| import OAI.Probability.LatticePassage.Model |
| import Mathlib.Probability.Independence.InfinitePi |
|
|
| namespace OAI |
|
|
| namespace CompanionFPP |
| open MeasureTheory ProbabilityTheory Set |
|
|
| noncomputable instance environmentProbability : IsProbabilityMeasure environmentLaw := |
| inferInstanceAs (IsProbabilityMeasure (Measure.infinitePi (fun _ : Edge => expMeasure 1))) |
|
|
| theorem edge_marginal (e : Edge) : |
| Measure.map (fun ω : Environment => ω e) environmentLaw = expMeasure 1 := |
| Measure.infinitePi_map_eval _ e |
|
|
| theorem independent_edge_weights : iIndepFun (fun e (ω : Environment) => ω e) environmentLaw := |
| iIndepFun_infinitePi (X := fun _ (x : ℝ) => x) (fun _ => measurable_id) |
|
|
| theorem exponential_positive : ∀ᵐ x ∂expMeasure 1, 0 < x := by |
| have hc : (expMeasure 1 (Iic 0)).toReal = 0 := by |
| rw [← measureReal_def, ← cdf_eq_real] |
| simp [cdf_expMeasure_eq (r := 1) (by norm_num)] |
| have hz : expMeasure 1 (Iic 0) = 0 := |
| (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) ENNReal.zero_ne_top).mp |
| (by simpa using hc) |
| simpa only [ae_iff, not_lt, Iic] using hz |
|
|
| theorem all_edge_weights_positive : ∀ᵐ ω ∂environmentLaw, ∀ e, 0 < ω e := by |
| rw [ae_all_iff] |
| intro e |
| exact (measurePreserving_eval_infinitePi (fun _ : Edge => expMeasure 1) e).quasiMeasurePreserving.ae exponential_positive |
|
|
| theorem all_path_costs_nonnegative : |
| ∀ᵐ ω ∂environmentLaw, ∀ x y (p : lattice.Walk x y), 0 ≤ pathCost ω p := by |
| filter_upwards [all_edge_weights_positive] with ω hω |
| intro x y p |
| exact pathCost_nonneg ω (fun e => (hω e).le) p |
|
|
| end CompanionFPP |
|
|
| end OAI |
|
|
