by |
| have hc := cheapChernoff_mul_slope hs hr |
| calc |
| _ = -(cheapChernoff shape rate * cheapSlope shape rate) * (n : ℝ) := by ring |
| _ = _ := by rw [hc]; ring |
| rw [he, Real.exp_nat_mul, Real.exp_neg, Real.exp_log (by norm_num : (0 : ℝ) < 2)] |
| rw [← div_pow, ← mul_pow] |
| norm_num |
| |
| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
| variable {τ : Edge → Ω → ℝ} {shape rate : ℝ} |
| |
| /-- All actual trails of n steps from p are exponentially unlikely to cost at |
| most c*n. This includes n=0, where the empty path has probability one. -/ |
| theorem GammaEnvironment.cheap_fixed_length_decay_real |
| (h : GammaEnvironment shape rate P τ) (hs : 0 < shape) (hr : 0 < rate) |
| (p : Vertex) (n : ℕ) : |
| P.real {ω | ∃ ds : List Step, ds.length = n ∧ |
| ((trace p ds).map traversalEdge).Nodup ∧ |
| pathCost (fun e => τ e ω) p ds ≤ cheapSlope shape rate * (n : ℝ)} ≤ |
| (1 / 2 : ℝ)^n := by |
| have hsp := cheapChernoff_pos hs hr |
| have hbase : 0 ≤ (rate / (rate + cheapChernoff shape rate)) ^ shape := |
| (Real.rpow_pos_of_pos (div_pos hr (add_pos hr hsp)) shape).le |
| have hden : ENNReal.ofReal (Real.exp |
| (-cheapChernoff shape rate * (cheapSlope shape rate * (n : ℝ)))) ≠ 0 := |
| ENNReal.ofReal_ne_zero_iff.mpr (Real.exp_pos _) |
| have hfin : ((4 ^ n : ℕ) : ℝ≥0∞) * |
| ((ENNReal.ofReal ((rate / (rate + cheapChernoff shape rate)) ^ shape)) ^ n / |
| ENNReal.ofReal (Real.exp |
| (-cheapChernoff shape rate * (cheapSlope shape rate * (n : ℝ))))) ≠ ⊤ := |
| ENNReal.mul_ne_top (by simp) |
| (ENNReal.div_ne_top (ENNReal.pow_ne_top ENNReal.ofReal_ne_top) hden) |
| have hb := ENNReal.toReal_mono hfin |
| (h.cheap_fixed_length_probability hs hr (cheapChernoff shape rate) hsp p n |
| (cheapSlope shape rate * (n : ℝ))) |
| simp only [ENNReal.toReal_mul, ENNReal.toReal_ofNat, ENNReal.toReal_div, |
| ENNReal.toReal_pow, ENNReal.toReal_ofReal hbase, |
| ENNReal.toReal_ofReal (Real.exp_pos _).le, Nat.cast_pow, Nat.cast_ofNat] at hb |
| change P.real _ ≤ _ at hb |
| rw [cheap_count_bound_eq hs hr n] at hb |
| exact hb |
| |
| end GammaFPP |
| |
| end OAI |
| |
