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
  s' : VSet.t
  x : V.t
  y : V.t
  z : V.t
  d : DisjointAdd x s0 s'
  e : EdgeOf G x y
  p0 : SPath G s0 y z
  IHp0 : Acc
           (fun
              x
               y : 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 x)) (pr1 (pr2 (pr1 x)))
              (pr2 (pr2 (pr1 x))) (pr1 (pr1 y)) (pr1 (pr2 (pr1 y)))
              (pr2 (pr2 (pr1 y))) (pr2 x) (pr2 y))
           {|
             pr1 := {| pr1 := s0; pr2 := {| pr1 := y; pr2 := z |} |};
             pr2 := p0
           |}
  s1 : VSet.t
  x1 : V.t
  y1 : V.t
  p1 : SPath G s1 x1 y1
  h1' : VSet.t
  h1'0 : V.t
  h1'1 : V.t
  h1 : sigma
         (fun H : VSet.t =>
          sigma
            (fun H0 : V.t =>
             sigma
               (fun H1 : V.t =>
                sigma (fun _ : SPath G h1' h1'0 h1'1 => SPath G H H0 H1))))
  h : SPath_direct_subterm G h1' h1'0 h1'1 (pr1 h1) 
        (pr1 (pr2 h1)) (pr1 (pr2 (pr2 h1))) (pr1 (pr2 (pr2 (pr2 h1))))
        (pr2 (pr2 (pr2 (pr2 h1))))
  ============================
   ({|
      pr1 := s1;
      pr2 :=
        {|
          pr1 := x1;
          pr2 :=
            {|
              pr1 := y1;
              pr2 :=
                {|
                  pr1 := s';
                  pr2 :=
                    {|
                      pr1 := x;
                      pr2 :=
                        {|
                          pr1 := z;
                          pr2 :=
                            {|
                              pr1 := p1;
                              pr2 := spath_step G s0 s' x y z d e p0
                            |}
                        |}
                    |}
                |}
            |}
        |}
    |} =
    {|
      pr1 := h1';
      pr2 := {| pr1 := h1'0; pr2 := {| pr1 := h1'1; pr2 := h1 |} |}
    |} ->
    let H := CoreTactics.block in
    let H0 := CoreTactics.block in
    Acc
      (fun
         x
          y : 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 x)) (pr1 (pr2 (pr1 x)))
         (pr2 (pr2 (pr1 x))) (pr1 (pr1 y)) (pr1 (pr2 (pr1 y)))
         (pr2 (pr2 (pr1 y))) (pr2 x) (pr2 y))
      {|
        pr1 := {| pr1 := s1; pr2 := {| pr1 := x1; pr2 := y1 |} |}; pr2 := p1
      |})


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