intros; unfold tmin in |- *; case (Req_EM_var (div_tau intruder evader) 0);
intro.
elim H; assumption.
unfold div_tau, Bx_R, Ax2_R in |- *; unfold dxdt, dydt in |- *;
unfold Rdiv in |- *; rewrite Rinv_mult_distr.
replace
(Rsqr (sind (heading intruder) * intruderSpeed) +
Rsqr (- evaderSpeed + cosd (heading intruder) * intruderSpeed))%R with
(Rsqr (intruderSpeed * cosd (heading intruder) - evaderSpeed) +
Rsqr (intruderSpeed * sind (heading intruder)))%R.
apply
Rmult_eq_reg_l
with
(Rsqr (intruderSpeed * cosd (heading intruder) - evaderSpeed) +
Rsqr (intruderSpeed * sind (heading intruder)))%R.
rewrite
(Rmult_comm
(Rsqr (intruderSpeed * cosd (heading intruder) - evaderSpeed) +
Rsqr (intruderSpeed * sind (heading intruder))))
; repeat rewrite Rmult_assoc;
rewrite <-
(Rinv_l_sym
(Rsqr (intruderSpeed * cosd (heading intruder) - evaderSpeed) +
Rsqr (intruderSpeed * sind (heading intruder))))
.
rewrite Rmult_1_r;
rewrite
(Rmult_comm
(Rsqr (intruderSpeed * cosd (heading intruder) - evaderSpeed) +
Rsqr (intruderSpeed * sind (heading intruder))))
; repeat rewrite Rmult_assoc;
rewrite <-
(Rinv_l_sym
(Rsqr (intruderSpeed * cosd (heading intruder) - evaderSpeed) +
Rsqr (intruderSpeed * sind (heading intruder))))
.
rewrite Rmult_1_r;
replace
(-
(-2 * (yt evader * (sind (heading intruder) * intruderSpeed)) +
2 * (yt intruder * (sind (heading intruder) * intruderSpeed)) -
2 * (cosd (heading intruder) * (xt evader * intruderSpeed)) +
2 * (cosd (heading intruder) * (xt intruder * intruderSpeed)) +
2 * (xt evader * evaderSpeed) - 2 * (xt intruder * evaderSpeed)))%R with
((yt evader * (sind (heading intruder) * intruderSpeed) -
yt intruder * (sind (heading intruder) * intruderSpeed) +
cosd (heading intruder) * (xt evader * intruderSpeed) -
cosd (heading intruder) * (xt intruder * intruderSpeed) -
xt evader * evaderSpeed + xt intruder * evaderSpeed) * 2)%R.
repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; ring.
discrR.
ring.
unfold div_tau in H; unfold dxdt, dydt in H; assumption.
assumption.
assumption.
cut
((intruderSpeed * cosd (heading intruder) - evaderSpeed)%R =
(- evaderSpeed + cosd (heading intruder) * intruderSpeed)%R).
cut
((intruderSpeed * sind (heading intruder))%R =
(sind (heading intruder) * intruderSpeed)%R).
intros; rewrite H0; rewrite H1; rewrite Rplus_comm; reflexivity.
ring.
ring.
discrR.
cut
((intruderSpeed * cosd (heading intruder) - evaderSpeed)%R =
(- evaderSpeed + cosd (heading intruder) * intruderSpeed)%R).
cut
((intruderSpeed * sind (heading intruder))%R =
(sind (heading intruder) * intruderSpeed)%R).
intros; rewrite <- H0; rewrite <- H1; rewrite Rplus_comm; assumption.
ring.
ring.
Qed.

(**********)
Lemma Ax2_R_0_div_tau :
forall intruder evader : State,
div_tau intruder evader = 0%R -> Ax2_R intruder = 0%R.