Library scc_SCCTarjan72_WP_parameter_dfs1_1

Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require int.MinMax.
Require list.List.
Require list.Length.
Require list.Mem.
Require map.Map.
Require map.Const.
Require list.Append.
Require list.Reverse.
Require list.NumOcc.

Definition unit := unit.

Axiom set : forall (a:Type), Type.
Parameter set_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (set a).
Existing Instance set_WhyType.

Parameter mem: forall {a:Type} {a_WT:WhyType a}, a -> (set a) -> Prop.

Definition infix_eqeq {a:Type} {a_WT:WhyType a} (s1:(set a)) (s2:(set
a)): Prop := forall (x:a), (mem x s1) <-> (mem x s2).

Axiom extensionality : forall {a:Type} {a_WT:WhyType a}, forall (s1:(set a))
(s2:(set a)), (infix_eqeq s1 s2) -> (s1 = s2).

Definition subset {a:Type} {a_WT:WhyType a} (s1:(set a)) (s2:(set
a)): Prop := forall (x:a), (mem x s1) -> (mem x s2).

Axiom subset_refl : forall {a:Type} {a_WT:WhyType a}, forall (s:(set a)),
(subset s s).

Axiom subset_trans : forall {a:Type} {a_WT:WhyType a}, forall (s1:(set a))
(s2:(set a)) (s3:(set a)), (subset s1 s2) -> ((subset s2 s3) -> (subset s1
s3)).

Parameter empty: forall {a:Type} {a_WT:WhyType a}, (set a).

Definition is_empty {a:Type} {a_WT:WhyType a} (s:(set a)): Prop :=
forall (x:a), ~ (mem x s).

Axiom empty_def1 : forall {a:Type} {a_WT:WhyType a}, (is_empty (empty : (set
a))).

Axiom mem_empty : forall {a:Type} {a_WT:WhyType a}, forall (x:a), ~ (mem x
(empty : (set a))).

Parameter add: forall {a:Type} {a_WT:WhyType a}, a -> (set a) -> (set a).

Axiom add_def1 : forall {a:Type} {a_WT:WhyType a}, 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_WT:WhyType a}, a -> (set a) -> (set a).

Axiom remove_def1 : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (y:a)
(s:(set a)), (mem x (remove y s)) <-> ((~ (x = y)) /\ (mem x s)).

Axiom add_remove : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (s:(set
a)), (mem x s) -> ((add x (remove x s)) = s).

Axiom remove_add : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (s:(set
a)), ((remove x (add x s)) = (remove x s)).

Axiom subset_remove : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (s:(set
a)), (subset (remove x s) s).

Parameter union: forall {a:Type} {a_WT:WhyType a}, (set a) -> (set a) -> (set
a).

Axiom union_def1 : forall {a:Type} {a_WT:WhyType a}, 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} {a_WT:WhyType a}, (set a) -> (set a) -> (set
a).

Axiom inter_def1 : forall {a:Type} {a_WT:WhyType a}, 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} {a_WT:WhyType a}, (set a) -> (set a) -> (set
a).

Axiom diff_def1 : forall {a:Type} {a_WT:WhyType a}, 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} {a_WT:WhyType a}, forall (s1:(set a))
(s2:(set a)), (subset (diff s1 s2) s1).

Parameter choose: forall {a:Type} {a_WT:WhyType a}, (set a) -> a.

Axiom choose_def : forall {a:Type} {a_WT:WhyType a}, forall (s:(set a)),
(~ (is_empty s)) -> (mem (choose s) s).

Parameter cardinal: forall {a:Type} {a_WT:WhyType a}, (set a) -> Z.

Axiom cardinal_nonneg : forall {a:Type} {a_WT:WhyType a}, forall (s:(set a)),
(0%Z <= (cardinal s))%Z.

Axiom cardinal_empty : forall {a:Type} {a_WT:WhyType a}, forall (s:(set a)),
((cardinal s) = 0%Z) <-> (is_empty s).

