Goal:
  
  G1 : t
  G2 : t
  l : labelling
  Gl := relabel_on G1 G2 l : t
  H : invariants G1
  preserves_edges : forall e : EdgeSet.elt,
                    EdgeSet.In e (E G1) -> EdgeSet.In e (E G2)
  HGl : correct_labelling G1 l
  H0 : invariants G2
  H1 : acyclic_no_loop G2
  embed : forall v1 v2 : VSet.elt,
          VSet.In v1 (V G1) ->
          VSet.In v2 (V G1) -> (lsp G2 v1 v2 <= lsp G1 v1 v2)%nbar
  s0 : VSet.t
  s' : VSet.t
  x : V.t
  y : V.t
  z : V.t
  d : DisjointAdd x s0 s'
  p : SPath Gl s0 y z
  yin : VSet.In z (V G1)
  IHp : VSet.In z (V G1) ->
        exists n : Z,
          lsp G2 y (first_in G1 Gl p) = Some n /\
          sweight p <= n + diff l (first_in G1 Gl p) z
  w : Z
  h : EdgeSet.In (x, w, y) (E Gl)
  s : V.t
  w0 : Z
  t : V.t
  e0G2 : EdgeSet.In (s, w0, t) (E G2)
  E : EdgeSet.mem (s, w0, t) (E G1) = false
  lbw : (Some (w0; e0G2).π1 <= lsp G2 s t)%nbar
  w1 : Z
  wpb : sweight p <= w1 + diff l (first_in G1 Gl p) z
  _Hyp_ : x = s
  _Hyp1_ : w = w0
  _Hyp2_ : y = t
  ============================
   ((lsp G2 x y + Some w1 <= lsp G2 x (first_in G1 Gl p))%nbar ->
    exists n : Z,
      lsp G2 x
        (first_in_clause_2 Gl (@first_in G1 Gl) s' x 
           (VSet.mem x (V G1)) z s0 y d (w; h) p) = 
      Some n /\
      w + sweight p <=
      n +
      diff l
        (first_in_clause_2 Gl (@first_in G1 Gl) s' x 
           (VSet.mem x (V G1)) z s0 y d (w; h) p) z)


DEBUG:Going to execute:
<coq-core.plugins.ltac::subst@1>