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(* 02110-1301 USA                                                     *)
Unset Standard Proposition Elimination Names.
(**********)
Definition dx (intruder evader : State) (t : R) : R :=
(xt intruder + t * intruderSpeed * cosd (heading intruder) -
(xt evader + t * evaderSpeed))%R.
Definition dy (intruder evader : State) (t : R) : R :=
(yt intruder + t * intruderSpeed * sind (heading intruder) - yt evader)%R.
Definition dxdt (intruder : State) : R :=
(intruderSpeed * cosd (heading intruder) - evaderSpeed)%R.
Definition dydt (intruder : State) : R :=
(intruderSpeed * sind (heading intruder))%R.
Definition RR (intruder evader : State) (t : R) : R :=
sqrt (Rsqr (dx intruder evader t) + Rsqr (dy intruder evader t)).
Definition div_tau (intruder evader : State) : R :=
(Rsqr (dxdt intruder) + Rsqr (dydt intruder))%R.
intros; elim (total_order_T r1 r2); intros.
elim a; intro.
right; red in |- *; intro; rewrite H in a0; elim (Rlt_irrefl r2 a0).
left; assumption.
right; red in |- *; intro.
rewrite H in b; elim (Rlt_irrefl r2 b).
Qed.
(***********)
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.
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.
intro; unfold Ax2_R in |- *; apply Rplus_le_le_0_compat; apply Rle_0_sqr.
Qed.
(**********)
Definition Bx_R (intruder evader : State) : R :=
(-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.
Definition C_R (intruder evader : State) : R :=
(Rsqr (yt intruder - yt evader) + Rsqr (xt intruder - xt evader))%R.
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.
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.
unfold div_tau, Ax2_R in |- *; intros; unfold dxdt, dydt in H;
rewrite Rplus_comm;
replace (- evaderSpeed + cosd (heading intruder) * intruderSpeed)%R with
(intruderSpeed * cosd (heading intruder) - evaderSpeed)%R;
[ rewrite (Rmult_comm (sind (heading intruder))); assumption | ring ].
Qed.
unfold div_tau, Ax2_R in |- *; intros; unfold dxdt, dydt in |- *;
replace (intruderSpeed * cosd (heading intruder) - evaderSpeed)%R with
(- evaderSpeed + cosd (heading intruder) * intruderSpeed)%R;
[ rewrite <- (Rmult_comm (sind (heading intruder))); rewrite Rplus_comm;
assumption
| ring ].
Qed.

(**********)
Lemma R_Rmin :
forall (intruder evader : State) (t : R),
div_tau intruder evader <> 0%R ->
Rsqr (RR intruder evader t) =
(Ax2_R intruder * Rsqr (t - tmin intruder evader) +
Rsqr_Rmin intruder evader)%R.