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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | 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
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