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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/ftc/DerivativeOps".
19 (* $Id: DerivativeOps.v,v 1.3 2004/04/23 10:00:58 lcf Exp $ *)
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.