intros.
cut
(tau intruder evader t =
(-
(dx intruder evader t * dxdt intruder +
dy intruder evader t * dydt intruder) / div_tau intruder evader)%R).
cut
(tmin intruder evader =
(-
(dxdt intruder * (xt intruder - xt evader) +
dydt intruder * (yt intruder - yt evader)) / div_tau intruder evader)%R).
intros.
rewrite H0; rewrite H1.
unfold Rdiv in |- *.
replace
(-
(dxdt intruder * (xt intruder - xt evader) +
dydt intruder * (yt intruder - yt evader)) * / div_tau intruder evader - t)%R
with
(-
(dxdt intruder * (xt intruder - xt evader + t * dxdt intruder) +
dydt intruder * (yt intruder - yt evader + t * dydt intruder)) /
div_tau intruder evader)%R.
cut ((xt intruder - xt evader + t * dxdt intruder)%R = dx intruder evader t).
cut ((yt intruder - yt evader + t * dydt intruder)%R = dy intruder evader t).
intros; rewrite H2; rewrite H3; unfold Rdiv in |- *;
rewrite (Rmult_comm (dx intruder evader t));
rewrite (Rmult_comm (dy intruder evader t)); reflexivity.
unfold dydt, dy in |- *; ring.
unfold dxdt, dx in |- *; ring.
unfold Rdiv in |- *; apply (Rmult_eq_reg_l (div_tau intruder evader)).
rewrite (Rmult_comm (div_tau intruder evader)); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; rewrite (Rmult_comm (div_tau intruder evader));
replace
((-
(dxdt intruder * (xt intruder - xt evader) +
dydt intruder * (yt intruder - yt evader)) * / div_tau intruder evader -
t) * div_tau intruder evader)%R with
(-
(dxdt intruder * (xt intruder - xt evader) +
dydt intruder * (yt intruder - yt evader)) * / div_tau intruder evader *
div_tau intruder evader - t * div_tau intruder evader)%R.
repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; unfold div_tau in |- *; unfold Rsqr in |- *; ring.
assumption.
symmetry  in |- *; unfold Rminus in |- *;
replace
(-
(dxdt intruder * (xt intruder + - xt evader) +
dydt intruder * (yt intruder + - yt evader)) *
/ div_tau intruder evader + - t)%R with
(-
(dxdt intruder * (xt intruder + - xt evader) +
dydt intruder * (yt intruder + - yt evader)) *
/ div_tau intruder evader + - t)%R.
replace (- (t * div_tau intruder evader))%R with
(- t * div_tau intruder evader)%R.
replace
(-
(dxdt intruder * (xt intruder + - xt evader) +
dydt intruder * (yt intruder + - yt evader)) * / div_tau intruder evader *
div_tau intruder evader)%R with
(-
(dxdt intruder * (xt intruder + - xt evader) +
dydt intruder * (yt intruder + - yt evader)) * / div_tau intruder evader *
div_tau intruder evader)%R.
apply Rmult_plus_distr_r.
ring.
ring.
ring.
assumption.
assumption.
unfold tmin in |- *; case (Req_EM_var (div_tau intruder evader) 0); intro.
rewrite e in H; elim H; reflexivity.
reflexivity.
unfold tau in |- *; case (Req_EM_var (div_tau intruder evader) 0); intro.
rewrite e in H; elim H; reflexivity.
reflexivity.
Qed.
(**********)
Definition Ax2_R (intruder : State) : R :=
(Rsqr (sind (heading intruder) * intruderSpeed) +
Rsqr (- evaderSpeed + cosd (heading intruder) * intruderSpeed))%R.

(**********)
Lemma Ax2_R_pos : forall intruder : State, (0 <= Ax2_R intruder)%R.
intro; unfold Ax2_R in |- *; apply Rplus_le_le_0_compat; apply Rle_0_sqr.