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The graph is represented by a pair (vertices
, successors
)
vertices
: this constant is the set of graph vertices
successors
: this function gives for each vertex the set of vertices directly joinable from it
Notice that this proof does not use paths.
module DfsWhitePathCompleteness 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) 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 (r s v: set vertex) = forall z. mem z s -> exists x l. mem x r /\ wpath x l z v predicate nbtw (r v: set vertex) = forall x x'. edge x x' -> mem x r -> mem x' (union r v) lemma nbtw_path: 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)
let rec dfs r v variant {(cardinal vertices - cardinal v), (cardinal r)} = requires {subset r vertices } requires {subset v vertices } ensures {subset result vertices } ensures {subset v result } ensures {subset r result } ensures {nbtw (diff result v) result && forall s. whiteaccess r s v -> subset s (diff result 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 assert {diff v' v == add x (diff v' (add x v)) }; let v'' = dfs r' v' in assert {diff v'' v == union (diff v'' v') (diff v' v) }; v'' end
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