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.

(* Specifications: *)
Lemma yp_derivable : derivable yp.
unfold yp in |- *; set (thetat_p := thetat intr); set (xi_p := xi intr);
set (yi_p := yi intr); reg.