Goal:
  
  G : t
  HI : invariants G
  HG : acyclic_no_loop G
  y_0 : V.t
  x_0 : V.t
  K : Z
  Hxs : lsp G x_0 y_0 = Some K
  x : V.t
  y : V.t
  z : V.t
  e : EdgeOf G' x y
  q : PathOf G' y z
  IHq : match from_G'_path q with
        | inl q' => weight q = weight q'
        | inr (q1, q2) => weight q <= weight q1 - K + weight q2
        end
  ============================
   match
     sumbool_rect
       (fun
          _ : {(y_0, - K, x_0) = (x, e.π1, y)} +
              {(y_0, - K, x_0) <> (x, e.π1, y)} =>
        (PathOf G x z + (PathOf G x y_0 × PathOf G x_0 z))%type)
       (fun XX : (y_0, - K, x_0) = (x, e.π1, y) =>
        inr
          (eq_rect_r
             (fun _pattern_value_ : V.t => PathOf G x _pattern_value_)
             (pathOf_refl G x) (fst_eq (fst_eq XX)),
           match from_G'_path q with
           | inl p =>
               eq_rect_r
                 (fun _pattern_value_ : V.t => PathOf G _pattern_value_ z) p
                 (snd_eq XX)
           | inr p => p.2
           end))
       (fun XX : (y_0, - K, x_0) <> (x, e.π1, y) =>
        sum_rect
          (fun _ : PathOf G y z + (PathOf G y y_0 × PathOf G x_0 z) =>
           (PathOf G x z + (PathOf G x y_0 × PathOf G x_0 z))%type)
          (fun IHq : PathOf G y z =>
           inl (pathOf_step G x y z (e.π1; EdgeSetFact.add_3 XX e.π2) IHq))
          (fun IHq : PathOf G y y_0 × PathOf G x_0 z =>
           inr
             (pathOf_step G x y y_0 (e.π1; EdgeSetFact.add_3 XX e.π2) IHq.1,
              IHq.2)) (from_G'_path q))
       (Edge.eq_dec (y_0, - K, x_0) (x, e.π1, y))
   with
   | inl q' => e.π1 + weight q = weight q'
   | inr (q1, q2) => e.π1 + weight q <= weight q1 - K + weight q2
   end


DEBUG:Going to execute:
destruct (Edge.eq_dec (y_0, - K, x_0) (x, e.π1, y)) as [XX| XX]

Goals:
  
  G : t
  HI : invariants G
  HG : acyclic_no_loop G
  y_0 : V.t
  x_0 : V.t
  K : Z
  Hxs : lsp G x_0 y_0 = Some K
  x : V.t
  y : V.t
  z : V.t
  e : EdgeOf G' x y
  q : PathOf G' y z
  IHq : match from_G'_path q with
        | inl q' => weight q = weight q'
        | inr (q1, q2) => weight q <= weight q1 - K + weight q2
        end
  XX : (y_0, - K, x_0) = (x, e.π1, y)
  ============================
   match
     sumbool_rect
       (fun
          _ : {(y_0, - K, x_0) = (x, e.π1, y)} +
              {(y_0, - K, x_0) <> (x, e.π1, y)} =>
        (PathOf G x z + (PathOf G x y_0 × PathOf G x_0 z))%type)
       (fun XX : (y_0, - K, x_0) = (x, e.π1, y) =>
        inr
          (eq_rect_r
             (fun _pattern_value_ : V.t => PathOf G x _pattern_value_)
             (pathOf_refl G x) (fst_eq (fst_eq XX)),
           match from_G'_path q with
           | inl p =>
               eq_rect_r
                 (fun _pattern_value_ : V.t => PathOf G _pattern_value_ z) p
                 (snd_eq XX)
           | inr p => p.2
           end))
       (fun XX : (y_0, - K, x_0) <> (x, e.π1, y) =>
        sum_rect
          (fun _ : PathOf G y z + (PathOf G y y_0 × PathOf G x_0 z) =>
           (PathOf G x z + (PathOf G x y_0 × PathOf G x_0 z))%type)
          (fun IHq : PathOf G y z =>
           inl (pathOf_step G x y z (e.π1; EdgeSetFact.add_3 XX e.π2) IHq))
          (fun IHq : PathOf G y y_0 × PathOf G x_0 z =>
           inr
             (pathOf_step G x y y_0 (e.π1; EdgeSetFact.add_3 XX e.π2) IHq.1,
              IHq.2)) (from_G'_path q)) (left XX)
   with
   | inl q' => e.π1 + weight q = weight q'
   | inr (q1, q2) => e.π1 + weight q <= weight q1 - K + weight q2
   end

  
  G : t
  HI : invariants G
  HG : acyclic_no_loop G
  y_0 : V.t
  x_0 : V.t
  K : Z
  Hxs : lsp G x_0 y_0 = Some K
  x : V.t
  y : V.t
  z : V.t
  e : EdgeOf G' x y
  q : PathOf G' y z
  IHq : match from_G'_path q with
        | inl q' => weight q = weight q'
        | inr (q1, q2) => weight q <= weight q1 - K + weight q2
        end
  XX : (y_0, - K, x_0) <> (x, e.π1, y)
  ============================
   match
     sumbool_rect
       (fun
          _ : {(y_0, - K, x_0) = (x, e.π1, y)} +
              {(y_0, - K, x_0) <> (x, e.π1, y)} =>
        (PathOf G x z + (PathOf G x y_0 × PathOf G x_0 z))%type)
       (fun XX : (y_0, - K, x_0) = (x, e.π1, y) =>
        inr
          (eq_rect_r
             (fun _pattern_value_ : V.t => PathOf G x _pattern_value_)
             (pathOf_refl G x) (fst_eq (fst_eq XX)),
           match from_G'_path q with
           | inl p =>
               eq_rect_r
                 (fun _pattern_value_ : V.t => PathOf G _pattern_value_ z) p
                 (snd_eq XX)
           | inr p => p.2
           end))
       (fun XX : (y_0, - K, x_0) <> (x, e.π1, y) =>
        sum_rect
          (fun _ : PathOf G y z + (PathOf G y y_0 × PathOf G x_0 z) =>
           (PathOf G x z + (PathOf G x y_0 × PathOf G x_0 z))%type)
          (fun IHq : PathOf G y z =>
           inl (pathOf_step G x y z (e.π1; EdgeSetFact.add_3 XX e.π2) IHq))
          (fun IHq : PathOf G y y_0 × PathOf G x_0 z =>
           inr
             (pathOf_step G x y y_0 (e.π1; EdgeSetFact.add_3 XX e.π2) IHq.1,
              IHq.2)) (from_G'_path q)) (right XX)
   with
   | inl q' => e.π1 + weight q = weight q'
   | inr (q1, q2) => e.π1 + weight q <= weight q1 - K + weight q2
   end


DEBUG:Going to execute:
simpl
DEBUG LTAC Evaluated term: (Edge.eq_dec (y_0, - K, x_0) (x, e.π1, y))