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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
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.Elements as E use import init_graph.GraphSetSucc predicate white_vertex (x: vertex) (v: set vertex) = not (mem x v) 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 lemma whitepath_id: forall x v. not mem x v -> whitepath x Nil x v lemma whitepath_sub: forall v v' x l y. subset v v' -> whitepath x l y v' -> whitepath x l y v lemma whitepath_cons: forall x x' y l v. not mem x v -> edge x x' -> whitepath x' l y v -> whitepath x (Cons x l) y v
let rec dfs r v variant {(cardinal vertices - cardinal v), (cardinal r)} = requires {subset r vertices } requires {subset v vertices } ensures {subset v result} ensures {subset result vertices} ensures {forall z. mem z (diff result v) -> exists y l. mem y r /\ whitepath y l z 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 let v'' = dfs r' v' in (*-------- nodeflip_whitepath ------------------*) assert {forall z. mem z (diff v'' v) -> z <> x -> (exists y l. (mem y r'/\ whitepath y l z v' )) \/ (exists y' l'. mem y' (successors x) /\ whitepath y' l' z v so exists l. whitepath x l z v)}; v'' let dfs_main roots = requires {subset roots vertices} ensures {forall z. mem z result -> exists y l. mem y roots /\ path y l z} dfs roots empty end
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