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(** * Using series definitions of cos and sin                    *)
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Definition sin_lb (a:R) : R := sin_approx a 3.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 cos_increasing_0 : forall x y:R, PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x < cos y -> x < y.