Axiom cardinal_add : forall {a:Type} {a_WT:WhyType a}, 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} {a_WT:WhyType a}, 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} {a_WT:WhyType a}, forall (s1:(set a))
(s2:(set a)), (subset s1 s2) -> ((cardinal s1) <= (cardinal s2))%Z.

Axiom subset_eq : forall {a:Type} {a_WT:WhyType a}, forall (s1:(set a))
(s2:(set a)), (subset s1 s2) -> (((cardinal s1) = (cardinal s2)) ->
(infix_eqeq s1 s2)).

Axiom cardinal1 : forall {a:Type} {a_WT:WhyType a}, forall (s:(set a)),
((cardinal s) = 1%Z) -> forall (x:a), (mem x s) -> (x = (choose s)).

Parameter elements: forall {a:Type} {a_WT:WhyType a}, (list a) -> (set a).

Axiom elements_mem : forall {a:Type} {a_WT:WhyType a}, forall (l:(list a))
(x:a), (list.Mem.mem x l) <-> (mem x (elements l)).

Axiom elements_Nil : forall {a:Type} {a_WT:WhyType a},
((elements Init.Datatypes.nil) = (empty : (set a))).

Axiom vertex : Type.
Parameter vertex_WhyType : WhyType vertex.
Existing Instance vertex_WhyType.

Parameter vertices: (set vertex).

Parameter successors: vertex -> (set vertex).

Axiom successors_vertices : forall (x:vertex), (mem x vertices) -> (subset
(successors x) vertices).

Definition edge (x:vertex) (y:vertex): Prop := (mem x vertices) /\ (mem y
(successors x)).

Inductive path: vertex -> (list vertex) -> vertex -> Prop :=
| Path_empty : forall (x:vertex), (path x Init.Datatypes.nil x)
| Path_cons : forall (x:vertex) (y:vertex) (z:vertex) (l:(list vertex)),
(edge x y) -> ((path y l z) -> (path x (Init.Datatypes.cons x l) z)).

Axiom path_right_extension : forall (x:vertex) (y:vertex) (z:vertex)
(l:(list vertex)), (path x l y) -> ((edge y z) -> (path x
(Init.Datatypes.app l (Init.Datatypes.cons y Init.Datatypes.nil)) z)).

Axiom path_right_inversion : forall (x:vertex) (z:vertex) (l:(list vertex)),
(path x l z) -> (((x = z) /\ (l = Init.Datatypes.nil)) \/ exists y:vertex,
exists l':(list vertex), (path x l' y) /\ ((edge y z) /\
(l = (Init.Datatypes.app l' (Init.Datatypes.cons y Init.Datatypes.nil))))).

Axiom path_trans : forall (x:vertex) (y:vertex) (z:vertex) (l1:(list vertex))
(l2:(list vertex)), (path x l1 y) -> ((path y l2 z) -> (path x
(Init.Datatypes.app l1 l2) z)).

Axiom empty_path : forall (x:vertex) (y:vertex), (path x Init.Datatypes.nil
y) -> (x = y).

Axiom path_decomposition : forall (x:vertex) (y:vertex) (z:vertex)
(l1:(list vertex)) (l2:(list vertex)), (path x
(Init.Datatypes.app l1 (Init.Datatypes.cons y l2)) z) -> ((path x l1 y) /\
(path y (Init.Datatypes.cons y l2) z)).

Inductive tree :=
| Null : tree
| Node : tree -> Z -> tree -> tree.
Axiom tree_WhyType : WhyType tree.
Existing Instance tree_WhyType.

Fixpoint size (t:tree) {struct t}: Z :=
match t with
| Null => 0%Z
| (Node l _ r) => ((1%Z + (size l))%Z + (size r))%Z
end.

Axiom size_pos : forall (t:tree), (0%Z <= (size t))%Z.

Axiom lmem_dec : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (l:(list a)),
(list.Mem.mem x l) \/ ~ (list.Mem.mem x l).

Axiom list_simpl_r : forall {a:Type} {a_WT:WhyType a}, forall (l1:(list a))
(l2:(list a)) (l:(list a)),
((Init.Datatypes.app l1 l) = (Init.Datatypes.app l2 l)) -> (l1 = l2).

