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 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 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 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)

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_decomposition:
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_decomposition_r:
forall x l z y v. whitepath x l z v -> (L.mem y l \/ y = z) -> exists l': list vertex. whitepath y l' z v

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

lemma whitepath_cons_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_cons_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 l'. edge x y /\ whitepath y l' z (add x 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

lemma whitepath_trans:
forall x l1 y l2 z v. whitepath x l1 y v -> whitepath y l2 z v -> whitepath x (l1 ++ l2) z v

lemma whitepath_Y:
forall x l z y x' l' v. whitepath x l z v -> (L.mem y l \/ y = z) -> whitepath x' l' y v -> exists l0. whitepath x' l0 z v

```

program

```
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 {whitepath_nodeflip roots visited result}
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
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};

(*-------- whitepath_NODEFLIP -------------------------------------------*)
(* case 1: x = z *)
assert {mem x r'};

(* case 2: whitepath x l z /\ x <> z *)
(* using lemma whitepath_whitepath_fst_not_twice *)
assert {forall l z. whitepath x l z visited -> x <> z ->
whiteaccess (successors x) z (add x visited) };
assert {forall l z. whitepath x l z visited -> x <> z -> nodeflip z (add x visited) r'};
assert {forall l z. whitepath x l z visited -> nodeflip z visited r'};

(* case 3: whiteaccess roots' z visited *)
(* case 3-1: not (whiteaccess roots' z r') *)
assert {forall x' l z. whitepath x' l z visited -> not whitepath x' l z r' ->
exists y. (L.mem y l \/ y = z) /\ nodeflip y visited r' };
assert {forall x' l z. whitepath x' l z visited -> not whitepath x' l z r' ->
exists y. (L.mem y l \/ y = z) /\ (y = x  \/ whiteaccess (successors x) y (add x visited)) };
assert {forall y. whiteaccess (successors x) y (add x visited) ->
exists l'. whitepath x l' y visited };
assert {forall x' l z. whitepath x' l z visited -> not whitepath x' l z r' ->
exists y l'. (L.mem y l \/ y = z) /\ whitepath x l' y visited };
assert {forall x' l z. whitepath x' l z visited -> not whitepath x' l z r' ->
exists l'. whitepath x l' z visited };
(* goto cases 1-2 *)
assert {forall x' l z. whitepath x' l z visited -> not whitepath x' l z r' ->
nodeflip z visited r };
(* case 3-2: whiteaccess roots' z r' *)
assert {forall x' l z. mem x' roots' -> whitepath x' l z r' ->
nodeflip z visited r};
r

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

end
```

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