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(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below    *)
Require Import ZArith.
Require Import Rbase.
Require int.Int.

(* Why3 assumption *)
Inductive list (a:Type) :=
  | Nil : list a
  | Cons : a -> (list a) -> list a.
Set Contextual Implicit.
Implicit Arguments Nil.
Unset Contextual Implicit.
Implicit Arguments Cons.

(* Why3 assumption *)
Set Implicit Arguments.
Fixpoint length (a:Type)(l:(list a)) {struct l}: Z :=
  match l with
  | Nil => 0%Z
  | (Cons _ r) => (1%Z + (length r))%Z
  end.
Unset Implicit Arguments.

Axiom Length_nonnegative : forall (a:Type), forall (l:(list a)),
  (0%Z <= (length l))%Z.

Axiom Length_nil : forall (a:Type), forall (l:(list a)),
  ((length l) = 0%Z) <-> (l = (Nil :(list a))).

Parameter set : forall (a:Type), Type.

Parameter mem: forall (a:Type), a -> (set a) -> Prop.
Implicit Arguments mem.

(* Why3 assumption *)
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).

(* Why3 assumption *)
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.

(* Why3 assumption *)
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 choose: forall (a:Type), (set a) -> a.
Implicit Arguments choose.

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

Parameter all: forall (a:Type), (set a).
Set Contextual Implicit.
Implicit Arguments all.
Unset Contextual Implicit.

Axiom all_def : forall (a:Type), forall (x:a), (mem x (all :(set a))).

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.

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

Parameter nth: forall (a:Type), Z -> (set a) -> a.
Implicit Arguments nth.

Axiom nth_injective : forall (a:Type), forall (s:(set a)) (i:Z) (j:Z),
  ((0%Z <= i)%Z /\ (i <  (cardinal s))%Z) -> (((0%Z <= j)%Z /\
  (j <  (cardinal s))%Z) -> (((nth i s) = (nth j s)) -> (i = j))).

Axiom nth_surjective : forall (a:Type), forall (s:(set a)) (x:a), (mem x
  s) -> exists i:Z, ((0%Z <= i)%Z /\ (i <  (cardinal s))%Z) -> (x = (nth i
  s)).

Parameter vertex : Type.

Parameter vertices: (set vertex).

Parameter edges: (set (vertex* vertex)%type).

(* Why3 assumption *)
Definition edge(x:vertex) (y:vertex): Prop := (mem (x, y) edges).

Axiom edges_def : forall (x:vertex) (y:vertex), (mem (x, y) edges) -> ((mem x
  vertices) /\ (mem y vertices)).

Parameter s: vertex.

Axiom s_in_graph : (mem s vertices).

Axiom vertices_cardinal_pos : (0%Z <  (cardinal vertices))%Z.

(* Why3 assumption *)
Set Implicit Arguments.
Fixpoint infix_plpl (a:Type)(l1:(list a)) (l2:(list a)) {struct l1}: (list
  a) :=
  match l1 with
  | Nil => l2
  | (Cons x1 r1) => (Cons x1 (infix_plpl r1 l2))
  end.
Unset Implicit Arguments.

Axiom Append_assoc : forall (a:Type), forall (l1:(list a)) (l2:(list a))
  (l3:(list a)), ((infix_plpl l1 (infix_plpl l2
  l3)) = (infix_plpl (infix_plpl l1 l2) l3)).

Axiom Append_l_nil : forall (a:Type), forall (l:(list a)), ((infix_plpl l
  (Nil :(list a))) = l).

Axiom Append_length : forall (a:Type), forall (l1:(list a)) (l2:(list a)),
  ((length (infix_plpl l1 l2)) = ((length l1) + (length l2))%Z).

(* Why3 assumption *)
Set Implicit Arguments.
Fixpoint mem1 (a:Type)(x:a) (l:(list a)) {struct l}: Prop :=
  match l with
  | Nil => False
  | (Cons y r) => (x = y) \/ (mem1 x r)
  end.
Unset Implicit Arguments.