Axiom list_assoc_cons : forall {a:Type} {a_WT:WhyType a},
forall (l1:(list a)) (l2:(list a)) (x:a),
((Init.Datatypes.app l1 (Init.Datatypes.cons x l2)) = (Init.Datatypes.app (Init.Datatypes.app l1 (Init.Datatypes.cons x Init.Datatypes.nil)) l2)).

Definition is_last {a:Type} {a_WT:WhyType a} (x:a) (s:(list a)): Prop :=
exists s':(list a),
(s = (Init.Datatypes.app s' (Init.Datatypes.cons x Init.Datatypes.nil))).

Parameter rank: forall {a:Type} {a_WT:WhyType a}, a -> (list a) -> Z.

Axiom rank_def : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (s:(list a)),
match s with
| Init.Datatypes.nil => ((rank x s) = (cardinal vertices))
| (Init.Datatypes.cons y s') => (((x = y) /\ ~ (list.Mem.mem x s')) ->
((rank x s) = (list.Length.length s'))) /\ ((~ ((x = y) /\
~ (list.Mem.mem x s'))) -> ((rank x s) = (rank x s')))
end.

Axiom rank_not_mem : forall {a:Type} {a_WT:WhyType a}, forall (x:a)
(s:(list a)), (~ (list.Mem.mem x s)) -> ((rank x s) = (cardinal vertices)).

Axiom rank_range : forall {a:Type} {a_WT:WhyType a}, forall (x:a)
(s:(list a)), (list.Mem.mem x s) -> ((0%Z <= (rank x s))%Z /\ ((rank x
s) < (list.Length.length s))%Z).

Axiom rank_min : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (y:a)
(s:(list a)), (is_last x s) -> ((list.Mem.mem y s) -> ((rank x
s) <= (rank y s))%Z).

Axiom rank_app_l : forall {a:Type} {a_WT:WhyType a}, forall (x:a)
(s:(list a)) (s':(list a)), (list.Mem.mem x s') -> ((~ (list.Mem.mem x
s)) -> ((rank x (Init.Datatypes.app s' s)) = ((rank x
s') + (list.Length.length s))%Z)).

Axiom rank_app_r : forall {a:Type} {a_WT:WhyType a}, forall (x:a)
(s:(list a)) (s':(list a)), (list.Mem.mem x s) -> ((rank x s) = (rank x
(Init.Datatypes.app s' s))).

Definition simplelist {a:Type} {a_WT:WhyType a} (l:(list a)): Prop :=
forall (x:a), ((list.NumOcc.num_occ x l) <= 1%Z)%Z.

Axiom simplelist_tl : forall {a:Type} {a_WT:WhyType a}, forall (x:a)
(l:(list a)), (simplelist (Init.Datatypes.cons x l)) -> ((simplelist l) /\
~ (list.Mem.mem x l)).

Axiom simplelist_split : forall {a:Type} {a_WT:WhyType a}, forall (x:a)
(l1:(list a)) (l2:(list a)) (l3:(list a)) (l4:(list a)),
((Init.Datatypes.app l1 (Init.Datatypes.cons x l2)) = (Init.Datatypes.app l3 (Init.Datatypes.cons x l4))) ->
((simplelist (Init.Datatypes.app l1 (Init.Datatypes.cons x l2))) ->
((l1 = l3) /\ (l2 = l4))).

Axiom simplelist_hd_max_rank : forall {a:Type} {a_WT:WhyType a}, forall (x:a)
(y:a) (s1:(list a)) (s2:(list a)), (s1 = (Init.Datatypes.cons x s2)) ->
((simplelist s1) -> ((list.Mem.mem y s2) -> ((rank y s1) < (rank x
s1))%Z)).

Axiom rank_before_suffix : forall {a:Type} {a_WT:WhyType a}, forall (x:a)
(y:a) (s1:(list a)) (s2:(list a)), (simplelist
(Init.Datatypes.app s1 s2)) -> ((is_last x s1) -> ((list.Mem.mem y s2) ->
((rank y (Init.Datatypes.app s1 s2)) < (rank x
(Init.Datatypes.app s1 s2)))%Z)).

