Section trajectory.
Unset Standard Proposition Elimination Names.
Variable intr : Trajectory.
Variable evad : EvaderTrajectory.
Variable T : TimeT.
unfold thetat in |- *; apply (cond_diff (theta intr)).
Qed.
Definition xe : R := x (tr evad) T.
Definition ye : R := y (tr evad) T.
Definition ve : R := v (h (tr evad)).
Definition r_vi : R := r (vi intr).
Definition rho_vi : R := rho (vi intr).
Definition Die (ti te : R) : R :=
sqrt
(Rsqr (xi intr ti - x (tr evad) te) + Rsqr (yi intr ti - y (tr evad) te)).
Definition conflict (t : R) : bool :=
match Rle_dec (Die t t) ConflictRange with
| left _ => true
| right _ => false
end.
intro; unfold xi, vi, thetat in |- *.
assert (H := cond_x intr t).
rewrite <- H.
apply pr_nu.
Qed.
intro; unfold yi, vi, thetat in |- *.
assert (H := cond_y intr t).
rewrite <- H.
apply pr_nu.
Qed.
Definition l : R := Die 0 T.
Definition d : R := Die 0 0.
Axiom
beta_exists :
{ beta : R |
xe = (l * cos beta + xi intr 0)%R /\
ye = (yi intr 0 - l * sin beta)%R /\
(0 <= beta + thetat intr 0 < 2 * PI)%R }.
Definition beta := let (a,_) := beta_exists in a.
unfold beta in |- *; case beta_exists; trivial.
Qed.
Definition Omega (a : R) : bool :=
match Rle_dec (PI / 2) a with
| left _ =>
match Rle_dec a (3 * (PI / 2)) with
| left _ => true
| right _ => false
end
| right _ => false
end.
Proof with trivial.
generalize beta_def; intro; elim H...
Qed.
Proof with trivial.
generalize beta_def; intro; elim H; intros _ H0; elim H0...
Qed.
Proof with trivial.
unfold d in |- *; unfold Die in |- *; rewrite Rsqr_sqrt...
replace (xi intr 0 - x (tr evad) 0)%R with
(xi intr 0 - xe + (xe - x (tr evad) 0))%R...
replace (yi intr 0 - y (tr evad) 0)%R with
(yi intr 0 - ye + (ye - y (tr evad) 0))%R...
repeat rewrite Rsqr_plus; cut ((ye - y (tr evad) 0)%R = 0%R)...
intro; rewrite H; repeat rewrite Rmult_0_r; rewrite Rsqr_0;
repeat rewrite Rplus_0_r; cut ((xe - x (tr evad) 0)%R = (ve * T)%R)...
intro; rewrite H0;
replace
(Rsqr (xi intr 0 - xe) + Rsqr (ve * T) + 2 * (xi intr 0 - xe) * (ve * T) +
Rsqr (yi intr 0 - ye))%R with
(Rsqr (ve * T) + (Rsqr (xi intr 0 - xe) + Rsqr (yi intr 0 - ye)) +
2 * (xi intr 0 - xe) * (ve * T))%R...
cut (Rsqr l = (Rsqr (xi intr 0 - xe) + Rsqr (yi intr 0 - ye))%R)...
intro; rewrite <- H1; rewrite xe_0; ring...
unfold l in |- *; unfold Die in |- *; rewrite Rsqr_sqrt;
[ unfold xe, ye in |- * | apply Rplus_le_le_0_compat; apply Rle_0_sqr ]...
ring...
unfold xe, ve in |- *; rewrite (tr_cond1 evad); ring...
unfold ye, ve in |- *; rewrite (tr_cond2 evad); unfold Rminus in |- *;
rewrite Rplus_opp_r...
ring...
ring...
apply Rplus_le_le_0_compat; apply Rle_0_sqr...
Qed.
Definition yp (t : R) : R :=
(- sin (thetat intr 0) * (xi intr t - xe) +
cos (thetat intr 0) * (yi intr t - ye))%R.
Hint Resolve xi_derivable yi_derivable: diff.
unfold yp in |- *; set (thetat_p := thetat intr); set (xi_p := xi intr);
set (yi_p := yi intr); reg.
apply (yi_derivable intr).
apply (xi_derivable intr).
Qed.
Hint Resolve yp_derivable: diff.
