The graph is represented by array (list of successors)
module GraphArraySucc use import int.Int use import array.Array use import list.List use import list.Mem use import list.NumOcc use import list.Append type graph = array (list int) predicate spl (lv: list int) = forall x: int. mem x lv -> num_occ x lv = 1 function order (g: graph) : int = length g predicate vertex (g: graph) (x: int) = 0 <= x < order g predicate out (g: graph) (x: int) = forall y: int. vertex g x -> mem y g[x] -> vertex g y predicate wf (g: graph) = forall x: int. vertex g x -> out g x predicate edge (x y: int) (g: graph) = vertex g x /\ mem y g[x] lemma mem_decidable: forall x: int, lv: list int. mem x lv \/ not mem x lv lemma spl_single: forall x: int. spl (Cons x Nil) lemma spl_expansion: forall x: int, l: list int. spl l -> not (mem x l) -> spl (l ++ (Cons x Nil)) lemma spl_sub: forall lv lv1 lv2: list int. lv = lv1 ++ lv2 -> spl lv -> spl lv1 /\ spl lv2 end module GraphArraySuccPath use import GraphArraySucc use import int.Int use import array.Array use import list.List use import list.Mem use import list.Append inductive path int (list int) int graph = | Path_empty: forall x: int, g: graph. vertex g x -> path x Nil x g | Path_cons: forall x y z: int, l: list int, g: graph. edge x y g -> path y l z g -> path x (Cons x l) z g predicate path_fst_not_twice (x z: int) (l: list int) (g: graph) = path x (Cons x l) z g /\ (not mem x l) lemma path_edge: forall x z: int, g: graph. path x (Cons x Nil) z g -> edge x z g lemma path_hd: forall x y z: int, l: list int, g: graph. path x (Cons y l) z g -> x = y lemma path_right_extension: forall x y z: int, l: list int, g: graph. wf g -> path x l y g -> edge y z g -> path x (l ++ Cons y Nil) z g lemma path_right_inversion: forall x z: int, l: list int, g: graph. path x l z g -> (x = z /\ l = Nil) \/ (exists y: int, l': list int. path x l' y g /\ edge y z g /\ l = l' ++ Cons y Nil) lemma path_trans: forall x y z: int, l1 l2: list int, g: graph. path x l1 y g -> path y l2 z g -> path x (l1 ++ l2) z g lemma empty_path: forall x z: int, g: graph. path x Nil z g -> x = z lemma path_decomposition: forall x y z: int, l1 l2: list int, g: graph. path x (l1 ++ Cons y l2) z g -> path x l1 y g /\ path y (Cons y l2) z g lemma path_vertex_l : forall x y z: int, l : list int, g : graph. wf g -> vertex g x -> path x l z g -> mem y l -> vertex g y lemma path_vertex_r : forall x z : int, l : list int, g : graph. wf g -> vertex g x -> path x l z g -> vertex g z lemma path_vertex_last_occ : forall x y z: int, l : list int, g: graph. path x l z g -> mem y l -> (exists l1 l2 : list int. l = l1 ++ (Cons y l2) /\ (path_fst_not_twice y z l2 g)) end module GraphArraySuccPathColored use import GraphArraySucc use import GraphArraySuccPath use import int.Int use import array.Array use import list.List use import list.Mem use import list.Append type color = WHITE | GRAY | BLACK predicate whitepath (x: int) (l: list int) (z: int) (g: graph) (c: array color) = path x l z g /\ c[x] = WHITE = c[z] /\ (forall y: int. mem y l -> c[y] = WHITE) predicate node_flip (x: int) (c1 c2: array color) = c1[x] = WHITE /\ c2[x] <> WHITE predicate whitepath_flip (x z: int) (l: list int) (g: graph) (c1 c2: array color) = whitepath x l z g c1 /\ not whitepath x l z g c2 lemma whitepath_trans: forall x y z: int, l1 l2: list int, g: graph, c: array color. whitepath x l1 y g c -> whitepath y l2 z g c -> whitepath x (l1 ++ l2) z g c lemma whitepath_mem_decomp: forall x y z: int, l: list int, g: graph, c: array color. whitepath x l z g c -> mem y l -> (exists l1 l2 : list int. l = l1 ++ (Cons y l2) /\ whitepath x l1 y g c /\ whitepath y (Cons y l2) z g c) lemma whitepath_mem_decomp_right: forall x y z: int, l: list int, g: graph, c: array color. wf g -> vertex g x -> whitepath x l z g c -> mem