Goal:
  
  G1 : t
  G2 : t
  on_edge : forall x y : V.t, EdgeOf G1 x y -> EdgeOf G2 x y
  weight_on_edge : forall (x y : V.t) (e : EdgeOf G1 x y),
                   (on_edge x y e).π1 >= e.π1
  x : V.t
  y : V.t
  p : PathOf G1 x y
  ============================
   (forall (x y y' : V.t) (e : EdgeOf G1 x y') (p : PathOf G1 y' y),
    weight (map_path p) >= weight p ->
    (on_edge x y' e).π1 + weight (map_path p) >= e.π1 + weight p)


DEBUG:Going to execute:
trivial; hnf; intros; (solve
            [ do
            ![ solve
             [ trivial | apply : eq_sym ; trivial ]
             | discriminate
             | contradiction
             | Ltac.split ]
            | case not_locked_false_eq_true; Ltac.assumption
            | match goal with
              | H:~ _ |- _ => solve [ case H; trivial ]
              end ])