Axiom mem_append : forall (a:Type), forall (x:a) (l1:(list a)) (l2:(list a)),
  (mem1 x (infix_plpl l1 l2)) <-> ((mem1 x l1) \/ (mem1 x l2)).

Axiom mem_decomp : forall (a:Type), forall (x:a) (l:(list a)), (mem1 x l) ->
  exists l1:(list a), exists l2:(list a), (l = (infix_plpl l1 (Cons x l2))).

(* Why3 assumption *)
Inductive path : vertex -> (list vertex) -> vertex -> Prop :=
  | Path_empty : forall (x:vertex), (path x (Nil :(list vertex)) x)
  | Path_cons : forall (x:vertex) (y:vertex) (z:vertex) (l:(list vertex)),
      (edge x y) -> ((path y l z) -> (path x (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 (infix_plpl l (Cons y
  (Nil :(list vertex)))) z)).

Axiom path_right_inversion : forall (x:vertex) (z:vertex) (l:(list vertex)),
  (path x l z) -> (((x = z) /\ (l = (Nil :(list vertex)))) \/
  exists y:vertex, exists lqt:(list vertex), (path x lqt y) /\ ((edge y z) /\
  (l = (infix_plpl lqt (Cons y (Nil :(list vertex))))))).

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
  (infix_plpl l1 l2) z)).

Axiom empty_path : forall (x:vertex) (y:vertex), (path x (Nil :(list vertex))
  y) -> (x = y).

Axiom path_decomposition : forall (x:vertex) (y:vertex) (z:vertex) (l1:(list
  vertex)) (l2:(list vertex)), (path x (infix_plpl l1 (Cons y l2)) z) ->
  ((path x l1 y) /\ (path y (Cons y l2) z)).

Parameter weight: vertex -> vertex -> Z.

(* Why3 assumption *)
Set Implicit Arguments.
Fixpoint path_weight(l:(list vertex)) (dst:vertex) {struct l}: Z :=
  match l with
  | Nil => 0%Z
  | (Cons x Nil) => (weight x dst)
  | (Cons x ((Cons y _) as r)) => ((weight x y) + (path_weight r dst))%Z
  end.
Unset Implicit Arguments.

Axiom path_weight_right_extension : forall (x:vertex) (y:vertex) (l:(list
  vertex)), ((path_weight (infix_plpl l (Cons x (Nil :(list vertex))))
  y) = ((path_weight l x) + (weight x y))%Z).

Axiom path_weight_decomposition : forall (y:vertex) (z:vertex) (l1:(list
  vertex)) (l2:(list vertex)), ((path_weight (infix_plpl l1 (Cons y l2))
  z) = ((path_weight l1 y) + (path_weight (Cons y l2) z))%Z).

Axiom path_in_vertices : forall (v1:vertex) (v2:vertex) (l:(list vertex)),
  (mem v1 vertices) -> ((path v1 l v2) -> (mem v2 vertices)).

Axiom long_path_decomposition : forall (l:(list vertex)) (v:vertex), (path s
  l v) -> (((cardinal vertices) <= (length l))%Z -> ((exists l1:(list
  vertex), (exists l2:(list vertex), (l = (infix_plpl l1 (Cons v l2))))) \/
  exists n:vertex, exists l1:(list vertex), exists l2:(list vertex),
  exists l3:(list vertex), (l = (infix_plpl l1 (Cons n (infix_plpl l2 (Cons n
  l3))))))).

Axiom simple_path : forall (v:vertex) (l:(list vertex)), (path s l v) ->
  exists lqt:(list vertex), (path s lqt v) /\
  ((length lqt) <  (cardinal vertices))%Z.

(* Why3 assumption *)
Definition negative_cycle(v:vertex): Prop := (mem v vertices) /\
  ((exists l1:(list vertex), (path s l1 v)) /\ exists l2:(list vertex),
  (path v l2 v) /\ ((path_weight l2 v) <  0%Z)%Z).

Require Import Why3. Ltac ae := why3 "alt-ergo".