Definition xp (t : R) : R :=
(cos (thetat intr 0) * (xi intr t - xe) +
sin (thetat intr 0) * (yi intr t - ye))%R.
unfold xp in |- *; set (thetat_p := thetat intr); set (xi_p := xi intr);
set (yi_p := yi intr); reg.
apply (yi_derivable intr).
apply (xi_derivable intr).
Qed.
Hint Resolve xp_derivable: diff.
Definition Rsqr_evader_distance (t : R) : R :=
(Rsqr (xi intr t - xe) + Rsqr (yi intr t - ye))%R.
unfold Rsqr_evader_distance in |- *.
set (xi_p := xi intr); set (yi_p := yi intr); reg.
apply (yi_derivable intr).
apply (xi_derivable intr).
Qed.
Hint Resolve Rsqr_evader_distance_derivable: diff.
Definition Rsqr_intruder_distance (t : R) : R :=
(Rsqr (xi intr t - xi intr 0) + Rsqr (yi intr t - yi intr 0))%R.
unfold Rsqr_intruder_distance in |- *; set (xi_p := xi intr);
set (yi_p := yi intr); reg.
apply (yi_derivable intr).
apply (xi_derivable intr).
Qed.
Hint Resolve Rsqr_intruder_distance_derivable: diff.
intro; unfold Rsqr_evader_distance in |- *; apply Rplus_le_le_0_compat;
apply Rle_0_sqr.
Qed.
intro; unfold Rsqr_intruder_distance in |- *; apply Rplus_le_le_0_compat;
apply Rle_0_sqr.
Qed.
Definition e (t : R) : R := sqrt (Rsqr_evader_distance t).
Definition i (t : R) : R := sqrt (Rsqr_intruder_distance t).
Proof with trivial.
unfold conflict in |- *; cut (e T = Die T T)...
intro; case (Rle_dec (Die T T) ConflictRange); intros;
[ rewrite H; exact r | elim diff_false_true ]...
Qed.
Proof with trivial.
intro; cut (e T = Die T T)...
intro; rewrite H0 in H; unfold conflict in |- *;
case (Rle_dec (Die T T) ConflictRange); intro...
elim n; assumption...
Qed.
rewrite xe_0; rewrite ye_0; unfold Rminus in |- *;
rewrite <- (Rplus_comm (- (l * sin beta))); repeat rewrite Rplus_assoc;
repeat rewrite Rplus_opp_r; repeat rewrite Rplus_0_r;
rewrite <- Rsqr_neg; repeat rewrite Rsqr_mult;
rewrite <- (Rmult_comm (Rsqr (sin beta))); rewrite sin2;
unfold Rminus in |- *; rewrite Rmult_plus_distr_r;
ring.
Qed.
intro; apply Rsqr_incr_0.
rewrite Rsqr_plus.
cut
(Rsqr l =
(Rsqr (e t) + Rsqr (i t) +
2 *
((xe - xi intr t) * (xi intr t - xi intr 0) +
(ye - yi intr t) * (yi intr t - yi intr 0)))%R).
intro.
rewrite H.
apply Rplus_le_compat_l.
repeat rewrite Rmult_assoc.
apply Rmult_le_compat_l.
left; prove_sup0.
unfold e, i in |- *.
unfold Rsqr_evader_distance, Rsqr_intruder_distance in |- *.
rewrite (Rsqr_neg (xi intr t - xe)).
rewrite (Rsqr_neg (yi intr t - ye)).
replace (- (xi intr t - xe))%R with (xe - xi intr t)%R.
replace (- (yi intr t - ye))%R with (ye - yi intr t)%R.
apply sqrt_cauchy.
ring.
ring.
rewrite L.
unfold e, i in |- *.
repeat rewrite Rsqr_sqrt.
unfold Rsqr_evader_distance, Rsqr_intruder_distance in |- *.
replace (xe - xi intr 0)%R with (xe - xi intr t + (xi intr t - xi intr 0))%R.
replace (ye - yi intr 0)%R with (ye - yi intr t + (yi intr t - yi intr 0))%R.
repeat rewrite Rsqr_plus.
rewrite (Rsqr_neg (xe - xi intr t)).
rewrite (Rsqr_neg (ye - yi intr t)).
replace (- (xe - xi intr t))%R with (xi intr t - xe)%R.
replace (- (ye - yi intr t))%R with (yi intr t - ye)%R.
ring.
ring.
ring.
ring.
ring.
apply Rsqr_intruder_distance_pos.
apply Rsqr_evader_distance_pos.
unfold l in |- *; unfold Die in |- *.
apply sqrt_positivity.
apply Rplus_le_le_0_compat; apply Rle_0_sqr.
unfold e, i in |- *; apply Rplus_le_le_0_compat; apply sqrt_positivity.
apply Rsqr_evader_distance_pos.
apply Rsqr_intruder_distance_pos.
