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7 (* ||T|| The HELM team. *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/ftc/DerivativeOps".
21 (* $Id: DerivativeOps.v,v 1.3 2004/04/23 10:00:58 lcf Exp $ *)
23 include "ftc/Derivative.ma".
29 (*#* **Algebraic Operations
31 We will now prove the main results about deriving functions built from
32 the algebraic operators#. #%\footnote{%Composition presents some
33 tricky questions, and is therefore discussed in a separated
36 [F'] and [G'] will be the derivatives, respectively, of [F] and [G].
38 We begin with some technical stuff that will be necessary for division.
41 inline "cic:/CoRN/ftc/DerivativeOps/a.var".
43 inline "cic:/CoRN/ftc/DerivativeOps/b.var".
45 inline "cic:/CoRN/ftc/DerivativeOps/Hab.var".
49 inline "cic:/CoRN/ftc/DerivativeOps/I.con".
53 inline "cic:/CoRN/ftc/DerivativeOps/F.var".
57 inline "cic:/CoRN/ftc/DerivativeOps/P.con".
63 inline "cic:/CoRN/ftc/DerivativeOps/Fbnd.var".
67 inline "cic:/CoRN/ftc/DerivativeOps/bnd_away_zero_square.con".
74 Hint Resolve bnd_away_zero_square: included.
78 Section Local_Results.
83 We can now derive all the usual rules for deriving constant and identity functions, sums, inverses and products of functions with a known derivative.
86 inline "cic:/CoRN/ftc/DerivativeOps/a.var".
88 inline "cic:/CoRN/ftc/DerivativeOps/b.var".
90 inline "cic:/CoRN/ftc/DerivativeOps/Hab'.var".
94 inline "cic:/CoRN/ftc/DerivativeOps/Hab.con".
96 inline "cic:/CoRN/ftc/DerivativeOps/I.con".
100 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_const.con".
102 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_id.con".
104 inline "cic:/CoRN/ftc/DerivativeOps/F.var".
106 inline "cic:/CoRN/ftc/DerivativeOps/F'.var".
108 inline "cic:/CoRN/ftc/DerivativeOps/G.var".
110 inline "cic:/CoRN/ftc/DerivativeOps/G'.var".
112 inline "cic:/CoRN/ftc/DerivativeOps/derF.var".
114 inline "cic:/CoRN/ftc/DerivativeOps/derG.var".
116 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_plus.con".
118 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_inv.con".
120 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_mult.con".
123 As was the case for continuity, the rule for the reciprocal function has a side condition.
128 inline "cic:/CoRN/ftc/DerivativeOps/Fbnd.var".
132 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_recip.con".
139 Hint Immediate derivative_imp_inc derivative_imp_inc': included.
143 Hint Resolve Derivative_I_const Derivative_I_id Derivative_I_plus
144 Derivative_I_inv Derivative_I_mult Derivative_I_recip: derivate.
151 inline "cic:/CoRN/ftc/DerivativeOps/a.var".
153 inline "cic:/CoRN/ftc/DerivativeOps/b.var".
155 inline "cic:/CoRN/ftc/DerivativeOps/Hab'.var".
159 inline "cic:/CoRN/ftc/DerivativeOps/Hab.con".
161 inline "cic:/CoRN/ftc/DerivativeOps/I.con".
165 inline "cic:/CoRN/ftc/DerivativeOps/F.var".
167 inline "cic:/CoRN/ftc/DerivativeOps/F'.var".
169 inline "cic:/CoRN/ftc/DerivativeOps/G.var".
171 inline "cic:/CoRN/ftc/DerivativeOps/G'.var".
173 inline "cic:/CoRN/ftc/DerivativeOps/derF.var".
175 inline "cic:/CoRN/ftc/DerivativeOps/derG.var".
178 From this lemmas the rules for the other algebraic operations follow directly.
181 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_minus.con".
183 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_scal.con".
185 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_nth.con".
187 inline "cic:/CoRN/ftc/DerivativeOps/Gbnd.var".
189 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_div.con".
196 Hint Resolve Derivative_I_minus Derivative_I_nth Derivative_I_scal
197 Derivative_I_div: derivate.
201 Section Derivative_Sums.
204 (*#* The derivation rules for families of functions are easily proved by
205 induction using the constant and addition rules.
208 inline "cic:/CoRN/ftc/DerivativeOps/a.var".
210 inline "cic:/CoRN/ftc/DerivativeOps/b.var".
212 inline "cic:/CoRN/ftc/DerivativeOps/Hab.var".
214 inline "cic:/CoRN/ftc/DerivativeOps/Hab'.var".
218 inline "cic:/CoRN/ftc/DerivativeOps/I.con".
222 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_Sum0.con".
224 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_Sumx.con".
226 inline "cic:/CoRN/ftc/DerivativeOps/Derivative_I_Sum.con".
233 Hint Resolve Derivative_I_Sum0 Derivative_I_Sum Derivative_I_Sumx: derivate.