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[helm.git] / matita / contribs / CoRN-Decl / transc / Trigonometric.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/transc/Trigonometric".
+
+include "CoRN.ma".
+
+(* $Id: Trigonometric.v,v 1.5 2004/04/23 10:01:08 lcf Exp $ *)
+
+include "transc/TaylorSeries.ma".
+
+(*#* *The Trigonometric Functions
+
+In this section, we explore the properties of the trigonometric functions which we previously defined.
+*)
+
+(* UNEXPORTED
+Section Lemmas
+*)
+
+(*#* First, we need a lemma on mappings. *)
+
+inline "cic:/CoRN/transc/Trigonometric/maps_translation.con".
+
+(* UNEXPORTED
+End Lemmas
+*)
+
+(* UNEXPORTED
+Section Sine_and_Cosine
+*)
+
+(*#* Sine, cosine and tangent at [Zero]. *)
+
+inline "cic:/CoRN/transc/Trigonometric/Sin_zero.con".
+
+inline "cic:/CoRN/transc/Trigonometric/Cos_zero.con".
+
+(* UNEXPORTED
+Hint Resolve Sin_zero Cos_zero: algebra.
+*)
+
+(* UNEXPORTED
+Opaque Sine Cosine.
+*)
+
+inline "cic:/CoRN/transc/Trigonometric/Tan_zero.con".
+
+(* UNEXPORTED
+Transparent Sine Cosine.
+*)
+
+(*#*
+Continuity of sine and cosine are trivial.
+*)
+
+inline "cic:/CoRN/transc/Trigonometric/Continuous_Sin.con".
+
+inline "cic:/CoRN/transc/Trigonometric/Continuous_Cos.con".
+
+(*#*
+The rules for the derivative of the sine and cosine function; we begin by proving that their defining sequences can be expressed in terms of one another.
+*)
+
+inline "cic:/CoRN/transc/Trigonometric/cos_sin_seq.con".
+
+inline "cic:/CoRN/transc/Trigonometric/sin_cos_seq.con".
+
+inline "cic:/CoRN/transc/Trigonometric/Derivative_Sin.con".
+
+inline "cic:/CoRN/transc/Trigonometric/Derivative_Cos.con".
+
+(* UNEXPORTED
+Hint Resolve Derivative_Sin Derivative_Cos: derivate.
+*)
+
+(* UNEXPORTED
+Hint Resolve Continuous_Sin Continuous_Cos: continuous.
+*)
+
+(* UNEXPORTED
+Section Sine_of_Sum
+*)
+
+(*#*
+We now prove the rule for the sine and cosine of the sum.  These rules
+have to be proved first as functional equalities, which is why we also
+state the results in a function form (which we won't do in other
+situations).
+
+%\begin{convention}% Let:
+ - [F := fun y => Sine[o] (FId{+} [-C-]y)];
+ - [G := fun y => (Sine{*} [-C-] (Cos y)) {+} (Cosine{*} [-C-] (Sin y))].
+
+%\end{convention}%
+*)
+
+(* begin hide *)
+
+inline "cic:/CoRN/transc/Trigonometric/Sine_and_Cosine/Sine_of_Sum/F.con" "Sine_and_Cosine__Sine_of_Sum__".
+
+inline "cic:/CoRN/transc/Trigonometric/Sine_and_Cosine/Sine_of_Sum/G.con" "Sine_and_Cosine__Sine_of_Sum__".
+
+inline "cic:/CoRN/transc/Trigonometric/Sine_and_Cosine/Sine_of_Sum/F'.con" "Sine_and_Cosine__Sine_of_Sum__".
+
+inline "cic:/CoRN/transc/Trigonometric/Sine_and_Cosine/Sine_of_Sum/G'.con" "Sine_and_Cosine__Sine_of_Sum__".
+
+(* end hide *)
+
+(* UNEXPORTED
+Opaque Sine Cosine.
+*)
+
+(* UNEXPORTED
+Opaque FAbs.
+*)
+
+(* UNEXPORTED
+Transparent FAbs.
+*)
+
+(* UNEXPORTED
+Opaque FAbs.
+*)
+
+(* UNEXPORTED
+Transparent FAbs.
+*)
+
+(* UNEXPORTED
+Opaque FAbs.
+*)
+
+(* UNEXPORTED
+Transparent FAbs.
+*)
+
+(* UNEXPORTED
+Opaque FAbs.
+*)
+
+(* UNEXPORTED
+Transparent FAbs.
+*)
+
+inline "cic:/CoRN/transc/Trigonometric/Sin_plus_Taylor_bnd_lft.con".
+
+(* UNEXPORTED
+Opaque FAbs.
+*)
+
+(* UNEXPORTED
+Transparent FAbs.
+*)
+
+(* UNEXPORTED
+Opaque FAbs.
+*)
+
+(* UNEXPORTED
+Transparent FAbs.
+*)
+
+(* UNEXPORTED
+Opaque FAbs.
+*)
+
+(* UNEXPORTED
+Transparent FAbs.
+*)
+
+(* UNEXPORTED
+Opaque FAbs.
+*)
+
+(* UNEXPORTED
+Transparent FAbs.
+*)
+
+inline "cic:/CoRN/transc/Trigonometric/Sin_plus_Taylor_bnd_rht.con".
+
+inline "cic:/CoRN/transc/Trigonometric/Sin_plus_eq.con".
+
+inline "cic:/CoRN/transc/Trigonometric/Sin_plus_der_lft.con".
+
+inline "cic:/CoRN/transc/Trigonometric/Sin_plus_der_rht.con".
+
+inline "cic:/CoRN/transc/Trigonometric/Sin_plus_fun.con".
+
+(* UNEXPORTED
+End Sine_of_Sum
+*)
+
+(* UNEXPORTED
+Opaque Sine Cosine.
+*)
+
+inline "cic:/CoRN/transc/Trigonometric/Cos_plus_fun.con".
+
+(* UNEXPORTED
+End Sine_and_Cosine
+*)
+