Axiom inter_com : forall {a:Type} {a_WT:WhyType a}, forall (s1:(set a))
(s2:(set a)), (infix_eqeq (inter s1 s2) (inter s2 s1)).

Axiom inter_add : forall {a:Type} {a_WT:WhyType a}, forall (s1:(set a))
(s2:(set a)) (x:a), (~ (mem x s2)) -> (infix_eqeq (inter (add x s1) s2)
(inter s1 s2)).

Axiom set_elt : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (s1:(set a))
(s2:(set a)) (s3:(set a)), (~ (mem x s3)) -> (infix_eqeq (union (add x s1)
(diff s2 s3)) (union s1 (diff (add x s2) s3))).

Axiom set_mem : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (s1:(set a))
(s2:(set a)) (s3:(set a)) (s4:(set a)), (infix_eqeq s1 (union s2 (diff s3
s4))) -> (((inter s2 s4) = (empty : (set a))) -> ((mem x s1) -> ~ (mem x
s4))).

Axiom inter_subset_inter : forall {a:Type} {a_WT:WhyType a}, forall (s1:(set
a)) (s2:(set a)) (s2':(set a)), ((inter s1 s2) = (empty : (set a))) ->
((subset s2' s2) -> ((inter s1 s2') = (empty : (set a)))).

Axiom subset_add : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (s:(set a))
(s':(set a)), (subset s' (add x s)) -> ((mem x s') \/ (subset s' s)).

Axiom union_add_l : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (s:(set
a)) (s':(set a)), (infix_eqeq (union (add x s) s') (add x (union s s'))).

Axiom union_add_r : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (s:(set
a)) (s':(set a)), (infix_eqeq (union s (add x s')) (add x (union s s'))).

Axiom elts_cons : forall {a:Type} {a_WT:WhyType a}, forall (x:a)
(l:(list a)), (infix_eqeq (elements (Init.Datatypes.cons x l)) (add x
(elements l))).

Axiom elts_app : forall {a:Type} {a_WT:WhyType a}, forall (s:(list a))
(s':(list a)), (infix_eqeq (elements (Init.Datatypes.app s s'))
(union (elements s) (elements s'))).

Axiom simplelist_app_inter : forall {a:Type} {a_WT:WhyType a},
forall (l1:(list a)) (l2:(list a)), (simplelist
(Init.Datatypes.app l1 l2)) -> ((inter (elements l1)
(elements l2)) = (empty : (set a))).

Axiom simplelist_length : forall {a:Type} {a_WT:WhyType a},
forall (s:(list a)), (simplelist s) ->
((list.Length.length s) = (cardinal (elements s))).

Parameter set_of: forall {a:Type} {a_WT:WhyType a}, (set (set a)) -> (set a).

Axiom set_of_empty : forall {a:Type} {a_WT:WhyType a}, ((set_of (empty : (set
(set a)))) = (empty : (set a))).

Axiom set_of_add : forall {a:Type} {a_WT:WhyType a}, forall (s:(set a))
(sx:(set (set a))), (infix_eqeq (set_of (add s sx)) (union s (set_of sx))).

Definition one_in_set_of {a:Type} {a_WT:WhyType a} (sccs:(set (set
a))): Prop := forall (x:a), (mem x (set_of sccs)) -> exists cc:(set a),
(mem cc sccs) /\ (mem x cc).

Axiom Induction : (forall (s:(set (set vertex))), (is_empty s) ->
(one_in_set_of s)) -> ((forall (s:(set (set vertex))), (one_in_set_of s) ->
forall (t:(set vertex)), (~ (mem t s)) -> (one_in_set_of (add t s))) ->
forall (s:(set (set vertex))), (one_in_set_of s)).

Axiom set_of_elt : forall (sccs:(set (set vertex))), (one_in_set_of sccs).

Axiom elt_set_of : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (cc:(set
a)) (sccs:(set (set a))), (mem x cc) -> ((mem cc sccs) -> (mem x
(set_of sccs))).

