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.

Lemma L_i : forall t : R, (i t <= l + e t)%R.
intro.