intros; unfold RR in |- *; unfold dx, dy in |- *; rewrite Rsqr_sqrt;
[ replace
(xt intruder + t * intruderSpeed * cosd (heading intruder) -
(xt evader + t * evaderSpeed))%R with
((- evaderSpeed + cosd (heading intruder) * intruderSpeed) * t +
(xt intruder - xt evader))%R;
[ replace
(yt intruder + t * intruderSpeed * sind (heading intruder) - yt evader)%R
with
(sind (heading intruder) * intruderSpeed * t +
(yt intruder - yt evader))%R;
[ repeat rewrite Rsqr_plus; repeat rewrite Rsqr_mult;
unfold Ax2_R in |- *;
apply
Rplus_eq_reg_l
with
(-
(Rsqr (sind (heading intruder) * intruderSpeed) +
Rsqr
(- evaderSpeed + cosd (heading intruder) * intruderSpeed)) *
Rsqr t)%R;
replace
(-
(Rsqr (sind (heading intruder) * intruderSpeed) +
Rsqr (- evaderSpeed + cosd (heading intruder) * intruderSpeed)) *
Rsqr t +
(Rsqr (- evaderSpeed + cosd (heading intruder) * intruderSpeed) *
Rsqr t + Rsqr (xt intruder - xt evader) +
2 *
((- evaderSpeed + cosd (heading intruder) * intruderSpeed) * t) *
(xt intruder - xt evader) +
(Rsqr (sind (heading intruder)) * Rsqr intruderSpeed * Rsqr t +
Rsqr (yt intruder - yt evader) +
2 * (sind (heading intruder) * intruderSpeed * t) *
(yt intruder - yt evader))))%R with
(Rsqr (xt intruder - xt evader) +
2 *
((- evaderSpeed + cosd (heading intruder) * intruderSpeed) * t) *
(xt intruder - xt evader) + Rsqr (yt intruder - yt evader) +
2 * (sind (heading intruder) * intruderSpeed * t) *
(yt intruder - yt evader))%R;
[ replace
(-
(Rsqr (sind (heading intruder) * intruderSpeed) +
Rsqr (- evaderSpeed + cosd (heading intruder) * intruderSpeed)) *
Rsqr t +
((Rsqr (sind (heading intruder) * intruderSpeed) +
Rsqr
(- evaderSpeed + cosd (heading intruder) * intruderSpeed)) *
Rsqr t + Bx_R intruder evader * t +
C_R intruder evader))%R with
(Bx_R intruder evader * t + C_R intruder evader)%R;
[ unfold C_R in |- *;
apply
Rplus_eq_reg_l
with
(-
(Rsqr (yt intruder - yt evader) +
Rsqr (xt intruder - xt evader)))%R;
replace
(-
(Rsqr (yt intruder - yt evader) +
Rsqr (xt intruder - xt evader)) +
(Rsqr (xt intruder - xt evader) +
2 *
((- evaderSpeed + cosd (heading intruder) * intruderSpeed) *
t) * (xt intruder - xt evader) +
Rsqr (yt intruder - yt evader) +
2 * (sind (heading intruder) * intruderSpeed * t) *
(yt intruder - yt evader)))%R with
(2 *
((- evaderSpeed + cosd (heading intruder) * intruderSpeed) *
t) * (xt intruder - xt evader) +
2 * (sind (heading intruder) * intruderSpeed * t) *
(yt intruder - yt evader))%R;
[ replace
(-
(Rsqr (yt intruder - yt evader) +
Rsqr (xt intruder - xt evader)) +
(Bx_R intruder evader * t +
(Rsqr (yt intruder - yt evader) +
Rsqr (xt intruder - xt evader))))%R with
(Bx_R intruder evader * t)%R;
[ unfold Bx_R in |- *; ring | ring ]
| ring ]
| ring ]
| rewrite (Rsqr_mult (sind (heading intruder)) intruderSpeed); ring ]
| ring ]
| ring ]
| apply Rplus_le_le_0_compat; apply Rle_0_sqr ].
Qed.
(**********)
Definition Rsqr_Rmin (intruder evader : State) : R :=
match Req_EM_var (Ax2_R intruder) 0 with
| left _ => 0%R
| right _ =>
((4 * Ax2_R intruder * C_R intruder evader -
Rsqr (Bx_R intruder evader)) / (4 * Ax2_R intruder))%R
end.

(**********)
Lemma tmin_0 :
forall intruder evader : State,
div_tau intruder evader <> 0%R ->
tmin intruder evader = (- Bx_R intruder evader / (2 * Ax2_R intruder))%R.