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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.

(**********)
Lemma Ax2_R_pos : forall intruder : State, (0 <= Ax2_R intruder)%R.