import Mathlib

structure DFSGraph where
  adj : Array (Array Nat)

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

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

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

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

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

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

-- Main function definition
def dfs_check_reachability (graph : DFSGraph) (start target : Nat)
    (h_precond : dfs_check_reachability_precond graph start target) : Bool :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def dfs_check_reachability_postcond (graph : DFSGraph) (start target : Nat) (result : Bool)
    (_ : dfs_check_reachability_precond graph start target) : Prop :=
  -- !benchmark @start postcond
  result = graph.reachable start target
  -- !benchmark @end postcond

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

-- Proof content
theorem dfs_check_reachability_postcond_satisfied (graph : DFSGraph) (start target : Nat)
    (h_precond : dfs_check_reachability_precond graph start target) :
    dfs_check_reachability_postcond graph start target
      (dfs_check_reachability graph start target h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof