add (normaly) the missing coq files

e0d31f8f · François Bobot · 890ca844 · e0d31f8f · e0d31f8f · e0d31f8f
Commit e0d31f8f authored Jan 31, 2012 by François Bobot
5 changed files
--- a/examples/programs/vstte12_bfs/vstte12_bfs_Graph_path_closure_1.v
+++ b/examples/programs/vstte12_bfs/vstte12_bfs_Graph_path_closure_1.v
+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require int.Int.
+Parameter set : forall (a:Type), Type.
+
+Parameter mem: forall (a:Type), a -> (set a) -> Prop.
+
+Implicit Arguments mem.
+
+Definition infix_eqeq (a:Type)(s1:(set a)) (s2:(set a)): Prop :=
+  forall (x:a), (mem x s1) <-> (mem x s2).
+Implicit Arguments infix_eqeq.
+
+Axiom extensionality : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (infix_eqeq s1 s2) -> (s1 = s2).
+
+Definition subset (a:Type)(s1:(set a)) (s2:(set a)): Prop := forall (x:a),
+  (mem x s1) -> (mem x s2).
+Implicit Arguments subset.
+
+Axiom subset_trans : forall (a:Type), forall (s1:(set a)) (s2:(set a))
+  (s3:(set a)), (subset s1 s2) -> ((subset s2 s3) -> (subset s1 s3)).
+
+Parameter empty: forall (a:Type), (set a).
+
+Set Contextual Implicit.
+Implicit Arguments empty.
+Unset Contextual Implicit.
+
+Definition is_empty (a:Type)(s:(set a)): Prop := forall (x:a), ~ (mem x s).
+Implicit Arguments is_empty.
+
+Axiom empty_def1 : forall (a:Type), (is_empty (empty:(set a))).
+
+Parameter add: forall (a:Type), a -> (set a) -> (set a).
+
+Implicit Arguments add.
+
+Axiom add_def1 : forall (a:Type), forall (x:a) (y:a), forall (s:(set a)),
+  (mem x (add y s)) <-> ((x = y) \/ (mem x s)).
+
+Parameter remove: forall (a:Type), a -> (set a) -> (set a).
+
+Implicit Arguments remove.
+
+Axiom remove_def1 : forall (a:Type), forall (x:a) (y:a) (s:(set a)), (mem x
+  (remove y s)) <-> ((~ (x = y)) /\ (mem x s)).
+
+Axiom subset_remove : forall (a:Type), forall (x:a) (s:(set a)),
+  (subset (remove x s) s).
+
+Parameter union: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments union.
+
+Axiom union_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (union s1 s2)) <-> ((mem x s1) \/ (mem x s2)).
+
+Parameter inter: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments inter.
+
+Axiom inter_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (inter s1 s2)) <-> ((mem x s1) /\ (mem x s2)).
+
+Parameter diff: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments diff.
+
+Axiom diff_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (diff s1 s2)) <-> ((mem x s1) /\ ~ (mem x s2)).
+
+Axiom subset_diff : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset (diff s1 s2) s1).
+
+Parameter cardinal: forall (a:Type), (set a) -> Z.
+
+Implicit Arguments cardinal.
+
+Axiom cardinal_nonneg : forall (a:Type), forall (s:(set a)),
+  (0%Z <= (cardinal s))%Z.
+
+Axiom cardinal_empty : forall (a:Type), forall (s:(set a)),
+  ((cardinal s) = 0%Z) <-> (is_empty s).
+
+Axiom cardinal_add : forall (a:Type), forall (x:a), forall (s:(set a)),
+  (~ (mem x s)) -> ((cardinal (add x s)) = (1%Z + (cardinal s))%Z).
+
+Axiom cardinal_remove : forall (a:Type), forall (x:a), forall (s:(set a)),
+  (mem x s) -> ((cardinal s) = (1%Z + (cardinal (remove x s)))%Z).
+
+Axiom cardinal_subset : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset s1 s2) -> ((cardinal s1) <= (cardinal s2))%Z.
+
+Parameter vertex : Type.
+
+Parameter succ: vertex -> (set vertex).
+
+
+Inductive path : vertex -> vertex -> Z -> Prop :=
+  | path_empty : forall (v:vertex), (path v v 0%Z)
+  | path_succ : forall (v1:vertex) (v2:vertex) (v3:vertex) (n:Z), (path v1 v2
+      n) -> ((mem v3 (succ v2)) -> (path v1 v3 (n + 1%Z)%Z)).
+
+Axiom path_nonneg : forall (v1:vertex) (v2:vertex) (n:Z), (path v1 v2 n) ->
+  (0%Z <= n)%Z.
+
+Axiom path_inversion : forall (v1:vertex) (v3:vertex) (n:Z), (0%Z <= n)%Z ->
+  ((path v1 v3 (n + 1%Z)%Z) -> exists v2:vertex, (path v1 v2 n) /\ (mem v3
+  (succ v2))).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem path_closure : forall (s:(set vertex)), (forall (x:vertex), (mem x
+  s) -> forall (y:vertex), (mem y (succ x)) -> (mem y s)) ->
+  forall (v1:vertex) (v2:vertex) (n:Z), (path v1 v2 n) -> ((mem v1 s) ->
+  (mem v2 s)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+induction 2; auto.
+intuition.
+eauto.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+
--- a/examples/programs/vstte12_bfs/vstte12_bfs_Graph_path_inversion_1.v
+++ b/examples/programs/vstte12_bfs/vstte12_bfs_Graph_path_inversion_1.v
+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require int.Int.
+Parameter set : forall (a:Type), Type.
+
+Parameter mem: forall (a:Type), a -> (set a) -> Prop.
+
+Implicit Arguments mem.
+
+Definition infix_eqeq (a:Type)(s1:(set a)) (s2:(set a)): Prop :=
+  forall (x:a), (mem x s1) <-> (mem x s2).
+Implicit Arguments infix_eqeq.
+
+Axiom extensionality : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (infix_eqeq s1 s2) -> (s1 = s2).
+
+Definition subset (a:Type)(s1:(set a)) (s2:(set a)): Prop := forall (x:a),
+  (mem x s1) -> (mem x s2).
+Implicit Arguments subset.
+
+Axiom subset_trans : forall (a:Type), forall (s1:(set a)) (s2:(set a))
+  (s3:(set a)), (subset s1 s2) -> ((subset s2 s3) -> (subset s1 s3)).
+
+Parameter empty: forall (a:Type), (set a).
+
+Set Contextual Implicit.
+Implicit Arguments empty.
+Unset Contextual Implicit.
+
+Definition is_empty (a:Type)(s:(set a)): Prop := forall (x:a), ~ (mem x s).
+Implicit Arguments is_empty.
+
+Axiom empty_def1 : forall (a:Type), (is_empty (empty:(set a))).
+
+Parameter add: forall (a:Type), a -> (set a) -> (set a).
+
+Implicit Arguments add.
+
+Axiom add_def1 : forall (a:Type), forall (x:a) (y:a), forall (s:(set a)),
+  (mem x (add y s)) <-> ((x = y) \/ (mem x s)).
+
+Parameter remove: forall (a:Type), a -> (set a) -> (set a).
+
+Implicit Arguments remove.
+
+Axiom remove_def1 : forall (a:Type), forall (x:a) (y:a) (s:(set a)), (mem x
+  (remove y s)) <-> ((~ (x = y)) /\ (mem x s)).
+
+Axiom subset_remove : forall (a:Type), forall (x:a) (s:(set a)),
+  (subset (remove x s) s).
+
+Parameter union: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments union.
+
+Axiom union_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (union s1 s2)) <-> ((mem x s1) \/ (mem x s2)).
+
+Parameter inter: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments inter.
+
+Axiom inter_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (inter s1 s2)) <-> ((mem x s1) /\ (mem x s2)).
+
+Parameter diff: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments diff.
+
+Axiom diff_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (diff s1 s2)) <-> ((mem x s1) /\ ~ (mem x s2)).
+
+Axiom subset_diff : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset (diff s1 s2) s1).
+
+Parameter cardinal: forall (a:Type), (set a) -> Z.
+
+Implicit Arguments cardinal.
+
+Axiom cardinal_nonneg : forall (a:Type), forall (s:(set a)),
+  (0%Z <= (cardinal s))%Z.
+
+Axiom cardinal_empty : forall (a:Type), forall (s:(set a)),
+  ((cardinal s) = 0%Z) <-> (is_empty s).
+
+Axiom cardinal_add : forall (a:Type), forall (x:a), forall (s:(set a)),
+  (~ (mem x s)) -> ((cardinal (add x s)) = (1%Z + (cardinal s))%Z).
+
+Axiom cardinal_remove : forall (a:Type), forall (x:a), forall (s:(set a)),
+  (mem x s) -> ((cardinal s) = (1%Z + (cardinal (remove x s)))%Z).
+
+Axiom cardinal_subset : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset s1 s2) -> ((cardinal s1) <= (cardinal s2))%Z.
+
+Parameter vertex : Type.
+
+Parameter succ: vertex -> (set vertex).
+
+
+Inductive path : vertex -> vertex -> Z -> Prop :=
+  | path_empty : forall (v:vertex), (path v v 0%Z)
+  | path_succ : forall (v1:vertex) (v2:vertex) (v3:vertex) (n:Z), (path v1 v2
+      n) -> ((mem v3 (succ v2)) -> (path v1 v3 (n + 1%Z)%Z)).
+
+Axiom path_nonneg : forall (v1:vertex) (v2:vertex) (n:Z), (path v1 v2 n) ->
+  (0%Z <= n)%Z.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem path_inversion : forall (v1:vertex) (v3:vertex) (n:Z),
+  (0%Z <= n)%Z -> ((path v1 v3 (n + 1%Z)%Z) -> exists v2:vertex, (path v1 v2
+  n) /\ (mem v3 (succ v2))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+inversion 2.
+apply False_ind; omega.
+intros; subst.
+assert (n0=n) by omega; subst.
+exists v2; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+
--- a/examples/programs/vstte12_bfs/vstte12_bfs_Graph_path_nonneg_1.v
+++ b/examples/programs/vstte12_bfs/vstte12_bfs_Graph_path_nonneg_1.v
+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require int.Int.
+Parameter set : forall (a:Type), Type.
+
+Parameter mem: forall (a:Type), a -> (set a) -> Prop.
+
+Implicit Arguments mem.
+
+Definition infix_eqeq (a:Type)(s1:(set a)) (s2:(set a)): Prop :=
+  forall (x:a), (mem x s1) <-> (mem x s2).
+Implicit Arguments infix_eqeq.
+
+Axiom extensionality : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (infix_eqeq s1 s2) -> (s1 = s2).
+
+Definition subset (a:Type)(s1:(set a)) (s2:(set a)): Prop := forall (x:a),
+  (mem x s1) -> (mem x s2).
+Implicit Arguments subset.
+
+Axiom subset_trans : forall (a:Type), forall (s1:(set a)) (s2:(set a))
+  (s3:(set a)), (subset s1 s2) -> ((subset s2 s3) -> (subset s1 s3)).
+
+Parameter empty: forall (a:Type), (set a).
+
+Set Contextual Implicit.
+Implicit Arguments empty.
+Unset Contextual Implicit.
+
+Definition is_empty (a:Type)(s:(set a)): Prop := forall (x:a), ~ (mem x s).
+Implicit Arguments is_empty.
+
+Axiom empty_def1 : forall (a:Type), (is_empty (empty:(set a))).
+
+Parameter add: forall (a:Type), a -> (set a) -> (set a).
+
+Implicit Arguments add.
+
+Axiom add_def1 : forall (a:Type), forall (x:a) (y:a), forall (s:(set a)),
+  (mem x (add y s)) <-> ((x = y) \/ (mem x s)).
+
+Parameter remove: forall (a:Type), a -> (set a) -> (set a).
+
+Implicit Arguments remove.
+
+Axiom remove_def1 : forall (a:Type), forall (x:a) (y:a) (s:(set a)), (mem x
+  (remove y s)) <-> ((~ (x = y)) /\ (mem x s)).
+
+Axiom subset_remove : forall (a:Type), forall (x:a) (s:(set a)),
+  (subset (remove x s) s).
+
+Parameter union: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments union.
+
+Axiom union_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (union s1 s2)) <-> ((mem x s1) \/ (mem x s2)).
+
+Parameter inter: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments inter.
+
+Axiom inter_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (inter s1 s2)) <-> ((mem x s1) /\ (mem x s2)).
+
+Parameter diff: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments diff.
+
+Axiom diff_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (diff s1 s2)) <-> ((mem x s1) /\ ~ (mem x s2)).
+
+Axiom subset_diff : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset (diff s1 s2) s1).
+
+Parameter cardinal: forall (a:Type), (set a) -> Z.
+
+Implicit Arguments cardinal.
+
+Axiom cardinal_nonneg : forall (a:Type), forall (s:(set a)),
+  (0%Z <= (cardinal s))%Z.
+
+Axiom cardinal_empty : forall (a:Type), forall (s:(set a)),
+  ((cardinal s) = 0%Z) <-> (is_empty s).
+
+Axiom cardinal_add : forall (a:Type), forall (x:a), forall (s:(set a)),
+  (~ (mem x s)) -> ((cardinal (add x s)) = (1%Z + (cardinal s))%Z).
+
+Axiom cardinal_remove : forall (a:Type), forall (x:a), forall (s:(set a)),
+  (mem x s) -> ((cardinal s) = (1%Z + (cardinal (remove x s)))%Z).
+
+Axiom cardinal_subset : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset s1 s2) -> ((cardinal s1) <= (cardinal s2))%Z.
+
+Parameter vertex : Type.
+
+Parameter succ: vertex -> (set vertex).
+
+
+Inductive path : vertex -> vertex -> Z -> Prop :=
+  | path_empty : forall (v:vertex), (path v v 0%Z)
+  | path_succ : forall (v1:vertex) (v2:vertex) (v3:vertex) (n:Z), (path v1 v2
+      n) -> ((mem v3 (succ v2)) -> (path v1 v3 (n + 1%Z)%Z)).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem path_nonneg : forall (v1:vertex) (v2:vertex) (n:Z), (path v1 v2 n) ->
+  (0%Z <= n)%Z.
+(* YOU MAY EDIT THE PROOF BELOW *)
+induction 1; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+
--- a/examples/programs/vstte12_bfs/vstte12_bfs_WP_BFS_WP_parameter_bfs_1.v
+++ b/examples/programs/vstte12_bfs/vstte12_bfs_WP_BFS_WP_parameter_bfs_1.v
+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require int.Int.
+Definition unit  := unit.
+
+Parameter qtmark : Type.
+
+Parameter at1: forall (a:Type), a -> qtmark -> a.
+
+Implicit Arguments at1.
+
+Parameter old: forall (a:Type), a -> a.
+
+Implicit Arguments old.
+
+Definition implb(x:bool) (y:bool): bool := match (x,
+  y) with
+  | (true, false) => false
+  | (_, _) => true
+  end.
+
+Parameter set : forall (a:Type), Type.
+
+Parameter mem: forall (a:Type), a -> (set a) -> Prop.
+
+Implicit Arguments mem.
+
+Definition infix_eqeq (a:Type)(s1:(set a)) (s2:(set a)): Prop :=
+  forall (x:a), (mem x s1) <-> (mem x s2).
+Implicit Arguments infix_eqeq.
+
+Axiom extensionality : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (infix_eqeq s1 s2) -> (s1 = s2).
+
+Definition subset (a:Type)(s1:(set a)) (s2:(set a)): Prop := forall (x:a),
+  (mem x s1) -> (mem x s2).
+Implicit Arguments subset.
+
+Axiom subset_trans : forall (a:Type), forall (s1:(set a)) (s2:(set a))
+  (s3:(set a)), (subset s1 s2) -> ((subset s2 s3) -> (subset s1 s3)).
+
+Parameter empty: forall (a:Type), (set a).
+
+Set Contextual Implicit.
+Implicit Arguments empty.
+Unset Contextual Implicit.
+
+Definition is_empty (a:Type)(s:(set a)): Prop := forall (x:a), ~ (mem x s).
+Implicit Arguments is_empty.
+
+Axiom empty_def1 : forall (a:Type), (is_empty (empty:(set a))).
+
+Parameter add: forall (a:Type), a -> (set a) -> (set a).
+
+Implicit Arguments add.
+
+Axiom add_def1 : forall (a:Type), forall (x:a) (y:a), forall (s:(set a)),
+  (mem x (add y s)) <-> ((x = y) \/ (mem x s)).
+
+Parameter remove: forall (a:Type), a -> (set a) -> (set a).
+
+Implicit Arguments remove.
+
+Axiom remove_def1 : forall (a:Type), forall (x:a) (y:a) (s:(set a)), (mem x
+  (remove y s)) <-> ((~ (x = y)) /\ (mem x s)).
+
+Axiom subset_remove : forall (a:Type), forall (x:a) (s:(set a)),
+  (subset (remove x s) s).
+
+Parameter union: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments union.
+
+Axiom union_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (union s1 s2)) <-> ((mem x s1) \/ (mem x s2)).
+
+Parameter inter: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments inter.
+
+Axiom inter_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (inter s1 s2)) <-> ((mem x s1) /\ (mem x s2)).
