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


DEBUG:Going to execute:
fun H =>
             match type of H with
             | sigma (fun x => _) =>
                 let H' := fresh H in
                 destruct H as [? H']; destruct_right_sigma H'
             | sigma _ =>
                 let H' := fresh H in
                 destruct H as [? H']; destruct_right_sigma H'
             | _ => idtac
             end