Qed.
unfold l in |- *; unfold Die in |- *; apply sqrt_positivity;
apply Rplus_le_le_0_compat; apply Rle_0_sqr.
Qed.
intro.
apply Rsqr_incr_0.
rewrite Rsqr_plus.
cut
(Rsqr (e t) =
(Rsqr l + Rsqr (i t) +
2 *
((xi intr t - xi intr 0) * (xi intr 0 - xe) +
(yi intr t - yi intr 0) * (yi intr 0 - ye)))%R).
intro.
rewrite H.
apply Rplus_le_compat_l.
repeat rewrite Rmult_assoc.
apply Rmult_le_compat_l.
left; prove_sup0.
cut (l = sqrt (Rsqr (xe - xi intr 0) + Rsqr (ye - yi intr 0))).
intro.
rewrite H0.
unfold i in |- *.
unfold Rsqr_intruder_distance in |- *.
rewrite (Rsqr_neg (xe - xi intr 0)).
rewrite (Rsqr_neg (ye - yi intr 0)).
replace (- (xe - xi intr 0))%R with (xi intr 0 - xe)%R.
replace (- (ye - yi intr 0))%R with (yi intr 0 - ye)%R.
rewrite <-
(Rmult_comm
(sqrt (Rsqr (xi intr t - xi intr 0) + Rsqr (yi intr t - yi intr 0))))
.
apply sqrt_cauchy.
ring.
ring.
apply Rsqr_inj.
apply l_is_pos.
apply sqrt_positivity; apply Rplus_le_le_0_compat; apply Rle_0_sqr.
rewrite Rsqr_sqrt.
apply L.
apply Rplus_le_le_0_compat; apply Rle_0_sqr.
rewrite L.
unfold e, i in |- *.
repeat rewrite Rsqr_sqrt.
unfold Rsqr_evader_distance, Rsqr_intruder_distance in |- *.
replace (xi intr t - xe)%R with (xi intr t - xi intr 0 + (xi intr 0 - xe))%R.
replace (yi intr t - ye)%R with (yi intr t - yi intr 0 + (yi intr 0 - ye))%R.
repeat rewrite Rsqr_plus.
rewrite (Rsqr_neg (xe - xi intr 0)).
rewrite (Rsqr_neg (ye - yi intr 0)).
replace (- (xe - xi intr 0))%R with (xi intr 0 - xe)%R.
replace (- (ye - yi intr 0))%R with (yi intr 0 - ye)%R.
ring.
ring.
ring.
ring.
ring.
apply Rsqr_intruder_distance_pos.
apply Rsqr_evader_distance_pos.
unfold e in |- *.
apply sqrt_positivity.
apply Rsqr_evader_distance_pos.
apply Rplus_le_le_0_compat.
apply l_is_pos.
unfold i in |- *.
apply sqrt_positivity.
apply Rsqr_intruder_distance_pos.
Qed.
intro.
apply Rsqr_incr_0.
rewrite Rsqr_plus.
cut
(Rsqr (i t) =
(Rsqr l + Rsqr (e t) +
2 *
((xi intr t - xe) * (xe - xi intr 0) + (yi intr t - ye) * (ye - yi intr 0)))%R).
intro.
rewrite H.
apply Rplus_le_compat_l.
repeat rewrite Rmult_assoc.
apply Rmult_le_compat_l.
left; prove_sup0.
cut (l = sqrt (Rsqr (xe - xi intr 0) + Rsqr (ye - yi intr 0))).
intro.
rewrite H0.
unfold e in |- *.
unfold Rsqr_evader_distance in |- *.
rewrite <-
(Rmult_comm (sqrt (Rsqr (xi intr t - xe) + Rsqr (yi intr t - ye))))
.
apply sqrt_cauchy.
apply Rsqr_inj.
apply l_is_pos.
apply sqrt_positivity; apply Rplus_le_le_0_compat; apply Rle_0_sqr.
rewrite Rsqr_sqrt.
apply L.
apply Rplus_le_le_0_compat; apply Rle_0_sqr.
rewrite L.
unfold e, i in |- *.
repeat rewrite Rsqr_sqrt.
unfold Rsqr_evader_distance, Rsqr_intruder_distance in |- *.
