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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 | import OAI.Probability.LatticePassage.Geodesic
namespace OAI
namespace CompanionFPP
private lemma sublist_length_of_cost {α : Type*} (f : α → ℝ)
{l₁ l₂ : List α} (h : l₁.Sublist l₂) (hpos : ∀ a ∈ l₂, 0 < f a)
(hcost : (l₂.map f).sum ≤ (l₁.map f).sum) : l₂.length ≤ l₁.length := by
induction h with
| slnil => simp
| @cons l₁ l₂ a h ih =>
have hnonneg : ∀ b ∈ l₂, 0 ≤ f b := fun b hb => (hpos b (by simp [hb])).le
have hle := (h.map f).sum_le_sum (by
intro b hb
obtain ⟨c,hc,rfl⟩ := List.mem_map.mp hb
exact hnonneg c hc)
have ha := hpos a (by simp)
simp only [List.map_cons, List.sum_cons] at hcost
linarith
| @cons_cons l₁ l₂ a h ih =>
have ht : ∀ b ∈ l₂, 0 < f b := fun b hb => hpos b (by simp [hb])
have hs : (l₂.map f).sum ≤ (l₁.map f).sum := by
simpa only [List.map_cons, List.sum_cons, add_le_add_iff_left] using hcost
simpa using ih ht hs
theorem minimizing_walk_isPath (ω : Environment) (hω : ∀ e, 0 < ω e)
{x y : Vertex} (p : lattice.Walk x y)
(hp : pathCost ω p = passageTime ω x y) : p.IsPath := by
classical
have hcost : pathCost ω p ≤ pathCost ω p.bypass := by
rw [hp]
exact passageTime_le_pathCost ω (fun e => (hω e).le) p.bypass
rw [pathCost_eq_darts_sum, pathCost_eq_darts_sum] at hcost
have hl := sublist_length_of_cost (fun d : lattice.Dart => edgeCost ω d.adj)
p.darts_bypass_sublist_darts (by intro d _; exact hω _) hcost
have he := p.length_le_bypass_length_iff.mp (by simpa using hl)
rw [← he]
exact p.bypass_isPath
theorem all_minimizing_walks_simple : ∀ᵐ ω ∂environmentLaw, ∀ x y : Vertex,
∀ p : lattice.Walk x y, pathCost ω p = passageTime ω x y → p.IsPath := by
filter_upwards [all_edge_weights_positive] with ω hω
exact fun _ _ p hp => minimizing_walk_isPath ω hω p hp
end CompanionFPP
end OAI
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