| 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 | |