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
  G : t
  s0 : VSet.t
  x : V.t
  pr1 : VSet.t
  pr0 : sigma (fun _ : V.t => V.t)
  y0 : SPath G (Init.pr1 {| pr1 := pr1; pr2 := pr0 |})
         (Init.pr1 (pr2 {| pr1 := pr1; pr2 := pr0 |}))
         (pr2 (pr2 {| pr1 := pr1; pr2 := pr0 |}))
  H2 : SPath_direct_subterm G
         (Init.pr1
            (Init.pr1 {| pr1 := {| pr1 := pr1; pr2 := pr0 |}; pr2 := y0 |}))
         (Init.pr1
            (pr2
               (Init.pr1 {| pr1 := {| pr1 := pr1; pr2 := pr0 |}; pr2 := y0 |})))
         (pr2
            (pr2
               (Init.pr1 {| pr1 := {| pr1 := pr1; pr2 := pr0 |}; pr2 := y0 |})))
         (Init.pr1
            (Init.pr1
               {|
                 pr1 := {| pr1 := s0; pr2 := {| pr1 := x; pr2 := x |} |};
                 pr2 := spath_refl G s0 x
               |}))
         (Init.pr1
            (pr2
               (Init.pr1
                  {|
                    pr1 := {| pr1 := s0; pr2 := {| pr1 := x; pr2 := x |} |};
                    pr2 := spath_refl G s0 x
                  |})))
         (pr2
            (pr2
               (Init.pr1
                  {|
                    pr1 := {| pr1 := s0; pr2 := {| pr1 := x; pr2 := x |} |};
                    pr2 := spath_refl G s0 x
                  |})))
         (pr2 {| pr1 := {| pr1 := pr1; pr2 := pr0 |}; pr2 := y0 |})
         (pr2
            {|
              pr1 := {| pr1 := s0; pr2 := {| pr1 := x; pr2 := x |} |};
              pr2 := spath_refl G s0 x
            |})
  ============================
   (Acc
      (fun
         x
          y : sigma
                (fun
                   index : sigma
                             (fun _ : VSet.t => sigma (fun _ : V.t => V.t))
                 =>
                 SPath G (Init.pr1 index) (Init.pr1 (pr2 index))
                   (pr2 (pr2 index))) =>
       SPath_direct_subterm G (Init.pr1 (Init.pr1 x))
         (Init.pr1 (pr2 (Init.pr1 x))) (pr2 (pr2 (Init.pr1 x)))
         (Init.pr1 (Init.pr1 y)) (Init.pr1 (pr2 (Init.pr1 y)))
         (pr2 (pr2 (Init.pr1 y))) (pr2 x) (pr2 y))
      {| pr1 := {| pr1 := pr1; pr2 := pr0 |}; pr2 := y0 |})


DEBUG:Going to execute:
destruct H as [? H']; destruct_right_sigma H'