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.

Lemma xpt_xp0 :
forall t : R,
(xp t - xp 0)%R =
(cos (thetat intr 0) * (xi intr t - xi intr 0) +
sin (thetat intr 0) * (yi intr t - yi intr 0))%R.