(***********)
Definition tau (intruder evader : State) (t : R) : R :=
  match Req_EM_var (div_tau intruder evader) 0 with
  | left _ => 0%R
  | right _ =>
      (-
       (dx intruder evader t * dxdt intruder +
        dy intruder evader t * dydt intruder) / div_tau intruder evader)%R
  end.Definition tmin (intruder evader : State) : R :=
  match Req_EM_var (div_tau intruder evader) 0 with
  | left _ => 0%R
  | right _ =>
      (-
       (dxdt intruder * (xt intruder - xt evader) +
        dydt intruder * (yt intruder - yt evader)) / 
       div_tau intruder evader)%R
  end.

(* Specifications: *)
Lemma tau_tmin : forall (intruder evader : State) (t : R), div_tau intruder evader <> 0%R -> tau intruder evader t = (tmin intruder evader - t)%R.