Axiom subset_set_of : forall (s:(set (set vertex))) (s':(set (set vertex))),
(subset s s') -> (subset (set_of s) (set_of s')).

Definition reachable (x:vertex) (y:vertex): Prop := exists l:(list vertex),
(path x l y).

Axiom reachable_trans : forall (x:vertex) (y:vertex) (z:vertex), (reachable x
y) -> ((reachable y z) -> (reachable x z)).

Axiom xset_path_xedge : forall (x:vertex) (y:vertex) (l:(list vertex))
(s:(set vertex)), (mem x s) -> ((~ (mem y s)) -> ((path x l y) ->
exists x':vertex, exists y':vertex, (mem x' s) /\ ((~ (mem y' s)) /\ ((edge
x' y') /\ ((reachable x x') /\ (reachable y' y)))))).

Definition in_same_scc (x:vertex) (y:vertex): Prop := (reachable x y) /\
(reachable y x).

Definition is_subscc (s:(set vertex)): Prop := forall (x:vertex) (y:vertex),
(mem x s) -> ((mem y s) -> (in_same_scc x y)).

Definition is_scc (s:(set vertex)): Prop := (~ (is_empty s)) /\ ((is_subscc
s) /\ forall (s':(set vertex)), (subset s s') -> ((is_subscc s') ->
(infix_eqeq s s'))).

Axiom same_scc_refl : forall (x:vertex), (in_same_scc x x).

Axiom same_scc_sym : forall (x:vertex) (z:vertex), (in_same_scc x z) ->
(in_same_scc z x).

Axiom same_scc_trans : forall (x:vertex) (y:vertex) (z:vertex), (in_same_scc
x y) -> ((in_same_scc y z) -> (in_same_scc x z)).

Axiom subscc_add : forall (x:vertex) (y:vertex) (cc:(set vertex)), (is_subscc
cc) -> ((mem x cc) -> ((in_same_scc x y) -> (is_subscc (add y cc)))).

Axiom scc_max : forall (x:vertex) (y:vertex) (cc:(set vertex)), (is_scc
cc) -> ((mem x cc) -> ((in_same_scc x y) -> (mem y cc))).

Inductive env :=
| mk_env : (set vertex) -> (list vertex) -> (set (set vertex)) -> Z ->
(map.Map.map vertex Z) -> env.
Axiom env_WhyType : WhyType env.
Existing Instance env_WhyType.

Definition num (v:env): (map.Map.map vertex Z) :=
match v with
| (mk_env x x1 x2 x3 x4) => x4
end.

Definition sn (v:env): Z := match v with
| (mk_env x x1 x2 x3 x4) => x3
end.

Definition sccs (v:env): (set (set vertex)) :=
match v with
| (mk_env x x1 x2 x3 x4) => x2
end.

Definition stack (v:env): (list vertex) :=
match v with
| (mk_env x x1 x2 x3 x4) => x1
end.

Definition blacks (v:env): (set vertex) :=
match v with
| (mk_env x x1 x2 x3 x4) => x
end.

Axiom stack_id : forall (s:(list vertex)) (b:(set vertex)) (ccs:(set (set
vertex))) (n:Z) (f:(map.Map.map vertex Z)), ((stack (mk_env b s ccs n
f)) = s).

Axiom blacks_id : forall (s:(list vertex)) (b:(set vertex)) (ccs:(set (set
vertex))) (n:Z) (f:(map.Map.map vertex Z)), ((blacks (mk_env b s ccs n
f)) = b).

Axiom sccs_id : forall (s:(list vertex)) (b:(set vertex)) (ccs:(set (set
vertex))) (n:Z) (f:(map.Map.map vertex Z)), ((sccs (mk_env b s ccs n
f)) = ccs).

Axiom sn_id : forall (s:(list vertex)) (b:(set vertex)) (ccs:(set (set
vertex))) (n:Z) (f:(map.Map.map vertex Z)), ((sn (mk_env b s ccs n
f)) = n).

Axiom num_id : forall (s:(list vertex)) (b:(set vertex)) (ccs:(set (set
vertex))) (n:Z) (f:(map.Map.map vertex Z)), ((num (mk_env b s ccs n
f)) = f).

