Goal:
  
  G1 : t
  G2 : t
  H : invariants G1
  ext := EdgeSet.diff (E G2) (E G1) : EdgeSet.t
  bs := border_set G1 ext : VSet.t
  H0 : invariants G2
  H1 : acyclic_no_loop G1
  H2 : acyclic_no_loop G2
  h : forall x y : VSet.elt,
      VSet.In x bs -> VSet.In y bs -> (lsp G2 x y <= lsp G1 x y)%nbar
  _s_ : VSet.t
  _s'_ : VSet.t
  x : V.t
  y : V.t
  z : V.t
  _d_ : DisjointAdd x _s_ _s'_
  e : EdgeOf G2 x y
  q : SPath G2 _s_ y z
  ih : VSet.In z (V G1) ->
       (Some (sweight q) <=
        lsp G2 y (first_in G1 G2 q) + lsp G1 (first_in G1 G2 q) z)%nbar
  hz : VSet.In z (V G1)
  Ee : EdgeSet.In (x, e.π1, y) (E G1)
  hx : VSet.In (x, e.π1, y)..s (V G1)
  hy : VSet.In (x, e.π1, y)..t (V G1)
  H3 : (lsp G1 x y + lsp G1 y z <= lsp G1 x z)%nbar
  ============================
   ((Some (sweight q) <= lsp G2 y y + lsp G1 y z)%nbar ->
    (Some (e.π1 + sweight q)%Z <= lsp G2 x x + lsp G1 x z)%nbar)


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 ])