y (l ++ (Cons z Nil)) -> exists l': list int. whitepath y l' z g c lemma whitepathflip_contains_node_flip: forall x z: int, l: list int, g: graph, c1 c2: array color. wf g -> vertex g x -> whitepath_flip x z l g c1 c2 -> exists y: int. vertex g y /\ mem y (l ++ (Cons z Nil)) /\ node_flip y c1 c2 lemma whitepathflip_contains_node_flip_whitepath: forall x z: int, l: list int, g: graph, c1 c2: array color. wf g -> vertex g x -> whitepath_flip x z l g c1 c2 -> exists y: int, l': list int. vertex g y /\ mem y (l ++ (Cons z Nil)) /\ node_flip y c1 c2 /\ whitepath y l' z g c1 lemma whitepath_whitepostfix : forall x z : int, l: list int, g: graph, c : array color. wf g -> vertex g x -> whitepath x l z g c -> x <> z -> (exists y: int, l': list int. edge x y g /\ whitepath y l' z g (set c x GRAY)) end module Dfs use import int.Int use import ref.Ref use import array.Array use import list.List use import list.Mem use import list.HdTlNoOpt use import list.NthNoOpt use import list.Append use import list.Reverse use import GraphArraySucc use import GraphArraySuccPath use import GraphArraySuccPathColored predicate white_monotony (g: graph) (c1 c2: array color) = forall x: int. vertex g x -> c2[x] = WHITE -> c1[x] = WHITE predicate whitepath_monotony (g: graph) (c1 c2: array color) = forall x z: int, l: list int. vertex g x -> whitepath x l z g c2 -> whitepath x l z g c1 predicate node_flip_whitepath (x: int) (g: graph) (c1 c2: array color) = forall z: int. node_flip z c1 c2 -> exists l: list int. whitepath x l z g c1 predicate whitepath_node_flip (x: int) (g: graph) (c1 c2: array color) = forall z: int, l: list int. whitepath x l z g c1 -> node_flip z c1 c2 predicate node_flip_whitepath_in_list (lv: list int) (g: graph) (c1 c2: array color) = forall z: int. node_flip z c1 c2 -> exists x: int, l': list int. mem x lv /\ whitepath x l' z g c1 predicate whitepath_in_list_node_flip (lv: list int) (g: graph) (c1 c2: array color) = forall x: int. mem x lv -> whitepath_node_flip x g c1 c2 predicate whitepath_flip_whitepath (x: int) (g: graph) (c1 c2: array color) = (*new*) forall y z: int, l: list int. whitepath_flip y z l g c1 c2 -> exists l': list int. whitepath x l' z g c1 predicate whitepath_flip_whitepath_in_list (lv: list int) (g: graph) (c1 c2: array color) = (*new*) forall x z: int, l: list int. whitepath_flip x z l g c1 c2 -> exists x': int, l': list int. mem x' lv /\ whitepath x' l' z g c1 predicate whitepath_cons (x: int) (g: graph) (c1 c2: array color) = forall y z: int, l: list int. mem y g[x] -> whitepath y l z g c2 -> whitepath x (Cons x l) z g c1 lemma flip_case : forall x z: int, c1 c2: array color. node_flip z c1 (set c2 x BLACK) -> node_flip z (set c1 x GRAY) c2 \/ z = x
let rec dfs (g: graph) (x: int) (c: array color) = requires {wf g /\ vertex g x /\ Array.length c = order g} requires {c[x] = WHITE} ensures {white_monotony g (old c) c} ensures {node_flip_whitepath x g (old c) c && whitepath_flip_whitepath x g (old c) c} ensures {whitepath_node_flip x g (old c) c} 'L0: c[x] <- GRAY; assert {whitepath_cons x g (at c 'L0) c}; 'L: let sons = ref (g[x]) in let ghost lv = ref Nil in while ( !sons <> Nil) do invariant {(reverse !lv) ++ !sons = g[x]} invariant {white_monotony g (at c 'L) c} invariant {whitepath_monotony g (at c 'L) c} invariant {whitepath_flip_whitepath_in_list !lv g (at c 'L) c} invariant {node_flip_whitepath_in_list !lv g (at c 'L) c} invariant {whitepath_in_list_node_flip !lv g (at c 'L) c} 'L1: match !sons with | Nil -> () | Cons y sons' -> if c[y] = WHITE then begin dfs g y c; end; sons := sons'; lv := Cons y !lv end done; c[x] <- BLACK let dfs_main (g: graph) = requires { wf g } let n = length (g) in let c = make n WHITE in for i = 0 to n - 1 do if c[i] = WHITE then dfs g i c done end
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