Definition wf_color (e:env) (grays:(set vertex)): Prop :=
match e with
| (mk_env b s ccs _ _) => (subset (union grays b) vertices) /\ ((infix_eqeq
(inter b grays) (empty : (set vertex))) /\ ((infix_eqeq (elements s)
(union grays (diff b (set_of ccs)))) /\ (subset (set_of ccs) b)))
end.

Definition wf_num (e:env) (grays:(set vertex)): Prop :=
match e with
| (mk_env b s ccs n f) => (forall (x:vertex), (((-1%Z)%Z <= (map.Map.get f
x))%Z /\ (((map.Map.get f x) < n)%Z /\
(n <= (cardinal vertices))%Z)) \/ ((map.Map.get f
x) = (cardinal vertices))) /\ ((n = (cardinal (union grays b))) /\
((forall (x:vertex), ((map.Map.get f x) = (cardinal vertices)) <-> (mem
x (set_of ccs))) /\ ((forall (x:vertex), ((map.Map.get f
x) = (-1%Z)%Z) <-> ~ (mem x (union grays b))) /\ forall (x:vertex)
(y:vertex), (list.Mem.mem x s) -> ((list.Mem.mem y s) ->
(((map.Map.get f x) < (map.Map.get f y))%Z <-> ((rank x s) < (rank y
s))%Z)))))
end.

Definition no_black_to_white (blacks1:(set vertex)) (grays:(set
vertex)): Prop := forall (x:vertex) (x':vertex), (edge x x') -> ((mem x
blacks1) -> (mem x' (union blacks1 grays))).

Definition wf_env (e:env) (grays:(set vertex)): Prop := let s := (stack e) in
((wf_color e grays) /\ ((wf_num e grays) /\ ((no_black_to_white (blacks e)
grays) /\ ((simplelist s) /\ ((forall (x:vertex) (y:vertex), (mem x
grays) -> ((list.Mem.mem y s) -> (((rank x s) <= (rank y s))%Z ->
(reachable x y)))) /\ forall (y:vertex), (list.Mem.mem y s) ->
exists x:vertex, (mem x grays) /\ (((rank x s) <= (rank y s))%Z /\
(reachable y x))))))).

Definition access_from (x:vertex) (s:(set vertex)): Prop :=
forall (y:vertex), (mem y s) -> (reachable x y).

Definition access_to (s:(set vertex)) (y:vertex): Prop := forall (x:vertex),
(mem x s) -> (reachable x y).

Definition rank_of_reachable (m:Z) (x:vertex) (s:(list vertex)): Prop :=
exists y:vertex, (list.Mem.mem y s) /\ ((m = (rank y s)) /\ (reachable x
y)).

Definition num_of_reachable (n:Z) (x:vertex) (e:env): Prop :=
exists y:vertex, (list.Mem.mem y (stack e)) /\ ((n = (map.Map.get (num e)
y)) /\ (reachable x y)).

Definition xedge_to (s1:(list vertex)) (s3:(list vertex)) (y:vertex): Prop :=
(exists s2:(list vertex), (s1 = (Init.Datatypes.app s2 s3)) /\
exists x:vertex, (list.Mem.mem x s2) /\ (edge x y)) /\ (list.Mem.mem y s3).

Definition subenv (e:env) (e':env): Prop := (exists s:(list vertex),
((stack e') = (Init.Datatypes.app s (stack e))) /\ (subset (elements s)
(blacks e'))) /\ ((subset (blacks e) (blacks e')) /\ ((subset (sccs e)
(sccs e')) /\ forall (x:vertex), (list.Mem.mem x (stack e)) ->
((map.Map.get (num e) x) = (map.Map.get (num e') x)))).

Axiom xedge_r : forall (x:vertex) (s1:(list vertex)) (s2:(list vertex))
(y:vertex), (xedge_to (Init.Datatypes.app s1 s2) s2 y) -> ((is_last x
s1) -> ((xedge_to (Init.Datatypes.app s1 s2) (Init.Datatypes.cons x s2)
y) \/ (edge x y))).

