DFS in graph - the whitepath theorem

The graph is represented by a pair (vertices, successors)

• vertices : this constant is the set of graph vertices
• successors : this function gives for each vertex the set of vertices directly joinable from it

This algorithm is depth-first-search in the graph. It picks randomly the son on which recursive call is done.
This theorem refers to the White-path theorem in book by Cormen et al.
The new visited vertices are exactly the ones reachable from roots by a white path w.r.t the previously visited set.
Fully automatic proof, with inductive definition of white paths and heavy use of the "by" connective.

Notice that this proof uses paths for the soundness part, but relies on the non-black-to-white property for its completeness.
This proof is close to the one in Cormen et al.; but is simpler since there is no need for the so-called Parenthesis theorem, nor for a time variable. The present proof follows from the structure of the recursion.

The formula "A && B" means "A /\ A -> B" .
The formula "A by B" means "B /\ B -> A"

module DfsWhitePathThm
  use import int.Int
  use import list.List
  use import list.Append
  use import list.Mem as L
  use import list.Elements as E
  use import init_graph.GraphSetSucc

  predicate white_vertex (x: vertex) (v: set vertex) =
    not (mem x v)
 
  inductive wpath vertex (list vertex) vertex (set vertex) =
  | WPath_empty:
      forall x v. white_vertex x v -> wpath x Nil x v
  | WPath_cons:
      forall x y l z v.
      white_vertex x v -> edge x y -> wpath y l z v -> wpath x (Cons x l) z v

  predicate whiteaccess (r s v: set vertex) =
    forall z. mem z s -> exists x l. mem x r /\ wpath x l z v

  lemma whiteaccess_var:
    forall r r' s v. subset r r' -> whiteaccess r s v -> whiteaccess r' s v

  lemma whiteaccess_covar1:
    forall r s s' v. subset s s' -> whiteaccess r s' v -> whiteaccess r s v

  lemma wpath_covar2:
    forall x l z v v'. subset v v' -> wpath x l z v' -> wpath x l z v

  lemma whiteaccess_covar2:
    forall r s v v'. subset v v' -> whiteaccess r s v' -> whiteaccess r s v

  lemma wpath_trans:
    forall x l y l' z v. wpath x l y v -> wpath y l' z v -> wpath x (l ++ l') z v

  lemma whiteaccess_trans:
    forall r r' s v. whiteaccess r r' v -> whiteaccess r' s v -> whiteaccess r s v

  lemma whiteaccess_cons:
    forall x s v. mem x vertices ->
      white_vertex x v -> whiteaccess (successors x) s v -> whiteaccess (add x empty) s v

  predicate nbtw (r v: set vertex) =
    forall x x'. edge x x' -> mem x r -> mem x' (union r v)

  lemma nbtw_path:
    forall v v'. nbtw (diff v' v) v' -> 
      forall x l z. mem x (diff v' v) -> wpath x l z v -> mem z (diff v' v)

	  
let rec dfs r v 
  variant {(cardinal vertices - cardinal v), (cardinal r)} =
  requires {subset r vertices }
  requires {subset v vertices }
  ensures {subset result vertices }
  ensures {subset v result }
  ensures {subset r result }
  ensures {nbtw (diff result v) result &&
           forall s. whiteaccess r s v <-> subset s (diff result v) }
  if is_empty r then v else
  let x = choose r in
  let r' = remove x r in
  if mem x v then dfs r' v else
  let v' = dfs (successors x) (add x v) in
  assert {whiteaccess (add x empty) (diff v' (add x v)) v
    by whiteaccess (successors x) (diff v' (add x v)) v
    by subset (diff v' (add x v)) (diff v' v)
    by subset v (add x v) };
  assert {whiteaccess (add x empty) (diff v' v) v
    by diff v' v == add x (diff v' (add x v))} ;
  let v'' = dfs r' v' in
  assert {diff v'' v == union (diff v'' v') (diff v' v) };
  assert {whiteaccess r' (diff v'' v') v
    by subset v v' };
  v''

end