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(** * Using series definitions of cos and sin                    *)
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Definition sin_ub (a:R) : R := sin_approx a 4.
Definition cos_lb (a:R) : R := cos_approx a 3.
Definition cos_ub (a:R) : R := cos_approx a 4.

Lemma sin_increasing_0 :
forall x y:R,
- (PI / 2) <= x ->
x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x < sin y -> x < y.