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.

(* Specifications: *)
Lemma l_is_pos : (0 <= l)%R.
unfold l in |- *; unfold Die in |- *; apply sqrt_positivity;
apply Rplus_le_le_0_compat; apply Rle_0_sqr.