Random DFS in graph

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

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

The algorithm is depth-first-search in the graph. It picks randomly the son on which recursive call is done.
The proof is mutually recursive on pre and post conditions. Fully automatic proof.

module DfsRandomSearch
  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

  lemma mem_decidable:
    forall x: 'a, l: list 'a. L.mem x l \/ not L.mem x l

  lemma list_suffix_fst_not_twice:
    forall x, l "induction" : list vertex. L.mem x l -> exists l1 l2. l = l1 ++ (Cons x l2) /\ not L.mem x l2

  predicate white_vertex (x: vertex) (v: set vertex) =
    not (mem x v)
 
  predicate white_monotony (v1 v2: set vertex) =
    forall x: vertex. white_vertex x v2 -> white_vertex x v1

  predicate nodeflip (x: vertex) (v1 v2: set vertex) =
    white_vertex x v1 /\ not (white_vertex x v2)

  predicate whitepath (x: vertex) (l: list vertex) (z: vertex) (v: set vertex) =
    path x l z /\ (forall y. L.mem y l -> white_vertex y v) /\ white_vertex z v

  predicate whiteaccess (r: set vertex) (z: vertex) (v: set vertex) =
    exists x l. mem x r /\ whitepath x l z v

  predicate nodeflip_whitepath (roots v1 v2: set vertex) =
    forall z. nodeflip z v1 v2 -> whiteaccess roots z v1

  predicate whitepath_nodeflip (roots v1 v2: set vertex) =
    forall x l z. mem x roots -> whitepath x l z v1 -> nodeflip z v1 v2

  predicate whitepath_fst_not_twice (x: vertex) (l: list vertex) (z: vertex) (v: set vertex) =
     whitepath x l z v /\ path_fst_not_twice x l z

  lemma whitepath_mem_decomp: 
    forall x l1 l2 z y v. whitepath x (l1 ++ (Cons y l2)) z v -> whitepath x l1 y v /\ whitepath y (Cons y l2) z v

  lemma whitepath_mem_decomp_right: 
    forall x l z y v. whitepath x l z v -> L.mem y (l ++ (Cons z Nil)) -> exists l': list vertex. whitepath y l' z v
    
  predicate path_fst_not_twice (x: vertex) (l: list vertex) (z: vertex) = 
   path x l z /\ 
   match l with
  | Nil -> true
  | Cons _ l' -> x <> z /\ not (L.mem x l')
   end

  lemma why_type_decidable:
    forall x y: 'a. x = y \/ x <> y
    
  lemma Lmem_decidable:
    forall x:vertex, l. L.mem x l \/ not (L.mem x l)

  lemma path_suffix_fst_not_twice:
    forall x z l "induction". path x l z -> 
      exists l1 l2. l = l1 ++ l2 /\ path_fst_not_twice x l2 z

  lemma path_path_fst_not_twice:
    forall x z l. path x l z -> 
      exists l'. path_fst_not_twice x l' z /\ subset (E.elements l') (E.elements l)

  lemma whitepath_whitepath_fst_not_twice: 
    forall x z l v. whitepath x l z v -> exists l'. whitepath_fst_not_twice x l' z v

  lemma path_not_nil_inversion:
    forall x z l. path x (Cons x l) z -> exists y. edge x y /\ path y l z

  lemma whitepath_not_nil_inversion:
    forall x z l v. whitepath x (Cons x l) z v -> exists y. edge x y /\ whitepath y l z v

  lemma whitepath_not_nil_fst_not_twice_inversion:
    forall x z l v. whitepath_fst_not_twice x (Cons x l) z v -> x <> z ->
    (exists y. edge x y /\ whitepath y l z (add x v))
    
  lemma whitepath_fst_not_twice_inversion :
    forall x z l v. whitepath_fst_not_twice x l z v -> x <> z -> 
    (exists y: vertex, l': list vertex. edge x y /\ l = Cons x l' /\ whitepath y l' z (add x v))

  lemma whitepath_inversion : 
    forall x z l v. whitepath x l z v -> x <> z -> 
    (exists y: vertex, l': list vertex. edge x y /\ whitepath y l' z (add x v))

  lemma nodeflip_monotony :
    forall x v0 v1 v2. subset v0 v1 -> nodeflip x v1 v2 -> nodeflip x v0 v2

  lemma whitepath_monotony :
    forall x l z v0 v1. subset v0 v1 -> whitepath x l z v1 -> whitepath x l z v0

  lemma whitepath_cons:
    forall x y l z v. edge x y -> whitepath y l z v -> not mem x v -> whitepath x (Cons x l) z v

  lemma whitepath_trans:
    forall x l y l' z v. whitepath x l y v -> whitepath y l' z v -> whitepath x (l ++ l') z v

  lemma whitepath_Y:
    forall x l z y x' l' v. whitepath x l z v -> L.mem y (l ++ (Cons z Nil)) -> whitepath x' l' y v -> exists l0. whitepath x' l0 z v

  lemma whitepath_flip_node_flip:
    forall x l z v1 v2. whitepath x l z v1 -> not whitepath x l z (union v2 v1) -> exists y. L.mem y (l ++ (Cons z Nil)) /\ nodeflip y v1 v2
	  

let rec dfs (roots: set vertex) (visited: set vertex): set vertex
  variant {(cardinal vertices - cardinal visited), (cardinal roots)} =
  requires {subset roots vertices }
  requires {subset visited vertices }
  ensures {subset visited result}
  ensures {subset result vertices}
  ensures {nodeflip_whitepath roots visited result}
    if is_empty roots then visited
    else
      let x = choose roots in
      let roots' = remove x roots in
      if mem x visited then
        dfs roots' visited
      else begin
        let r' = dfs (successors x) (add x visited) in
        let r = dfs roots' r' in
        (*-------- nodeflip_whitepath ----------------------------*)
        assert {forall z. nodeflip z visited r -> nodeflip z visited r' \/ nodeflip z r' r}; 
        (*-------- case 1 ----------*)
        assert {forall z. nodeflip z visited r' -> z = x \/ nodeflip z (add x visited) r' };
        (* case 1-1: nodeflip z visited r' /\ z = x *)
            assert {whitepath x Nil x visited};
        (* case 1-2: nodeflip z visited r' /\ z <> x *)
        assert {forall z. nodeflip z (add x visited) r' -> whiteaccess (successors x) z (add x visited)};
        assert {forall x' l z. whitepath x' l z (add x visited) -> whitepath x' l z visited};
        assert {forall z x' l. edge x x' -> whitepath x' l z visited -> whitepath x (Cons x l) z visited};
        assert {forall z. nodeflip z (add x visited) r' -> whiteaccess roots z visited};
        (*-------- case 2 ------------*)
        assert {forall z. nodeflip z r' r -> whiteaccess roots' z r'};
        assert {forall z x' l. whitepath x' l z r' -> whitepath x' l z visited};

	r
     end

let dfs_main (roots: set vertex) : set vertex =
   requires {subset roots vertices}
   dfs roots empty

end