Proof with trivial.
unfold Alpha in |- *;
case
(Rle_dec
((Rsqr (ve evad * T) + Rsqr (l intr evad T) - Rsqr AlertRange) /
(2 * ve evad * T)) (l intr evad T * cos (beta intr evad T)));
intros...
elim diff_true_false...
unfold Rdiv in n; cut (0 < 2 * ve evad * T)%R...
intro...
cut
(l intr evad T * cos (beta intr evad T) <
(Rsqr (ve evad * T) + Rsqr (l intr evad T) - Rsqr AlertRange) *
/ (2 * ve evad * T))%R...
intro;
generalize
(Rmult_lt_compat_r (2 * ve evad * T)
(l intr evad T * cos (beta intr evad T))
((Rsqr (ve evad * T) + Rsqr (l intr evad T) - Rsqr AlertRange) *
/ (2 * ve evad * T)) H0 H1); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym...
rewrite Rmult_1_r; intro;
generalize
(Rplus_lt_compat_l
(Rsqr AlertRange -
2 * l intr evad T * ve evad * T * cos (beta intr evad T))
(l intr evad T * (cos (beta intr evad T) * (2 * (ve evad * T))))
(Rsqr (ve evad * T) + Rsqr (l intr evad T) - Rsqr AlertRange) H2);
replace
(Rsqr AlertRange - 2 * l intr evad T * ve evad * T * cos (beta intr evad T) +
l intr evad T * (cos (beta intr evad T) * (2 * (ve evad * T))))%R with
(Rsqr AlertRange)...
replace
(Rsqr AlertRange - 2 * l intr evad T * ve evad * T * cos (beta intr evad T) +
(Rsqr (ve evad * T) + Rsqr (l intr evad T) - Rsqr AlertRange))%R with
(Rsqr (ve evad * T) + Rsqr (l intr evad T) -
2 * ve evad * T * l intr evad T * cos (beta intr evad T))%R...
rewrite <- d_l_beta...
intro; apply Rsqr_incrst_0...
left; apply AlertRange_pos...
unfold d in |- *; unfold Die in |- *; apply sqrt_positivity;
apply Rplus_le_le_0_compat; apply Rle_0_sqr...
ring...
ring...
repeat apply prod_neq_R0...
discrR...
red in |- *; intro; generalize (TypeSpeed_pos (h (tr evad))); intro;
unfold ve in H2; rewrite H2 in H3; elim (Rlt_irrefl 0 H3)...
cut (0 < T)%R...
intro...
red in |- *; intro; rewrite H3 in H2; elim (Rlt_irrefl 0 H2)...
apply Rlt_le_trans with MinT...
apply MinT_is_pos...
apply (cond_1 T)...
auto with real...
repeat apply Rmult_lt_0_compat...
prove_sup0...
unfold ve in |- *; apply (TypeSpeed_pos (h (tr evad)))...
apply Rlt_le_trans with MinT...
apply MinT_is_pos...
apply (cond_1 T)...
Qed.

Lemma Alpha_d_AlertRange_3 :
(AlertRange < d intr evad)%R -> Alpha (beta intr evad T) = false.