(* Why3 goal *)
Theorem key_lemma_1 : forall (v:vertex) (n:Z), (forall (l:(list vertex)),
  (path s l v) -> (((length l) <  (cardinal vertices))%Z ->
  (n <= (path_weight l v))%Z)) -> ((exists l:(list vertex), (path s l v) /\
  ((path_weight l v) <  n)%Z) -> exists u:vertex, (negative_cycle u)).
intros v n hpath.
intros (ln, (pathln, lnneg)).
generalize pathln lnneg; clear pathln lnneg.
cut (length ln <= length ln)%Z. 2: omega.
cut (0 <= length ln)%Z. 2: apply Length_nonnegative.
generalize (length ln) at 1 3.
intros z hz; generalize ln. clear ln.
pattern z; apply Z_lt_induction; auto.
clear z hz; intros z IH.
intros ln hlen pathln lnneg.
assert (h: (cardinal vertices <= length ln)%Z) by ae.
destruct (long_path_decomposition ln v pathln h) as
  [(l1, (l2, eq)) | (u, (l1, (l2, (l3, eq))))].
(* s --> v --> v *)
rewrite eq in pathln.
generalize (path_decomposition _ _ _ _ _ pathln).
intros (h1, pathuv).
assert (case: (path_weight (Cons v l2) v < 0 \/
              (path_weight (Cons v l2) v >= 0))%Z) by omega.
destruct case.
exists v; ae.
assert (hpath': path s l1 v) by ae.
assert (length ln = length l1 + 1 + length l2)%Z.
  rewrite eq. rewrite Append_length. ae.
assert (0 <= length l2)%Z.
  apply Length_nonnegative.
assert (smaller: (0 <= length l1 < z)%Z).
  split. apply Length_nonnegative. omega.
apply (IH (length l1) smaller l1); auto.
omega.
clear H0 H1 smaller IH hpath.
assert (path_weight ln v =
        path_weight l1 v + path_weight (Cons v l2) v)%Z.
  rewrite eq.
  rewrite path_weight_decomposition. ae.
ae.
(* s --> u --> u --> v *)
rewrite eq in pathln.
generalize (path_decomposition _ _ _ _ _ pathln).
intros (h1, pathuv).
replace (Cons u (infix_plpl l2 (Cons u l3)))
   with (infix_plpl (Cons u l2) (Cons u l3)) in pathuv by ae.
generalize (path_decomposition _ _ _ _ _ pathuv).
intros (h2, pathuv2).
assert (case: (path_weight (Cons u l2) u < 0 \/
              (path_weight (Cons u l2) u >= 0))%Z) by omega.
destruct case.
exists u; ae.
pose (l' := (infix_plpl l1 (Cons u l3))).
assert (hpath': path s l' v) by ae.
assert (length ln = length l1 + 1 + length l2 + 1 + length l3)%Z.
  rewrite eq.
  repeat rewrite Append_length.
  replace (length (Cons u (infix_plpl l2 (Cons u l3))))
            with (1 + length (infix_plpl l2 (Cons u l3)))%Z by ae.
  rewrite Append_length. ae.
assert (length l' = length l1 + 1 + length l3)%Z.
  subst l'.
  rewrite Append_length.
  ae.
assert (0 <= length l2)%Z.
  apply Length_nonnegative.
assert (smaller: (0 <= length l' < z)%Z).
  split. apply Length_nonnegative. omega.
apply (IH (length l') smaller l'); auto.
omega.
clear H0 H1 H2 smaller IH hpath.
assert (path_weight ln v =
        path_weight l1 u + path_weight (Cons u l2) u +
        path_weight (Cons u l3) v)%Z.
  rewrite eq.
  rewrite path_weight_decomposition.
  replace (Cons u (infix_plpl l2 (Cons u l3)))
   with (infix_plpl (Cons u l2) (Cons u l3)) by ae.
  rewrite path_weight_decomposition. ae.
assert (path_weight l' v = path_weight l1 u + path_weight (Cons u l3) v)%Z.
  ae.
omega.
Qed.