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(** * Definition of arctangent                           *)
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(** ** Definition of arctangent as the reciprocal function of tangent and proof of this status *)

(* Specifications: *)
Lemma tan_1_gt_1 : tan 1 > 1.
Lemma sin_lt_x x : 0 < x -> sin x < x.
Definition frame_tan y : {x | 0 < x < PI/2 /\ Rabs y < tan x}.
Lemma ub_opp : forall x, x < PI/2 -> -PI/2 < -x.
Lemma pos_opp_lt : forall x, 0 < x -> -x < x.
Definition pre_atan (y : R) : {x : R | -PI/2 < x < PI/2 /\ tan x = y}.