Goal:
  
  G1 : t
  G2 : t
  on_edge : forall x y : V.t, EdgeOf G1 x y -> EdgeOf G2 x y
  weight_on_edge : forall (x y : V.t) (e : EdgeOf G1 x y),
                   (on_edge x y e).π1 <= e.π1
  x : V.t
  y : V.t
  p : PathOf G1 x y
  ============================
   (weight (map_path p) <= weight p)


DEBUG:Going to execute:
<coq-equations.plugin::refine_ho@0> $1
DEBUG LTAC Evaluated term: (fun_elim ?P ?f ?f0 x y p)

Goals:
  
  G1 : t
  G2 : t
  on_edge : forall x y : V.t, EdgeOf G1 x y -> EdgeOf G2 x y
  weight_on_edge : forall (x y : V.t) (e : EdgeOf G1 x y),
                   (on_edge x y e).π1 <= e.π1
  x : V.t
  y : V.t
  p : PathOf G1 x y
  ============================
   (forall x : V.t,
    (fun (x0 y0 : V.t) (p0 : PathOf G1 x0 y0) (x1 : PathOf G2 x0 y0) =>
     weight x1 <= weight p0) x x (pathOf_refl G1 x) 
      (pathOf_refl G2 x))

  
  G1 : t
  G2 : t
  on_edge : forall x y : V.t, EdgeOf G1 x y -> EdgeOf G2 x y
  weight_on_edge : forall (x y : V.t) (e : EdgeOf G1 x y),
                   (on_edge x y e).π1 <= e.π1
  x : V.t
  y : V.t
  p : PathOf G1 x y
  ============================
   (forall (x y y' : V.t) (e : EdgeOf G1 x y') (p : PathOf G1 y' y),
    (fun (x0 y0 : V.t) (p0 : PathOf G1 x0 y0) (x1 : PathOf G2 x0 y0) =>
     weight x1 <= weight p0) y' y p (map_path p) ->
    (fun (x0 y0 : V.t) (p0 : PathOf G1 x0 y0) (x1 : PathOf G2 x0 y0) =>
     weight x1 <= weight p0) x y (pathOf_step G1 x y' y e p)
      (pathOf_step G2 x y' y (on_edge x y' e) (map_path p)))


DEBUG:Going to execute:
cbv beta

Goals:
  
  G1 : t
  G2 : t
  on_edge : forall x y : V.t, EdgeOf G1 x y -> EdgeOf G2 x y
  weight_on_edge : forall (x y : V.t) (e : EdgeOf G1 x y),
                   (on_edge x y e).π1 <= e.π1
  x : V.t
  y : V.t
  p : PathOf G1 x y
  ============================
   (forall x : V.t, weight (pathOf_refl G2 x) <= weight (pathOf_refl G1 x))

  
  G1 : t
  G2 : t
  on_edge : forall x y : V.t, EdgeOf G1 x y -> EdgeOf G2 x y
  weight_on_edge : forall (x y : V.t) (e : EdgeOf G1 x y),
                   (on_edge x y e).π1 <= e.π1
  x : V.t
  y : V.t
  p : PathOf G1 x y
  ============================
   (forall (x y y' : V.t) (e : EdgeOf G1 x y') (p : PathOf G1 y' y),
    weight (map_path p) <= weight p ->
    weight (pathOf_step G2 x y' y (on_edge x y' e) (map_path p)) <=
    weight (pathOf_step G1 x y' y e p))


DEBUG:Going to execute:
simp map_path
Level 6: evaluation returns
idtac of type tacvalue

Level 6: evaluation returns
idtac of type tacvalue

Level 1: evaluation returns
idtac of type tacvalue

Level 1: evaluation returns
idtac of type tacvalue