replace (xi intr t - xi intr 0)%R with (xi intr t - xe + (xe - xi intr 0))%R.
replace (yi intr t - yi intr 0)%R with (yi intr t - ye + (ye - yi intr 0))%R.
repeat rewrite Rsqr_plus.
ring.
ring.
ring.
apply Rsqr_evader_distance_pos.
apply Rsqr_intruder_distance_pos.
unfold i in |- *.
apply sqrt_positivity.
apply Rsqr_intruder_distance_pos.
apply Rplus_le_le_0_compat.
apply l_is_pos.
unfold e in |- *.
apply sqrt_positivity.
apply Rsqr_evader_distance_pos.
Qed.
Proof with trivial.
generalize (dtheta_rho intr); intros; split...
unfold thetat in |- *; cut (derivable (fun y : R => (- rho_vi * y)%R));
[ intro X | reg ]...
cut
(forall z : R,
derive_pt (fun y : R => (- rho_vi * y)%R) z (X z) = (- rho_vi)%R);
[ intro | intro; reg ]...
apply Rplus_le_reg_l with (- theta intr 0)%R...
replace (- theta intr 0 + (theta intr 0 - rho_vi * t))%R with
(- rho_vi * t - - rho_vi * 0)%R; [ idtac | ring ]...
replace (- theta intr 0 + theta intr t)%R with
(theta intr t - theta intr 0)%R; [ idtac | ring ]...
apply
(IAF_var (theta intr) (fun y : R => (- rho_vi * y)%R) 0 t
(cond_diff (theta intr)) X H0)...
intros; rewrite (H1 c); unfold rho_vi in |- *; unfold vi in |- *; elim (H c);
intros H4 _; assumption...
unfold thetat in |- *; cut (derivable (fun y : R => (rho_vi * y)%R));
[ intro X | reg ]...
cut (forall z : R, derive_pt (fun y : R => (rho_vi * y)%R) z (X z) = rho_vi);
[ intro | intro; reg ]...
apply Rplus_le_reg_l with (- theta intr 0)%R...
replace (- theta intr 0 + (theta intr 0 + rho_vi * t))%R with
(rho_vi * t - rho_vi * 0)%R; [ idtac | ring ]...
replace (- theta intr 0 + theta intr t)%R with
(theta intr t - theta intr 0)%R; [ idtac | ring ]...
apply
(IAF_var (fun y : R => (rho_vi * y)%R) (theta intr) 0 t X
(cond_diff (theta intr)) H0)...
intros; rewrite (H1 c); unfold rho_vi in |- *; unfold vi in |- *; elim (H c)...
Qed.
Proof with trivial.
intros; apply Rmult_le_pos...
left; unfold rho_vi in |- *; apply rho_pos...
Qed.
Proof with trivial.
intros; apply Rmult_le_compat_l...
left; unfold rho_vi in |- *; apply rho_pos...
Qed.
intro; unfold yp in |- *.
assert (H := xi_derivable intr t).
assert (H0 := yi_derivable intr t).
assert (H1 := thetat_derivable t).
set (xi_p := xi intr) in H |- *.
set (yi_p := yi intr) in H0 |- *.
set (thetat_p := thetat intr) in H1 |- *.
reg.
assert (H2 := cond_x intr t).
assert (H3 := cond_y intr t).
replace (derive_pt xi_p t H) with
(derive_pt (x intr) t (cond_diff (x intr) t)); [ idtac | apply pr_nu ].
replace (derive_pt yi_p t H0) with
(derive_pt (y intr) t (cond_diff (y intr) t)); [ idtac | apply pr_nu ].
rewrite H2; rewrite H3.
rewrite sin_minus.
unfold fct_cte, vi, thetat_p, thetat in |- *.
ring.
Qed.
Proof with trivial.
intros...
rewrite Dyp...
generalize (theta_inv t H); intro; elim H1; intros...
cut (thetat intr t - thetat intr 0 <= rho_vi * t)%R...
cut (- (rho_vi * t) <= thetat intr t - thetat intr 0)%R...
intros...
split...
replace (- (vi intr * sin (rho_vi * t)))%R with
(vi intr * - sin (rho_vi * t))%R...
apply Rmult_le_compat_l...
left; apply TypeSpeed_pos...
rewrite <- (sin_neg (rho_vi * t))...
apply sin_incr_1...
apply Ropp_ge_le_contravar; apply Rle_ge...
apply Rle_trans with 0%R...
