import Mathlib

structure BFSGraph where
  adj : Array (Array Nat)

def BFSGraph.well_formed (g : BFSGraph) : Prop :=
  ∀ u, u < g.adj.size →
    ∀ v, v ∈ g.adj.getD u #[] → v < g.adj.size

def BFSGraph.size (g : BFSGraph) : Nat :=
  g.adj.size

def BFSGraph.has_edge (g : BFSGraph) (u v : Nat) : Prop :=
  u < g.adj.size ∧
  v ∈ g.adj.getD u #[]

def BFSGraph.is_path (g : BFSGraph) (p : List Nat) : Prop :=
  p.length > 0 ∧
  ∀ i, i + 1 < p.length →
    g.has_edge (p.getD i 0) (p.getD (i + 1) 0)

def BFSGraph.reachable (g : BFSGraph) (start target : Nat) : Prop :=
  ∃ p, g.is_path p ∧ p.head? = some start ∧ p.getLast? = some target

def BFSGraph.is_shortest_distance (g : BFSGraph) (start target : Nat) (d : Nat) : Prop :=
  (∃ p, g.is_path p ∧ p.head? = some start ∧ p.getLast? = some target ∧ p.length = d + 1) ∧
  (∀ p, g.is_path p → p.head? = some start → p.getLast? = some target → p.length ≥ d + 1)

-- Precondition definitions
@[reducible, simp]
def bfs_shortest_path_precond (graph : BFSGraph) (start target : Nat) : Prop :=
  -- !benchmark @start precond
  graph.well_formed ∧ graph.size ≤ 10000 ∧ start < graph.size ∧ target < graph.size
  -- !benchmark @end precond

-- !benchmark @start auxcode
-- !benchmark @end auxcode

-- Main function definition
def bfs_shortest_path (graph : BFSGraph) (start target : Nat)
    (h_precond : bfs_shortest_path_precond graph start target) : Option Nat :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def bfs_shortest_path_postcond (graph : BFSGraph) (start target : Nat) (result : Option Nat)
    (_ : bfs_shortest_path_precond graph start target) : Prop :=
  -- !benchmark @start postcond
  match result with
  | some dist => graph.is_shortest_distance start target dist
  | none => ¬ graph.reachable start target
  -- !benchmark @end postcond

-- !benchmark @start lemma
-- !benchmark @end lemma

-- Proof content
theorem bfs_shortest_path_postcond_satisfied (graph : BFSGraph) (start target : Nat)
    (h_precond : bfs_shortest_path_precond graph start target) :
    bfs_shortest_path_postcond graph start target
      (bfs_shortest_path graph start target h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof