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 Dfs use import int.Int use import ref.Ref use import array.Array use import set.Fset 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 list.Elements as E use import GraphArraySucc use import GraphArraySuccPath type color = WHITE | GRAY | BLACK predicate nodeflip (x: int) (c1 c2: array color) = c1[x] = WHITE /\ c2[x] <> WHITE predicate whitepath (x: int) (l: list int) (z: int) (g: graph) (c: array color) = path x l z g /\ (forall y: int. mem y l -> c[y] = WHITE) /\ c[z] = WHITE predicate whiteaccess (lv: list int) (z: int) (g: graph) (c: array color) = exists x l. mem x lv /\ whitepath x l z g c predicate path_fst_not_twice (x: int) (l: list int) (z: int) (g: graph) = path x l z g /\ match l with | Nil -> true | Cons _ l' -> x <> z /\ not (mem x l') end lemma path_suffix_fst_not_twice: forall x z g l "induction". path x l z g -> exists l1 l2. l = l1 ++ l2 /\ path_fst_not_twice x l2 z g lemma path_path_fst_not_twice: forall x z l g. path x l z g -> exists l'. path_fst_not_twice x l' z g /\ subset (E.elements l') (E.elements l) predicate whitepath_fst_not_twice (x: int) (l: list int) (z: int) (g: graph) (c: array color) = whitepath x l z g c /\ path_fst_not_twice x l z g lemma whitepath_decomposition: forall x l1 l2 z y g c. whitepath x (l1 ++ (Cons y l2)) z g c -> whitepath x l1 y g c /\ whitepath y (Cons y l2) z g c lemma whitepath_decomp: forall x l z y g c. whitepath x l z g c -> mem y l -> exists l': list int. whitepath y l' z g c lemma whitepath_mem_decomposition_r: forall x l z y g c. wf g -> vertex g x -> whitepath x l z g c -> (mem y l \/ y = z) -> exists l': list int. whitepath y l' z g c lemma whitepath_whitepath_fst_not_twice: forall x z l g c. whitepath x l z g c -> exists l'. whitepath_fst_not_twice x l' z g c lemma path_cons_inversion: forall x z l g. path x (Cons x l) z g -> exists y. edge x y g /\ path y l z g lemma whitepath_cons_inversion: forall x z l g c. whitepath x (Cons x l) z g c -> exists y. edge x y g /\ whitepath y l z g c lemma whitepath_cons_fst_not_twice_inversion: forall x z l g c. whitepath_fst_not_twice x (Cons x l) z g c -> x <> z -> (exists y. edge x y g /\ whitepath y l z g (set c x GRAY)) lemma whitepath_fst_not_twice_inversion : forall x z l g c. whitepath_fst_not_twice x l z g c -> x <> z -> (exists y l'. edge x y g /\ whitepath y l' z g (set c x GRAY)) predicate nodeflip_whitepath (roots: list int) (g: graph) (c1 c2: array color) = forall z. nodeflip z c1 c2 -> whiteaccess roots z g c1 predicate whitepath_nodeflip (roots: list int) (g: graph) (c1 c2: array color) = forall x l z. mem x roots -> whitepath x l z g c1 -> nodeflip z c1 c2 lemma whitepath_trans: forall x l1 y l2 z g c. whitepath x l1 y g c -> whitepath y l2 z g c -> whitepath x (l1 ++ l2) z g c lemma whitepath_Y: forall x l z y x' l' g c. whitepath x l z g c -> (mem y l \/ y = z) -> whitepath x' l' y g c -> exists l0. whitepath x' l0 z g c 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 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
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 {whitepath_nodeflip (Cons x Nil) g (old c) c} ensures {nodeflip_whitepath (Cons x Nil) g (old c) c} 'L0: c[x] <- GRAY; assert {whitepath_cons x g (at c 'L0) c}; (* nodeflip_whitepath *) 'L: let sons = ref (g[x]) in let ghost lv = ref Nil in while ( !sons <> Nil) do invariant {(reverse !lv) ++ !sons = g[x]} invariant {forall y. mem y !sons -> vertex g y} invariant {white_monotony g (at c 'L) c} invariant {whitepath_monotony g (at c 'L) c} invariant {whitepath_nodeflip !lv g (at c 'L) c} invariant {nodeflip_whitepath !lv g (at c 'L) c} 'L1: match !sons with | Nil -> () | Cons y sons' -> (*--------- whitepath_nodeflip ------------*) assert {forall l z. whitepath y l z g (at c 'L) -> not whitepath y l z g c -> exists z'. (mem z' l \/ z' = z) /\ nodeflip z' (at c 'L) c}; assert {forall l z. whitepath y l z g (at c 'L) -> not whitepath y l z g c -> exists y' l' z'. (mem z' l \/ z' = z) /\ mem y' !lv /\ whitepath y' l' z' g (at c 'L)}; assert {forall l z. whitepath y l z g (at c 'L) -> not whitepath y l z g c -> exists y' l'. mem y' !lv /\ whitepath y' l' z g (at c 'L)}; assert {forall l z. whitepath y l z g (at c 'L) -> not whitepath y l z g c -> nodeflip z (at c 'L) c}; if c[y] = WHITE then begin dfs g y c; end; sons := sons'; lv := Cons y !lv end done; 'L2: (*------- nodeflip_whitepath -----------*) assert {whitepath x Nil x g (at c 'L0)}; assert {nodeflip_whitepath (Cons x Nil) g (at c 'L0) (at c 'L)}; assert {forall z. nodeflip z (at c 'L) c -> exists l. whitepath x l z g (at c 'L0)}; assert {forall z. nodeflip z (at c 'L0) c -> exists l. whitepath x l z g (at c 'L0)}; (*------- whitepath_nodeflip -----------*) assert {forall z l. whitepath x l z g (at c 'L0) -> x <> z -> nodeflip z (at c 'L0) c}; assert {nodeflip x (at c 'L0) c}; c[x] <- BLACK; assert {forall z. nodeflip z (at c 'L0) c <-> nodeflip z (at c 'L0) (at c 'L2)} 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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