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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.
Proof with trivial.
intros; unfold Rsqr_Rmin in |- *; case (Req_EM_var (Ax2_R intruder) 0); intro...
elim H; apply div_tau_0_Ax2_R...
rewrite Rsqr_minus; rewrite (tmin_0 intruder evader H); rewrite Rsqr_div...
unfold Rdiv in |- *; rewrite Rsqr_R; rewrite <- Rsqr_neg; rewrite Rsqr_mult;
replace (Rsqr 2) with 4%R...
rewrite (Rinv_mult_distr 2 (Ax2_R intruder))...
rewrite (Rinv_mult_distr 4 (Ax2_R intruder))...
rewrite Rinv_mult_distr...
unfold Rminus in |- *; do 2 rewrite Rmult_plus_distr_l...
rewrite <-
(Rmult_comm (Rsqr (Bx_R intruder evader) * (/ 4 * / Rsqr (Ax2_R intruder))))
; unfold Rsqr in |- *;
rewrite (Rinv_mult_distr (Ax2_R intruder) (Ax2_R intruder))...
repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym...
rewrite Rmult_1_r...
repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l...
rewrite
(Rmult_plus_distr_r (4 * (Ax2_R intruder * C_R intruder evader))
(- (Bx_R intruder evader * Bx_R intruder evader))
(/ 4 * / Ax2_R intruder))...
replace
(Ax2_R intruder *
- (2 * (t * (- Bx_R intruder evader * (/ 2 * / Ax2_R intruder)))))%R with
(Bx_R intruder evader * t)%R...
replace
(4 * (Ax2_R intruder * C_R intruder evader) * (/ 4 * / Ax2_R intruder))%R
with (C_R intruder evader)...
ring...
rewrite (Rmult_comm (/ 4))...
repeat rewrite Rmult_assoc...
rewrite (Rmult_comm 4)...
repeat rewrite Rmult_assoc...
rewrite <- Rinv_l_sym...
rewrite Rmult_1_r; rewrite (Rmult_comm (C_R intruder evader));
repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym...
symmetry  in |- *; apply Rmult_1_l...
discrR...
rewrite (Rmult_comm (Ax2_R intruder));
replace
(- (2 * (t * (- Bx_R intruder evader * (/ 2 * / Ax2_R intruder)))))%R with
(-2 * t * - Bx_R intruder evader * / 2 * / Ax2_R intruder)%R...
repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym...
rewrite Rmult_1_r; repeat rewrite <- Rmult_assoc;
replace (-2 * t * - Bx_R intruder evader)%R with
(t * Bx_R intruder evader * 2)%R...
repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym...
ring...
discrR...
ring...
ring...
discrR...
unfold Rsqr in |- *; apply prod_neq_R0...
discrR...
discrR...
apply prod_neq_R0...
Qed.
intros; unfold dxdt, dx in |- *; ring.
Qed.
intros; unfold dydt, dy in |- *; ring.
Qed.
intros; unfold RR in |- *; unfold div_tau in H;
generalize
(Rplus_eq_R0 (Rsqr (dxdt intruder)) (Rsqr (dydt intruder))
(Rle_0_sqr (dxdt intruder)) (Rle_0_sqr (dydt intruder)) H);
intro; elim H0; intros; generalize (Rsqr_eq_0 (dxdt intruder) H1);
intro; generalize (Rsqr_eq_0 (dydt intruder) H2);
intro; rewrite <- (dx_dxdt intruder evader t1);
rewrite <- (dx_dxdt intruder evader t2);
rewrite <- (dy_dydt intruder evader t1);
rewrite <- (dy_dydt intruder evader t2); rewrite H3;
rewrite H4; repeat rewrite Rmult_0_r; repeat rewrite Rplus_0_r;
reflexivity.
Qed.
intros; unfold RR in |- *; apply sqrt_positivity; apply Rplus_le_le_0_compat;
apply Rle_0_sqr.
Qed.
intros; case (Req_EM_var (div_tau intruder evader) 0); intro.
right; apply (R_equal_when_zero intruder evader (tmin intruder evader) t e).
apply Rsqr_incr_0.
repeat rewrite R_Rmin.
apply Rplus_le_compat_r; apply Rmult_le_compat_l.
apply Ax2_R_pos.
unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rsqr_0; apply Rle_0_sqr.
assumption.
assumption.
apply RR_pos.
apply RR_pos.
Qed.
intros; case (Req_EM_var (div_tau intruder evader) 0); intro;
[ right; apply R_equal_when_zero; assumption
| rewrite (tau_tmin intruder evader t1 n);
replace (t1 + (tmin intruder evader - t1))%R with (tmin intruder evader);
[ apply derivative_eq_zero_tmin | ring ] ].
Qed.
intros; case (Req_EM_var (div_tau intruder evader) 0); intro.
right; apply (R_equal_when_zero intruder evader t2 t1 e).
apply Rsqr_incr_0.
repeat rewrite R_Rmin.
apply Rplus_le_compat_r; apply Rmult_le_compat_l.
apply Ax2_R_pos.
case (Rtotal_order (tmin intruder evader) t1); intro.
generalize (Rle_lt_trans t2 (tmin intruder evader) t1 H H1); intro;
elim (Rlt_irrefl t1 (Rle_lt_trans t1 t2 t1 H0 H2)).
elim H1; intros.
rewrite H2 in H; generalize (Rle_antisym t1 t2 H0 H); intro; rewrite H3;
right; reflexivity.
rewrite Rsqr_neg; rewrite (Rsqr_neg (t1 - tmin intruder evader));
replace (- (t2 - tmin intruder evader))%R with (tmin intruder evader - t2)%R.
replace (- (t1 - tmin intruder evader))%R with (tmin intruder evader - t1)%R.
apply Rsqr_incr_1.
unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_ge_le_contravar;
apply Rle_ge; assumption.
apply Rplus_le_reg_l with t2; rewrite Rplus_0_r; rewrite Rplus_comm;
unfold Rminus in |- *; repeat rewrite Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_r; assumption.
apply Rplus_le_reg_l with t1; rewrite Rplus_0_r; rewrite Rplus_comm;
unfold Rminus in |- *; repeat rewrite Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_r; apply Rle_trans with t2;
assumption.
ring.
ring.
assumption.
assumption.
apply RR_pos.
apply RR_pos.
Qed.
intros; case (Req_EM_var (div_tau intruder evader) 0); intro;
[ right; apply R_equal_when_zero; assumption
| rewrite (tau_tmin intruder evader t n) in H;
apply asymptotic_decrease_tmin;
[ apply Rplus_le_reg_l with (- t)%R; repeat rewrite <- Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_l; rewrite Rplus_comm;
assumption
| apply (Rplus_le_compat_l t t1 t2 H0) ] ].
Qed.

Lemma asymptotic_increase_tmin :
forall (intruder evader : State) (t1 t2 : R),
(tmin intruder evader <= t1)%R ->
(t1 <= t2)%R -> (RR intruder evader t1 <= RR intruder evader t2)%R.