rewrite <- Ropp_0...
apply Ropp_ge_le_contravar; apply Rle_ge; apply (rho_t_pos t H)...
left; apply PI2_RGT_0...
apply Rle_trans with (- (rho_vi * t))%R...
apply Ropp_ge_le_contravar...
apply Rle_ge...
apply Rle_trans with (rho_vi * t)%R...
ring...
apply Rmult_le_compat_l...
left; apply TypeSpeed_pos...
apply sin_incr_1...
apply Rle_trans with (- (rho_vi * t))%R...
apply Ropp_ge_le_contravar; apply Rle_ge...
apply Rle_trans with (rho_vi * t)%R...
apply Rle_trans with 0%R...
rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; left;
apply PI2_RGT_0...
apply (rho_t_pos t H)...
apply Rplus_le_reg_l with (thetat intr 0)...
unfold Rminus in |- *...
rewrite (Rplus_comm (thetat intr t))...
repeat rewrite <- Rplus_assoc...
rewrite Rplus_opp_r...
rewrite Rplus_0_l...
apply Rplus_le_reg_l with (thetat intr 0)...
unfold Rminus in |- *...
rewrite (Rplus_comm (thetat intr t))...
repeat rewrite <- Rplus_assoc...
rewrite Rplus_opp_r...
rewrite Rplus_0_l...
Qed.
unfold yp in |- *; rewrite xe_0; rewrite ye_0;
rewrite <- (Rplus_comm (xi intr 0)); unfold Rminus in |- *;
repeat rewrite Ropp_plus_distr; repeat rewrite <- Rplus_assoc;
repeat rewrite Rplus_opp_r; repeat rewrite Rplus_0_l;
rewrite sin_plus; ring.
Qed.
Definition hy (t : R) : R := (r_vi * cos (rho_vi * t))%R.
unfold hy in |- *; reg.
Qed.
Proof with trivial.
intro; unfold hy in |- *; reg...
replace (vi intr:R) with (r_vi * rho_vi)%R.
ring.
unfold rho_vi, r_vi in |- *; unfold r in |- *.
field.
assert (H := rho_pos (vi intr)); red in |- *; intro; rewrite H0 in H;
elim (Rlt_irrefl _ H)...
Qed.
Proof with trivial.
intros; split...
apply (IAF_var yp hy 0 t yp_derivable hy_derivable H)...
intros; elim H1; intros...
generalize
(Dyp0_PI2 c H2
(Rle_trans (rho_vi * c) (rho_vi * t) (PI / 2) (rho_t_le c t H3) H0));
intro...
elim H4; intros...
cut (derive_pt hy c (hy_derivable c) = (- (vi intr * sin (rho_vi * c)))%R)...
intro...
rewrite <- H7 in H5...
rewrite Dhy; ring...
replace (hy 0 - hy t)%R with ((- hy)%F t - (- hy)%F 0)%R...
apply
(IAF_var (- hy)%F yp 0 t (derivable_opp hy hy_derivable) yp_derivable H)...
intros; elim H1; intros...
generalize
(Dyp0_PI2 c H2
(Rle_trans (rho_vi * c) (rho_vi * t) (PI / 2) (rho_t_le c t H3) H0));
intro...
elim H4; intros...
cut ((- derive_pt hy c (hy_derivable c))%R = (vi intr * sin (rho_vi * c))%R)...
intro; rewrite <- H7 in H6...
replace (derive_pt (- hy) c (derivable_opp hy hy_derivable c)) with
(derive_pt (- hy) c (derivable_pt_opp hy c (hy_derivable c)));
[ idtac | apply pr_nu ]...
rewrite (derive_pt_opp hy c (hy_derivable c))...
rewrite Dhy; ring...
unfold opp_fct in |- *; ring...
Qed.
