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DFS in graph - soundness of the whitepath theorem

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.
This theorem refers to the whitepath theorem in Cormen et al.
The new visited vertices are reachable by a white path w.r.t the old visited set.
Fully automatic proof, with inductive definition of white paths, mutable set of visited vertices, black/white color. The proof is similar to the functional one.

Notice that this proof uses paths.

```
module DfsWhitePathSoundness
use import int.Int
use import list.List
use import list.Append
use import list.Mem as L
use import list.Length
use import list.HdTlNoOpt
use import list.Reverse
use import list.Elements as E
use import array.Array
use import ref.Ref
use import init_graph.GraphListArraySucc

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 (roots b v: set vertex) =
forall z. mem z b -> exists x l. mem x roots /\ wpath x l z v

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

lemma nbtw_wpath:
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)

lemma path_wpathempty:
forall y l z. wpath y l z empty -> path y l z

lemma nbtw_path:
forall v. nbtw v v ->
forall x l z. mem x v -> path x l z -> mem z v

```

program

```  type color = WHITE | GRAY | BLACK

let rec dfs1 x (ghost v) col =
requires {mem x vertices }
requires {col[x] = WHITE /\ Array.length col = cardinal vertices}
requires {forall x. mem x !v <-> mem x vertices /\ col[x] <> WHITE}
ensures {forall x. mem x !v <-> mem x vertices /\ col[x] <> WHITE}
ensures {subset !(old v) !v }
ensures {mem x !v}
ensures {nbtw (diff !v !(old v)) !v &&
forall s. whiteaccess (add x empty) s !(old v) -> subset s (diff !v !(old v)) }

'L0:
let ghost v0 = !v in
assert {not mem x v0};
col[x] <- GRAY; v := add x !v;
let sons = ref (successors x) in
let ghost dejavu = ref Nil in
while !sons <> Nil do
invariant {subset (elements !sons) vertices}
invariant {subset !v vertices}
invariant {forall z. mem z !v <-> mem z vertices /\ col[z] <> WHITE}
invariant {subset (add x v0) !v}
invariant {subset (elements !dejavu) !v}
invariant {nbtw (diff !v (add x v0)) !v}
invariant {successors x = (reverse !dejavu) ++ !sons}
let y = hd !sons in
if col[y] = WHITE then dfs1 y v col;
sons := tl !sons;
dejavu := Cons y !dejavu;
done;
'L1:
let ghost v' = !v in
assert {diff v' v0 == add x (diff v' (add x v0))};
col[x] <- BLACK

let dfs_main roots =
requires {subset (elements roots) vertices}
ensures {subset (elements roots) (elements result)}
ensures {nbtw (elements result) (elements result)
so forall x l z. mem x (elements roots) -> path x l z -> mem z (elements result)}
let ghost v = ref empty in let col = make (cardinal vertices) WHITE in
let xs = ref roots in
let ghost dejavu = ref Nil in
while !xs <> Nil do
invariant {subset (elements !xs) vertices}
invariant {subset !v vertices}
invariant {forall z. mem z !v <-> mem z vertices /\ col[z] <> WHITE}
invariant {nbtw !v !v}
invariant {subset (elements !dejavu) !v}
invariant {roots = (reverse !dejavu) ++ !xs}
let y = hd !xs in
if col[y] = WHITE then dfs1 y v col;
xs := tl !xs;
dejavu := Cons y !dejavu;
done;
assert {subset (elements roots) !v};
let res = ref Nil in
for x = 0 to (cardinal vertices) - 1  do
invariant {forall y. L.mem y !res <-> 0 <= y < x /\ col[y] <> WHITE}
if col[x] <> WHITE then res := Cons x !res
done;
assert {elements !res == !v};
!res

```

```
```

extracted program

```
(* let rec dfs1 x col =
col[x] <- GRAY;
let sons = ref (successors x) in
while !sons <> Nil do
let y = hd !sons in
if col[y] = WHITE then dfs1 y col;
sons := tl !sons;
done;
col[x] <- BLACK

let dfs_main roots =
let col = make (cardinal vertices) WHITE in
let xs = ref roots in
while !xs <> Nil do
let y = hd !xs in
if col[y] = WHITE then dfs1 y col;
xs := tl !xs;
done;
let res = ref Nil in
for x = 0 to (cardinal vertices) - 1  do
if col[x] <> WHITE then res := Cons x !res
done;
!res
*)

end
```

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