{c vertices}
Proof adapted from Dowek and Munoz
[vertices] constant: set of graph vertices
[successor] function: for each vertex x, this function gives the set or vertices directly joinable from x
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 white_monotony (v1 v2: set vertex) = forall x: vertex. white_vertex x v2 -> white_vertex x v1 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 lemma whitepath_mem_decomp: 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_decomp_right: 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 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 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 predicate nodeflip_whitepath (roots v1 v2: set vertex) = forall z. nodeflip z v1 v2 -> exists x l. mem x roots /\ whitepath x l z v1 predicate whitepath_nodeflip (roots v1 v2: set vertex) = forall x l z. mem x roots -> whitepath x l z v1 -> nodeflip z v1 v2 lemma why_type_decidable: forall x y: 'a. x = y \/ x <> y lemma Lmem_decidable: forall x:vertex, l. L.mem x l \/ not (L.mem x l) 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) 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_not_nil_inversion: forall x z l. path x (Cons x l) z -> exists y. edge x y /\ path y l z lemma whitepath_not_nil_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_not_nil_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)) lemma whitepath_inversion : forall x z l v. whitepath x l z v -> x <> z -> (exists y: vertex, l': list vertex. edge x y /\ whitepath y l' z (add x v)) lemma nodeflip_monotony : forall x v0 v1 v2. subset v0 v1 -> nodeflip x v1 v2 -> nodeflip x v0 v2 lemma whitepath_monotony : forall x l z v0 v1. subset v0 v1 -> whitepath x l z v1 -> whitepath x l z v0 let rec random_search (roots: set vertex) (visited: set vertex): set vertex variant {(cardinal vertices - cardinal visited), (cardinal roots)} = requires {subset roots vertices } requires {subset visited vertices } ensures {subset visited result} (* *) ensures {whitepath_nodeflip roots visited result} ensures {nodeflip_whitepath roots visited 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 (* using lemma whitepath_whitepath_fst_not_twice *) let r = random_search (union roots' (successors x)) (add x visited) in (*-------------- whitepath_nodeflip -------------*) (* case 1: whitepath x l z /\ x = z *) (* assert {mem x r}; *) (* case 2: whitepath x l z /\ x <> z *) (* assert {forall l z. whitepath x l z visited -> x <> z -> exists y l'. edge x y /\ whitepath y l' z (add x visited)}; same as lemma whitepath_inversion *) (* assert {forall l z. whitepath x l z visited -> x <> z -> nodeflip z (add x visited) r}; *) assert {forall l z. whitepath x l z visited -> nodeflip z visited r}; (* case 3: whitepath roots' l z *) (* case 3-1: whitepath roots' l z /\ mem x l *) (* assert {forall x' l z. mem x' roots' -> whitepath x' l z visited -> L.mem x (l ++ Cons z Nil) -> exists l'. whitepath x l' z visited}; *) (* goto cases 1-2 *) (* case 3-2: whitepath roots' l z /\ not mem x l *) assert {forall x' l z. mem x' roots' -> whitepath x' l z visited -> not L.mem x (l ++ Cons z Nil) -> whitepath x' l z (add x visited)}; (*----------- nodeflip_whitepath -----------*) assert {forall z. nodeflip z visited r -> nodeflip z (add x visited) r \/ z = x}; (* case 1: nodeflip z visited r /\ z = x *) assert {whitepath x Nil x visited}; (* case 2: nodeflip z visited r /\ z <> x *) assert {forall x' l z. edge x x' -> whitepath x' l z (add x visited) -> whitepath x (Cons x l) z visited}; r end let random_search_main (roots: set vertex) = requires {subset roots vertices} random_search roots (empty) end
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