intros; generalize (ypt t H H0); intro; elim H1; intros; split;
[ unfold hy in H2; unfold Rminus in H2; rewrite yp0 in H2;
rewrite Rmult_0_r in H2; rewrite cos_0 in H2; rewrite Rmult_1_r in H2;
generalize
(Rplus_le_compat_r (l * sin (thetat intr 0 + beta))
(r_vi * cos (rho_vi * t) + - r_vi)
(yp t + - (l * sin (thetat intr 0 + beta))) H2);
replace (r_vi * cos (rho_vi * t) + - r_vi)%R with
(r_vi * (cos (rho_vi * t) - 1))%R;
[ repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
intro; rewrite <- (Rplus_comm (r_vi * (cos (rho_vi * t) - 1)));
rewrite (Rplus_comm beta); assumption
| ring ]
| unfold hy in H3; unfold Rminus in H3; rewrite yp0 in H3;
rewrite Rmult_0_r in H3; rewrite cos_0 in H3; rewrite Rmult_1_r in H3;
generalize
(Rplus_le_compat_r (l * sin (thetat intr 0 + beta))
(yp t + - (l * sin (thetat intr 0 + beta)))
(r_vi + - (r_vi * cos (rho_vi * t))) H3);
replace (r_vi + - (r_vi * cos (rho_vi * t)))%R with
(- (r_vi * (cos (rho_vi * t) - 1)))%R;
[ repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
intro; unfold Rminus in |- *; rewrite (Rplus_comm beta);
rewrite <- (Rplus_comm (- (r_vi * (cos (rho_vi * t) + -1))));
assumption
| unfold Rminus in |- *; rewrite Rmult_plus_distr_l; ring ] ].
Qed.
intro; unfold xp in |- *.
assert (H := xi_derivable intr t).
assert (H0 := yi_derivable intr t).
assert (H1 := thetat_derivable t).
set (xi_p := xi intr) in H |- *.
set (yi_p := yi intr) in H0 |- *.
set (thetat_p := thetat intr) in H1 |- *.
reg.
assert (H2 := cond_x intr t).
assert (H3 := cond_y intr t).
replace (derive_pt yi_p t H0) with
(derive_pt (y intr) t (cond_diff (y intr) t)); [ idtac | apply pr_nu ].
replace (derive_pt xi_p t H) with
(derive_pt (x intr) t (cond_diff (x intr) t)); [ idtac | apply pr_nu ].
unfold vi in |- *.
rewrite H2; rewrite H3; rewrite cos_minus; unfold thetat_p, thetat; ring.
Qed.
intros; rewrite Dxp; apply Rmult_le_compat_l;
[ left; apply TypeSpeed_pos
| generalize (theta_inv t H); intro; elim H1; intros;
case (Rcase_abs (thetat intr t - thetat intr 0));
intro;
[ cut (thetat intr 0 - thetat intr t <= rho_vi * t)%R;
[ cut (- (rho_vi * t) <= thetat intr 0 - thetat intr t)%R;
[ intros; rewrite <- (cos_neg (thetat intr t - thetat intr 0));
replace (- (thetat intr t - thetat intr 0))%R with
(thetat intr 0 - thetat intr t)%R;
[ apply cos_decr_1;
[ left;
generalize
(Ropp_lt_gt_contravar (thetat intr t - thetat intr 0) 0 r);
rewrite Ropp_0;
replace (- (thetat intr t - thetat intr 0))%R with
(thetat intr 0 - thetat intr t)%R;
[ intro; assumption | ring ]
| apply Rle_trans with (rho_vi * t)%R; assumption
| apply (rho_t_pos t H)
| assumption
| assumption ]
| ring ]
| apply Rplus_le_reg_l with (thetat intr t + rho_vi * t)%R;
repeat rewrite Rplus_assoc; rewrite Rplus_opp_r;
rewrite Rplus_0_r;
replace
(thetat intr t + (rho_vi * t + (thetat intr 0 - thetat intr t)))%R
with (thetat intr 0 + rho_vi * t)%R;
[ assumption | ring ] ]
| apply Rplus_le_reg_l with (thetat intr t - rho_vi * t)%R;
unfold Rminus in |- *; repeat rewrite Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_r; rewrite Rplus_comm;
repeat rewrite Rplus_assoc; rewrite Rplus_opp_l;
rewrite Rplus_0_r; rewrite Rplus_comm; assumption ]
| generalize (Rge_le (thetat intr t - thetat intr 0) 0 r); intro;
cut (thetat intr t - thetat intr 0 <= rho_vi * t)%R;
[ cut (- (rho_vi * t) <= thetat intr t - thetat intr 0)%R;
[ intros; apply cos_decr_1;
[ assumption
| apply Rle_trans with (rho_vi * t)%R; assumption
| apply (rho_t_pos t H)
| assumption
| assumption ]
| apply Rplus_le_reg_l with (thetat intr 0); unfold Rminus in |- *;
rewrite (Rplus_comm (thetat intr t));
repeat rewrite <- Rplus_assoc; rewrite Rplus_opp_r;
rewrite Rplus_0_l; assumption ]
| apply Rplus_le_reg_l with (thetat intr 0); unfold Rminus in |- *;
rewrite (Rplus_comm (thetat intr t)); repeat rewrite <- Rplus_assoc;
rewrite Rplus_opp_r; rewrite Rplus_0_l; assumption ] ] ].
