assert (lo : 0 <= x) by (apply Rlt_le; assumption).
assert (up : x <= 4) by (apply Rlt_le, Rlt_trans with (1:=int2); lra).
destruct (pre_sin_bound _ 0 lo up) as [t _]; clear lo up.
apply Rlt_le_trans with (2:= t); clear t.
unfold sin_approx; simpl sum_f_R0; unfold sin_term; simpl.
match goal with |- _ < ?a =>
replace a with (x * (1 - x^2/6)) by (simpl; field)
end.
assert (t' : x ^ 2 <= 4). {
replace 4 with (2 ^ 2) by field.
apply (pow_incr x 2); split; apply Rlt_le; assumption.
}
apply Rmult_lt_0_compat;[assumption | lra ].
Qed.

Lemma sin_PI2 : sin (PI / 2) = 1.
replace (PI / 2) with PI2 by (unfold PI; field).