(**********)
Definition init_tr (h : TypeSpeed) : Trajectory :=
  mkTrajectory
    (mkDifferential (fun t : R => (h * t)%R) (init_tr_derivable_x h))
    (mkDifferential (fun t : R => 0%R) (init_tr_derivable_y h))
    (mkDifferential (fun t : R => 0%R) (init_tr_derivable_y h))
    (mkDifferential (fun t : R => 0%R) (init_tr_derivable_y h)) h
    (init_tr_cond_x h) (init_tr_cond_y h) init_tr_cond_phi
    (init_tr_cond_theta h).

Lemma dtheta_rho : forall (tr : Trajectory) (t : R), (- rho (h tr) <= derive_pt (theta tr) t (cond_diff (theta tr) t) <= rho (h tr))%R.