Qed.
unfold xp in |- *; rewrite xe_0; rewrite ye_0;
rewrite <- (Rplus_comm (xi intr 0)); unfold Rminus in |- *;
repeat rewrite Ropp_plus_distr; repeat rewrite <- Rplus_assoc;
repeat rewrite Rplus_opp_r; repeat rewrite Rplus_0_l;
rewrite cos_plus; ring.
Qed.
Definition hx (t : R) : R := (r_vi * sin (rho_vi * t))%R.
unfold hx in |- *; reg.
Qed.
Proof with trivial.
intro; unfold hx in |- *; reg...
replace (vi intr:R) with (r_vi * rho_vi)%R.
ring.
unfold rho_vi, r_vi in |- *; unfold r in |- *.
field.
assert (H := rho_pos (vi intr)); red in |- *; intro; rewrite H0 in H;
elim (Rlt_irrefl _ H)...
Qed.
intros; replace (hx t) with (hx t - hx 0)%R.
apply (IAF_var xp hx 0 t xp_derivable hx_derivable H); intros; rewrite Dhx;
elim H1; intros;
apply
(Dx0_PI c H2 (Rle_trans (rho_vi * c) (rho_vi * t) PI (rho_t_le c t H3) H0)).
unfold hx in |- *; rewrite Rmult_0_r; rewrite sin_0; unfold Rminus in |- *;
rewrite Rmult_0_r; rewrite Ropp_0; apply Rplus_0_r.
Qed.
intros; apply Rmult_le_pos;
[ left; unfold r_vi in |- *; unfold r in |- *; unfold Rdiv in |- *;
apply Rmult_lt_0_compat;
[ apply (TypeSpeed_pos (vi intr))
| apply Rinv_0_lt_compat; apply (rho_pos (vi intr)) ]
| apply sin_ge_0; [ apply (rho_t_pos t H) | assumption ] ].
Qed.
intros; generalize (xpt t H H0); unfold hx in |- *; intro;
apply Rle_trans with (r_vi * sin (rho_vi * t))%R;
[ apply (r_sin_pos t H H0) | assumption ].
Qed.
intros; generalize (xpt t H H0); intro; unfold Rminus, hx in H1;
rewrite xp0 in H1;
generalize
(Rplus_le_compat_l (- l * cos (thetat intr 0 + beta))
(r_vi * sin (rho_vi * t)) (xp t + - (- l * cos (thetat intr 0 + beta)))
H1); rewrite <- (Rplus_comm (r_vi * sin (rho_vi * t)));
rewrite (Rplus_comm (- l * cos (thetat intr 0 + beta)));
repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
rewrite (Rplus_comm beta); unfold Rminus in |- *;
intro;
replace (- (l * cos (thetat intr 0 + beta)))%R with
(- l * cos (thetat intr 0 + beta))%R; [ assumption | ring ].
Qed.
Proof with trivial.
intro x; unfold Omega in |- *...
case (Rle_dec (PI / 2) x); intro...
case (Rle_dec x (3 * (PI / 2))); intros...
split...
elim diff_false_true...
intro; elim diff_false_true...
Qed.
intro; rewrite cos_minus; rewrite cos_PI; rewrite sin_PI; ring.
Qed.
Proof with trivial.
intro; generalize (Omega_defeq (beta + thetat intr 0) H); intro; rewrite xp0;
replace (- l * cos (thetat intr 0 + beta))%R with
(l * - cos (thetat intr 0 + beta))%R...
apply Rmult_le_pos...
apply l_is_pos...
rewrite <- neg_cos_var; unfold Rminus in |- *; elim H0; intros;
apply cos_ge_0...
apply Rplus_le_reg_l with PI; replace (PI + - (PI / 2))%R with (PI / 2)%R...
rewrite (Rplus_comm PI); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l;
rewrite Rplus_0_r; rewrite <- (Rplus_comm beta)...
pattern PI at 2 in |- *; rewrite double_var; ring...
apply Rplus_le_reg_l with PI; replace (PI + PI / 2)%R with (3 * (PI / 2))%R...
rewrite (Rplus_comm PI); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l;
rewrite Rplus_0_r; rewrite <- (Rplus_comm beta)...
pattern PI at 2 in |- *; rewrite double_var; ring...
ring...
