assert (cc : continuity (fun r =>- cos r)). {
apply continuity_opp, continuity_cos.
}
assert (cvp : 0 < cos (7/8)). {
assert (int78 : -2 <= 7/8 <= 2) by (split; lra).
destruct int78 as [lower upper].
case (pre_cos_bound _ 0 lower upper).
unfold cos_approx; simpl sum_f_R0; unfold cos_term.
intros cl _; apply Rlt_le_trans with (2 := cl); simpl.
lra.
}
assert (cun : cos (7/4) < 0). {
replace (7/4) with (7/8 + 7/8) by field.
rewrite cos_plus.
apply Rlt_minus; apply Rsqr_incrst_1.
- exact sin_gt_cos_7_8.
- apply Rlt_le; assumption.
- apply Rlt_le; apply Rlt_trans with (1 := cvp); exact sin_gt_cos_7_8.
}
apply IVT; auto; lra.
Qed.
Definition PI2 := proj1_sig PI_2_aux.
Definition PI := 2 * PI2.

Lemma cos_pi2 : cos PI2 = 0.
unfold PI2; case PI_2_aux; simpl.