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
  H4' : VSet.t
  H4'0 : V.t
  H4 : sigma
         (fun H : V.t =>
          sigma
            (fun H0 : VSet.t =>
             sigma
               (fun H1 : V.t =>
                sigma
                  (fun H2 : V.t =>
                   sigma (fun _ : SPath G H4' H4'0 H => SPath G H0 H1 H2)))))
  H2 : SPath_direct_subterm G H4' H4'0 (pr1 H4) (pr1 (pr2 H4))
         (pr1 (pr2 (pr2 H4))) (pr1 (pr2 (pr2 (pr2 H4))))
         (pr1 (pr2 (pr2 (pr2 (pr2 H4))))) (pr2 (pr2 (pr2 (pr2 (pr2 H4)))))
  ============================
   (forall (s0 s' : VSet.t) (x y z : V.t) (d : DisjointAdd x s0 s')
      (e : EdgeOf G x y) (a0 : SPath G s0 y z),
    Acc
      (fun
         x0
          y0 : sigma
                 (fun
                    index : sigma
                              (fun _ : VSet.t => sigma (fun _ : V.t => V.t))
                  => SPath G (pr1 index) (pr1 (pr2 index)) (pr2 (pr2 index)))
       =>
       SPath_direct_subterm G (pr1 (pr1 x0)) (pr1 (pr2 (pr1 x0)))
         (pr2 (pr2 (pr1 x0))) (pr1 (pr1 y0)) (pr1 (pr2 (pr1 y0)))
         (pr2 (pr2 (pr1 y0))) (pr2 x0) (pr2 y0))
      {|
        pr1 := {| pr1 := s0; pr2 := {| pr1 := y; pr2 := z |} |}; pr2 := a0
      |} ->
    forall (pr1 : VSet.t) (pr0 pr2 : V.t) (y1 : SPath G pr1 pr0 pr2),
    {|
      pr1 := pr1;
      pr2 :=
        {|
          pr1 := pr0;
          pr2 :=
            {|
              pr1 := pr2;
              pr2 :=
                {|
                  pr1 := s';
                  pr2 :=
                    {|
                      pr1 := x;
                      pr2 :=
                        {|
                          pr1 := z;
                          pr2 :=
                            {|
                              pr1 := y1;
                              pr2 := spath_step G s0 s' x y z d e a0
                            |}
                        |}
                    |}
                |}
            |}
        |}
    |} = {| pr1 := H4'; pr2 := {| pr1 := H4'0; pr2 := H4 |} |} ->
    let H := CoreTactics.block in
    let H0 := CoreTactics.block in
    Acc
      (fun
         x0
          y0 : 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 x0))
         (Init.pr1 (Init.pr2 (Init.pr1 x0)))
         (Init.pr2 (Init.pr2 (Init.pr1 x0))) (Init.pr1 (Init.pr1 y0))
         (Init.pr1 (Init.pr2 (Init.pr1 y0)))
         (Init.pr2 (Init.pr2 (Init.pr1 y0))) (Init.pr2 x0) 
         (Init.pr2 y0))
      {|
        pr1 := {| pr1 := pr1; pr2 := {| pr1 := pr0; pr2 := pr2 |} |};
        pr2 := y1
      |})


DEBUG:Going to execute:
fun id id' =>
             hnf in id;
              match type of id with
              | sigma _ =>
                  let xn := fresh id "'" in
                  destruct id as [xn id]; DepElim.decompose_exists xn id;
                   cbv beta iota delta [TODO projection TODO projection]
                    in id, id'; DepElim.decompose_exists id id'
              | _ =>
                  cbv beta iota delta [TODO projection TODO projection]
                   in id, id'
              end