| import OAI.Probability.LatticePassage.ExponentialMoment |
| import Mathlib.Combinatorics.SimpleGraph.Paths |
|
|
| namespace OAI |
|
|
| namespace CompanionFPP |
| open MeasureTheory ProbabilityTheory Real Finset |
| open scoped ENNReal |
|
|
| variable {V : Type*} {G : SimpleGraph V} |
|
|
| def pathEdges {x y : V} : G.Walk x y → List G.edgeSet |
| | .nil => [] |
| | .cons h p => ⟨s(_, _), h⟩ :: pathEdges p |
|
|
| lemma pathEdges_val {x y : V} (p : G.Walk x y) : |
| (pathEdges p).map Subtype.val = p.edges := by |
| induction p with |
| | nil => rfl |
| | cons h p ih => simp [pathEdges, ih] |
|
|
| lemma pathEdges_length {x y : V} (p : G.Walk x y) : |
| (pathEdges p).length = p.length := by |
| induction p with |
| | nil => rfl |
| | cons h p ih => simp [pathEdges, ih] |
|
|
| lemma pathEdges_nodup {x y : V} {p : G.Walk x y} (hp : p.IsTrail) : |
| (pathEdges p).Nodup := by |
| apply List.Nodup.of_map (f := Subtype.val) |
| rw [pathEdges_val] |
| exact hp.edges_nodup |
|
|
| lemma pathCost_eq_edgeList_sum (ω : Weights G) {x y : V} (p : G.Walk x y) : |
| pathCost ω p = ((pathEdges p).map ω).sum := by |
| induction p with |
| | nil => rfl |
| | cons h p ih => simp [pathEdges, pathCost, edgeCost, ih] |
|
|
| theorem cheap_trail_probability {x y : Vertex} (p : lattice.Walk x y) (hp : p.IsTrail) |
| (t : ℝ) : environmentLaw {ω : Environment | pathCost ω p ≤ t} ≤ |
| ENNReal.ofReal (1 / 6 : ℝ) ^ p.length / ENNReal.ofReal (exp (-5 * t)) := by |
| classical |
| have hn := pathEdges_nodup hp |
| have hc : (pathEdges p).toFinset.card = p.length := by |
| rw [List.toFinset_card_of_nodup hn, pathEdges_length] |
| have he : ∀ ω : Environment, (∑ e ∈ (pathEdges p).toFinset, ω e) = pathCost ω p := by |
| intro ω |
| rw [pathCost_eq_edgeList_sum] |
| exact List.sum_toFinset _ hn |
| have h := cheap_edge_set_probability (pathEdges p).toFinset t |
| simpa only [he, hc] using h |
|
|
| end CompanionFPP |
|
|
| end OAI |
|
|
