# Random search 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 a ministep of a random search in the graph. It is generic to any search strategy.
The proof is adapted from Dowek and Munoz and is fully automatic.

```
module RandomSearch
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
use import init_graph.GraphSetSuccPath

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 whitereachable (x z: vertex) (v: set vertex) =
exists l. whitepath x l z v

predicate whiteaccess (r s: set vertex) (v: set vertex) =
forall z. mem z s -> exists x. mem x r /\ whitereachable x 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 ++ (Cons z Nil)) -> 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))

```

### program

```
let rec random_search roots visited
variant {(cardinal vertices - cardinal visited), (cardinal roots)} =
requires {subset roots vertices }
requires {subset visited vertices }
ensures {subset visited result}
ensures {forall x l z. mem x roots -> whitepath x l z visited -> mem z result }
if is_empty roots then
visited
else
let x = choose roots in
let roots' = remove x roots in
if mem x visited then
random_search roots' visited
else begin
let r = random_search (union roots' (successors x)) (add x visited) in
(* -------------- whitepath_nodeflip ------------ *)
(* case 1: whitepath roots' l z /\ not (L.mem x l \/ z = x) *)
assert {forall y l z. mem y roots' -> whitepath y l z visited -> not (L.mem x l \/ x = z)
-> whitepath y l z (add x visited)};
(* case 2: whitepath roots' l z /\ (L.mem x l \/ z = x) *)
assert {forall y l z. whitepath y l z visited -> (L.mem x l \/ z = x)
-> exists l'. whitepath x l' z visited};
(* case 2-1: whitepath x l z visited /\ z = x *)
assert {forall z. z = x -> mem z r};
(* case 2-2: whitepath x l z visited /\ z <> x *)
(* using lemma whitepath_whitepath_fst_not_twice *)
assert {forall l z. z <> x -> whitepath x l z visited
-> exists x' l'. edge x x' /\ whitepath x' l' z (add x visited) };
r
end

let random_search_main (roots: set vertex) =
requires {subset roots vertices}
random_search roots (empty)

end
```

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