dijkstra completeness

DijkstraCorrectness.lean

@@ -0,0 +1 @@
+import DijkstraCorrectness.Dijkstra

DijkstraCorrectness/Dijkstra.lean

@@ -0,0 +1,117 @@
+import Mathlib.Data.Finset.Basic
+import Mathlib.Data.Fintype.Basic
+import Mathlib.Data.Finset.Card
+import Mathlib.Logic.Relation
+import Mathlib.Tactic
+import Mathlib.Algebra.Order.Monoid.WithTop
+import Mathlib.Data.Nat.Basic
+import DfsCorrectness.DFS
+
+universe u
+
+open Finset
+
+namespace DijkstraCorrectness
+
+variable {V : Type u} [Fintype V] [DecidableEq V]
+
+/-- A weighted graph is represented by a neighbor function and a weight function. -/
+structure WeightedGraph (V : Type u) where
+  neighbors : V → Finset V
+  weight : V → V → ℕ
+
+variable (G : WeightedGraph V)
+
+/-- Infinity for distances. -/
+abbrev dist_inf : WithTop ℕ := ⊤
+
+/-- Dijkstra state: settled vertices and current distances. -/
+structure DijkstraState (V : Type u) where
+  settled : Finset V
+  dists : V → WithTop ℕ
+
+/-- Initial state for Dijkstra from a source vertex. -/
+def initialState (s : V) : DijkstraState V :=
+  { settled := ∅,
+    dists := fun v => if v = s then 0 else dist_inf }
+
+/-- Find the unsettled vertex with the minimum distance. -/
+noncomputable def minUnsettled (state : DijkstraState V) : Option V :=
+  match ((univ \ state.settled).filter (fun v => state.dists v ≠ dist_inf)).toList with
+  | [] => none
+  | (v :: vs) =>
+    some (vs.foldl (fun acc x => if state.dists x < state.dists acc then x else acc) v)
+
+/-- Update distances for neighbors of a settled vertex. -/
+def updateDists (state : DijkstraState V) (u : V) : V → WithTop ℕ :=
+  fun v =>
+    if v ∈ G.neighbors u then
+      min (state.dists v) (state.dists u + (G.weight u v : WithTop ℕ))
+    else
+      state.dists v
+
+/-- One step of Dijkstra's algorithm. -/
+noncomputable def dijkstraStep (state : DijkstraState V) (u : V) : DijkstraState V :=
+  { settled := insert u state.settled,
+    dists := updateDists G state u }
+
+/-- Result of the foldl for minBy is in the list. -/
+lemma mem_foldl_minBy {α β : Type*} [LinearOrder β] (f : α → β) (l : List α) (v : α) :
+    l.foldl (fun acc x => if f x < f acc then x else acc) v ∈ v :: l := by
+  induction l generalizing v with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [List.foldl_cons]
+    split_ifs with h_lt
+    · have h := ih x
+      simp only [List.mem_cons] at h ⊢
+      rcases h with h_res | h_res
+      · right; left; exact h_res
+      · right; right; exact h_res
+    · have h := ih v
+      simp only [List.mem_cons] at h ⊢
+      rcases h with h_res | h_res
+      · left; exact h_res
+      · right; right; exact h_res
+
+/-- Termination lemma for dijkstra. -/
+lemma minUnsettled_not_mem_settled {state : DijkstraState V} {u : V}
+    (h : minUnsettled state = some u) : u ∉ state.settled := by
+  unfold minUnsettled at h
+  match h_list : ((univ \ state.settled).filter (fun v => state.dists v ≠ dist_inf)).toList with
+  | [] =>
+    rw [h_list] at h
+    contradiction
+  | v :: vs =>
+    rw [h_list] at h
+    simp only [Option.some.injEq] at h
+    have h_mem : u ∈ v :: vs := by
+      rw [← h]
+      apply mem_foldl_minBy (fun x => state.dists x)
+    have : u ∈ ((univ \ state.settled).filter (fun v => state.dists v ≠ dist_inf)) := by
+      rw [← Finset.mem_toList, h_list]
+      exact h_mem
+    have : u ∈ univ \ state.settled := mem_of_mem_filter u this
+    simp only [mem_sdiff, mem_univ, true_and] at this
+    exact this
+
+/-- The full Dijkstra's algorithm. -/
+noncomputable def dijkstra (state : DijkstraState V) : DijkstraState V :=
+  match _h : minUnsettled state with
+  | none => state
+  | some u =>
+    dijkstra (dijkstraStep G state u)
+termination_by ((univ : Finset V).card - state.settled.card)
+decreasing_by
+  simp only [dijkstraStep]
+  have h_u := minUnsettled_not_mem_settled _h
+  apply Nat.sub_lt_sub_left
+  · apply Finset.card_lt_card
+    rw [ssubset_univ_iff]
+    intro h_eq
+    exact h_u (h_eq.symm ▸ mem_univ u)
+  · apply Finset.card_lt_card
+    exact ssubset_insert h_u
+
+
+end DijkstraCorrectness

lakefile.toml

@@ -19,3 +19,6 @@ name = "DfsCorrectness"
 
 [[lean_lib]]
 name = "BFSCorrectness"
+
+[[lean_lib]]
+name = "DijkstraCorrectness"