Axiom subscc_after_last_gray : forall (x:vertex) (e:env) (g:(set vertex))
(s2:(list vertex)) (s3:(list vertex)), (wf_env e (add x g)) ->
match e with
| (mk_env b s _ _ _) => (s = (Init.Datatypes.app s2 s3)) -> ((is_last x
s2) -> ((subset (elements s2) (add x b)) -> (is_subscc (elements s2))))
end.

Axiom wf_color_post_cond_split : forall (s2:(set vertex)) (s3:(set vertex))
(g:(set vertex)) (b:(set vertex)) (sccs1:(set vertex)), (infix_eqeq
(union s2 s3) (union g (diff b sccs1))) -> (((inter s2 s3) = (empty : (set
vertex))) -> (((inter g s2) = (empty : (set vertex))) -> (infix_eqeq s3
(union g (diff b (union s2 sccs1)))))).

Axiom wf_color_sccs : forall (e:env) (g:(set vertex)), (wf_color e g) ->
(infix_eqeq (set_of (sccs e)) (diff (union (blacks e) g)
(elements (stack e)))).

Axiom wf_color_3_cases : forall (x:vertex) (e:env) (g:(set vertex)),
(wf_color e g) -> ((mem x (set_of (sccs e))) \/ ((mem x
(elements (stack e))) \/ ~ (mem x (union (blacks e) g)))).

Require Import mathcomp.ssreflect.ssreflect.
Ltac compress :=
repeat (match goal with [H1: ?A, H2: ?A |- _] => clear H2 end).

Theorem WP_parameter_dfs1 : forall (x:vertex) (e:(set vertex))
(e1:(list vertex)) (e2:(set (set vertex))) (e3:Z) (e4:(map.Map.map vertex
Z)) (grays:(set vertex)), ((mem x vertices) /\ ((access_to grays x) /\
((~ (mem x (union e grays))) /\ ((wf_env (mk_env e e1 e2 e3 e4) grays) /\
forall (cc:(set vertex)), (mem cc e2) <-> ((subset cc e) /\ (is_scc
cc)))))) -> let o := (add x grays) in forall (o1:(set vertex))
(o2:(list vertex)) (o3:(set (set vertex))) (o4:Z) (o5:(map.Map.map vertex
Z)), let o6 := (mk_env o1 o2 o3 o4 o5) in (((o1 = e) /\ ((o3 = e2) /\
((o2 = (Init.Datatypes.cons x e1)) /\ ((o4 = (e3 + 1%Z)%Z) /\
(o5 = (map.Map.set e4 x e3)))))) -> let o7 := (successors x) in (((subset
o7 vertices) /\ ((forall (x1:vertex), (mem x1 o7) -> (access_to o x1)) /\
((wf_env o6 o) /\ forall (cc:(set vertex)), (mem cc o3) <-> ((subset cc
o1) /\ (is_scc cc))))) -> forall (result:Z) (result1:(set vertex))
(result2:(list vertex)) (result3:(set (set vertex))) (result4:Z)
(result5:(map.Map.map vertex Z)), ((wf_env (mk_env result1 result2 result3
result4 result5) o) /\ ((forall (cc:(set vertex)), (mem cc result3) <->
((subset cc result1) /\ (is_scc cc))) /\ ((forall (x1:vertex), (mem x1
o7) -> (result <= (map.Map.get result5 x1))%Z) /\
(((result = (cardinal vertices)) \/ exists x1:vertex, (mem x1 o7) /\
(num_of_reachable result x1 (mk_env result1 result2 result3 result4
result5))) /\ ((forall (y:vertex), (xedge_to result2 o2 y) ->
(result <= (map.Map.get result5 y))%Z) /\ ((subset o7 (union result1 o)) /\
(subenv o6 (mk_env result1 result2 result3 result4 result5)))))))) ->
forall (result6:(list vertex)) (result7:(list vertex)),
(((Init.Datatypes.app result6 result7) = result2) /\ ((list.Mem.mem x
result2) -> exists s':(list vertex),
(result6 = (Init.Datatypes.app s' (Init.Datatypes.cons x Init.Datatypes.nil))))) ->
(((exists s':(list vertex),
(result6 = (Init.Datatypes.app s' (Init.Datatypes.cons x Init.Datatypes.nil)))) /\
((result7 = e1) /\ (subset (elements result6) (add x result1)))) ->
((is_subscc (elements result6)) -> ((result < e3)%Z -> exists y:vertex,
(mem y grays) /\ ((list.Mem.mem y result2) /\ (((map.Map.get result5
y) < (map.Map.get result5 x))%Z /\ (reachable x y)))))))).

