Proof with trivial.
intros hyp_evad H...
cut (0 <= d intr evad)%R...
cut (0 <= RR (measure2state intr 0) (measure2state (tr evad) 0) T)%R...
intros...
generalize
(Rsqr_incr_1 _ (RR (measure2state intr 0) (measure2state (tr evad) 0) T) H2
H1 H0); intro...
rewrite R_T in H3...
rewrite (d_l_beta intr evad T) in H3...
unfold ve in H3...
rewrite hyp_evad in H3...
rewrite Rsqr_minus in H3...
repeat rewrite Rsqr_mult in H3...
rewrite cos2 in H3...
generalize
(Rplus_le_compat_l (- Rsqr (l intr evad T) - Rsqr V * Rsqr T)
(Rsqr V * Rsqr T + Rsqr (l intr evad T) -
2 * V * T * l intr evad T * cos (beta intr evad T))
(Rsqr (l intr evad T) *
(1 - Rsqr (sin (beta intr evad T + thetat intr 0))) +
Rsqr V * Rsqr T -
2 * (l intr evad T * cos (beta intr evad T + thetat intr 0)) * (V * T) +
Rsqr (l intr evad T) * Rsqr (sin (beta intr evad T + thetat intr 0))) H3)...
replace
(- Rsqr (l intr evad T) - Rsqr V * Rsqr T +
(Rsqr V * Rsqr T + Rsqr (l intr evad T) -
2 * V * T * l intr evad T * cos (beta intr evad T)))%R with
(2 * V * T * l intr evad T * - cos (beta intr evad T))%R...
replace
(- Rsqr (l intr evad T) - Rsqr V * Rsqr T +
(Rsqr (l intr evad T) * (1 - Rsqr (sin (beta intr evad T + thetat intr 0))) +
Rsqr V * Rsqr T -
2 * (l intr evad T * cos (beta intr evad T + thetat intr 0)) * (V * T) +
Rsqr (l intr evad T) * Rsqr (sin (beta intr evad T + thetat intr 0))))%R
with
(2 * V * T * l intr evad T * - cos (beta intr evad T + thetat intr 0))%R...
intro...
rewrite <- (Ropp_involutive (cos (beta intr evad T)))...
rewrite <- (Ropp_involutive (cos (beta intr evad T + thetat intr 0)))...
apply Ropp_ge_le_contravar...
apply Rle_ge...
apply Rmult_le_reg_l with (2 * V * T * l intr evad T)%R...
apply Rmult_lt_0_compat...
prove_sup0...
(*apply TypeSpeed_pos. useless since 8.4 thanks to beta/delta in apply*)
apply Rlt_le_trans with MinT; [ apply MinT_is_pos | apply (cond_1 T) ]...
ring...
ring...
(*Rewrite hyp_evad.*)
apply RR_pos...
unfold d in |- *; unfold Die in |- *; apply sqrt_positivity...
apply Rplus_le_le_0_compat; apply Rle_0_sqr...
Qed.

(* Specifications: *)
Lemma R_T_d_diff_1 :
h (tr evad) = V ->
(cos (beta intr evad T + thetat intr 0) <= cos (beta intr evad T))%R ->
(d intr evad <= RR (measure2state intr 0) (measure2state (tr evad) 0) T)%R.