Qed.
intros; generalize (xpt t H H0); intro; generalize (xp0_pos H1); intro;
unfold hx in H2;
generalize
(Rplus_le_compat_r (xp 0) (r_vi * sin (rho_vi * t)) (xp t - xp 0) H2);
unfold Rminus in |- *; repeat rewrite Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_r; intro;
apply (plus_le_is_le (r_vi * sin (rho_vi * t)) (xp 0) (xp t) H3 H4).
Qed.
intro; unfold Rsqr_evader_distance, xp, yp in |- *; repeat rewrite Rsqr_plus;
ring_simplify; repeat rewrite Rsqr_mult;
rewrite <- (Rsqr_neg (sin (thetat intr 0)));
pattern (Rsqr (xi intr t - xe)) at 1 in |- *.
replace (Rsqr (xi intr t - xe)) with
((Rsqr (sin (thetat intr 0)) + Rsqr (cos (thetat intr 0))) *
Rsqr (xi intr t - xe))%R.
pattern (Rsqr (yi intr t - ye)) at 1 in |- *.
replace (Rsqr (yi intr t - ye)) with
((Rsqr (sin (thetat intr 0)) + Rsqr (cos (thetat intr 0))) *
Rsqr (yi intr t - ye))%R.
ring.
rewrite sin2_cos2 in |- *;  ring.
rewrite sin2_cos2 in |- *;  ring.
Qed.
intro; unfold xp in |- *; ring.
Qed.
intro; unfold yp in |- *; ring.
Qed.
intro; unfold Rsqr_intruder_distance in |- *; rewrite xpt_xp0;
rewrite ypt_yp0; repeat rewrite Rsqr_plus; ring_simplify;
repeat rewrite Rsqr_mult;
rewrite <- Rsqr_neg.
pattern (Rsqr (xi intr t - xi intr 0)) at 1 in |- *.
replace (Rsqr (xi intr t - xi intr 0)) with
((Rsqr (sin (thetat intr 0)) + Rsqr (cos (thetat intr 0))) *
Rsqr (xi intr t - xi intr 0))%R.
pattern (Rsqr (yi intr t - yi intr 0)) at 1 in |- *.
replace (Rsqr (yi intr t - yi intr 0)) with
((Rsqr (sin (thetat intr 0)) + Rsqr (cos (thetat intr 0))) *
Rsqr (yi intr t - yi intr 0))%R.
ring.
rewrite sin2_cos2 in |- *;  ring.
rewrite sin2_cos2 in |- *;  ring.
Qed.
intro; generalize (isometric_evader t); intro; split;
[ rewrite H; pattern (Rsqr (yp t)) at 1 in |- *;
rewrite <- (Rplus_0_l (Rsqr (yp t))); apply Rplus_le_compat_r;
apply Rle_0_sqr
| rewrite H; pattern (Rsqr (xp t)) at 1 in |- *;
rewrite <- (Rplus_0_r (Rsqr (xp t))); apply Rplus_le_compat_l;
apply Rle_0_sqr ].
Qed.
intros; cut (i t <= vi intr * t)%R;
[ intros; unfold Rminus in |- *; rewrite Rplus_comm;
apply Rplus_le_reg_l with (vi intr * t)%R; repeat rewrite <- Rplus_assoc;
rewrite Rplus_opp_r; rewrite Rplus_0_l; rewrite Rplus_comm;
apply Rle_trans with (e t + i t)%R;
[ apply L_intruder | apply Rplus_le_compat_l; assumption ]
| unfold i in |- *; unfold Rsqr_intruder_distance in |- *;
apply (YCNGFTYS intr t H) ].
Qed.
intros; cut (2 * r_vi * sin (rho_vi * (t / 2)) <= i t)%R;
[ intro; intros; unfold Rminus in |- *; rewrite Rplus_comm;
apply Rplus_le_reg_l with l; repeat rewrite <- Rplus_assoc;
rewrite Rplus_opp_r; rewrite Rplus_0_l; apply Rle_trans with (i t);
[ assumption | apply L_i ]
| unfold i in |- *; unfold Rsqr_intruder_distance in |- *;
apply (YCNGSTYS intr t H H0) ].
Qed.

Lemma xp_pos :
forall t : R,
(0 <= t)%R ->
(rho_vi * t <= PI)%R -> Omega (beta + thetat intr 0) = true -> (0 <= xp t)%R.