Proof.

move=> x blacks stack sccs sn num grays [r11 [r12 [r13 [i11a i12a]]]].
move=> g0 b0 s0 sccs0 n0 num0 e0 [_ [_ [s0_is_xstack [incr_sn num_x_n]]]].
move=> roots0 [r'1 [r'2 [[wf_color0 [wf_num0 [_ [simpl_s0 _]]]] h12]]].
move: {+} wf_num0 => [_ [card0 _]].
move=> n1 b1 s1 sccs1 sn1 num1.
move=> [hwf_env1 a].
move: {+}hwf_env1 => [hwf_color1 [wf_num1 _]].
move: {+}hwf_color1 => [_ [_ [elts_s1 _]]].
move: a => [_ [_ [reach1 [_ [roots0_nwhite subenv01]]]]].
move: {+}subenv01 => [[s' [s1_decomp _]] _]; simpl in s1_decomp.
move=> s2 s3 [s1_split s2_split].
move=> [_ [s3_eq_stack subset_b1]].
move=> _ n1_lt_n.
compress.

have n1_neq_infty: (n1 <> cardinal vertices).
- apply Z.lt_neq.
move :i11a => [[sub _] [[_ [card _]] _]].
apply (Z.lt_le_trans _ (cardinal (union grays blacks)) _).
+ by rewrite -card.
+ by apply cardinal_subset.

move :reach1 => [_ | [x1 [x1_in_roots0 x1_reach_n1]]] //=.
move :x1_reach_n1 => [y [y_in_s1 [n1_is_ynum1 x1_reach_y]]].
have x_reach_y: reachable x y.
- move :x1_reach_y => [l path_x1_l_y].
by exists (x :: l)%list; apply (Path_cons x x1 y l).
have x_in_s0: Mem.mem x s0.
- by rewrite s0_is_xstack; simpl; left.
simpl in y_in_s1.
have x_in_s1: Mem.mem x s1.
- by rewrite s1_decomp; apply Append.mem_append;right.
have sn_is_xnum0: sn = Map.get num0 x.
- by rewrite num_x_n Map.Select_eq.
have sn_is_xnum1: sn = Map.get num1 x.
- by move: {+}subenv01 => [_ [_ [_ num01]]]; rewrite -num01.
move: {+}wf_num1 => [_ [_ [_ [_ numrank1]]]].
have rkyx: (rank y s1 < rank x s1)%Z.
- apply numrank1.
+ by [].
+ by [].
+ by rewrite -sn_is_xnum1 -n1_is_ynum1.
move :hwf_env1; simpl; move => [_ [_ [_ [_ [_ back_to_grays]]]]].
simpl in back_to_grays.
move/back_to_grays :y_in_s1 => [y' [y'_in_xgrays [y'_le_y y_reach_y']]].

have y'_lt_x: (rank y' s1 < rank x s1)%Z.
- apply (Z.le_lt_trans _ (rank y s1) _).
+ by apply y'_le_y.
+ by [].
have y'_in_grays: mem y' grays.
- move/add_def1 :y'_in_xgrays => [y'_is_x | y'_in_grays]; last by [].
move :y'_lt_x; rewrite y'_is_x; move => x_lt_x.
by apply Z.lt_irrefl in x_lt_x.
have y'_in_s1: Mem.mem y' s1.
- by apply/elements_mem /elts_s1 /union_def1; left.
exists y'.
split; first by [].
split; first by [].
split; first by apply numrank1.
by apply (reachable_trans _ y _).

Qed.