+
+Parameter diff: forall (a:Type), (set a) -> (set a) -> (set a).
+
+Implicit Arguments diff.
+
+Axiom diff_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (diff s1 s2)) <-> ((mem x s1) /\ ~ (mem x s2)).
+
+Axiom subset_diff : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset (diff s1 s2) s1).
+
+Parameter cardinal: forall (a:Type), (set a) -> Z.
+
+Implicit Arguments cardinal.
+
+Axiom cardinal_nonneg : forall (a:Type), forall (s:(set a)),
+  (0%Z <= (cardinal s))%Z.
+
+Axiom cardinal_empty : forall (a:Type), forall (s:(set a)),
+  ((cardinal s) = 0%Z) <-> (is_empty s).
+
+Axiom cardinal_add : forall (a:Type), forall (x:a), forall (s:(set a)),
+  (~ (mem x s)) -> ((cardinal (add x s)) = (1%Z + (cardinal s))%Z).
+
+Axiom cardinal_remove : forall (a:Type), forall (x:a), forall (s:(set a)),
+  (mem x s) -> ((cardinal s) = (1%Z + (cardinal (remove x s)))%Z).
+
+Axiom cardinal_subset : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset s1 s2) -> ((cardinal s1) <= (cardinal s2))%Z.
+
+Parameter vertex : Type.
+
+Parameter succ: vertex -> (set vertex).
+
+
+Inductive path : vertex -> vertex -> Z -> Prop :=
+  | path_empty : forall (v:vertex), (path v v 0%Z)
+  | path_succ : forall (v1:vertex) (v2:vertex) (v3:vertex) (n:Z), (path v1 v2
+      n) -> ((mem v3 (succ v2)) -> (path v1 v3 (n + 1%Z)%Z)).
+
+Axiom path_nonneg : forall (v1:vertex) (v2:vertex) (n:Z), (path v1 v2 n) ->
+  (0%Z <= n)%Z.
+
+Axiom path_inversion : forall (v1:vertex) (v3:vertex) (n:Z), (0%Z <= n)%Z ->
+  ((path v1 v3 (n + 1%Z)%Z) -> exists v2:vertex, (path v1 v2 n) /\ (mem v3
+  (succ v2))).
+
+Axiom path_closure : forall (s:(set vertex)), (forall (x:vertex), (mem x
+  s) -> forall (y:vertex), (mem y (succ x)) -> (mem y s)) ->
+  forall (v1:vertex) (v2:vertex) (n:Z), (path v1 v2 n) -> ((mem v1 s) ->
+  (mem v2 s)).
+
+Definition shortest_path(v1:vertex) (v2:vertex) (n:Z): Prop := (path v1 v2
+  n) /\ forall (m:Z), (m <  n)%Z -> ~ (path v1 v2 m).
+
+Inductive ref (a:Type) :=
+  | mk_ref : a -> ref a.
+Implicit Arguments mk_ref.
+
+Definition contents (a:Type)(u:(ref a)): a :=
+  match u with
+  | (mk_ref contents1) => contents1
+  end.
+Implicit Arguments contents.
+
+Definition bag (a:Type) := (ref (set a)).
+
+Definition inv(s:vertex) (t:vertex) (visited:(set vertex)) (current:(set
+  vertex)) (next:(set vertex)) (d:Z): Prop := (subset current visited) /\
+  ((forall (x:vertex), (mem x current) -> (shortest_path s x d)) /\
+  ((subset next visited) /\ ((forall (x:vertex), (mem x next) ->
+  (shortest_path s x (d + 1%Z)%Z)) /\ ((forall (x:vertex) (m:Z), (path s x
+  m) -> ((m <= d)%Z -> (mem x visited))) /\ ((forall (x:vertex), (mem x
+  visited) -> exists m:Z, (path s x m) /\ (m <= (d + 1%Z)%Z)%Z) /\
+  ((forall (x:vertex), (shortest_path s x (d + 1%Z)%Z) -> ((mem x next) \/
+  ~ (mem x visited))) /\ ((mem t visited) -> ((mem t current) \/ (mem t
+  next))))))))).
+
+Definition closure(visited:(set vertex)) (current:(set vertex)) (next:(set
+  vertex)) (x:vertex): Prop := (mem x visited) -> ((~ (mem x current)) ->
+  ((~ (mem x next)) -> forall (y:vertex), (mem y (succ x)) -> (mem y