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19 (* $Id: DerivativeOps.v,v 1.3 2004/04/23 10:00:58 lcf Exp $ *)
21 include "ftc/Derivative.ma".
27 (*#* **Algebraic Operations
29 We will now prove the main results about deriving functions built from
30 the algebraic operators#. #%\footnote{%Composition presents some
31 tricky questions, and is therefore discussed in a separated
34 [F'] and [G'] will be the derivatives, respectively, of [F] and [G].
36 We begin with some technical stuff that will be necessary for division.
39 alias id "a" = "cic:/CoRN/ftc/DerivativeOps/Lemmas/a.var".
41 alias id "b" = "cic:/CoRN/ftc/DerivativeOps/Lemmas/b.var".
43 alias id "Hab" = "cic:/CoRN/ftc/DerivativeOps/Lemmas/Hab.var".
47 inline procedural "cic:/CoRN/ftc/DerivativeOps/Lemmas/I.con" "Lemmas__" as definition.
51 alias id "F" = "cic:/CoRN/ftc/DerivativeOps/Lemmas/F.var".
55 inline procedural "cic:/CoRN/ftc/DerivativeOps/Lemmas/P.con" "Lemmas__" as definition.
61 alias id "Fbnd" = "cic:/CoRN/ftc/DerivativeOps/Lemmas/Fbnd.var".
65 inline procedural "cic:/CoRN/ftc/DerivativeOps/bnd_away_zero_square.con" as lemma.
72 Hint Resolve bnd_away_zero_square: included.
81 We can now derive all the usual rules for deriving constant and identity functions, sums, inverses and products of functions with a known derivative.
84 alias id "a" = "cic:/CoRN/ftc/DerivativeOps/Local_Results/a.var".
86 alias id "b" = "cic:/CoRN/ftc/DerivativeOps/Local_Results/b.var".
88 alias id "Hab'" = "cic:/CoRN/ftc/DerivativeOps/Local_Results/Hab'.var".
92 inline procedural "cic:/CoRN/ftc/DerivativeOps/Local_Results/Hab.con" "Local_Results__" as definition.
94 inline procedural "cic:/CoRN/ftc/DerivativeOps/Local_Results/I.con" "Local_Results__" as definition.
98 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_const.con" as lemma.
100 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_id.con" as lemma.
102 alias id "F" = "cic:/CoRN/ftc/DerivativeOps/Local_Results/F.var".
104 alias id "F'" = "cic:/CoRN/ftc/DerivativeOps/Local_Results/F'.var".
106 alias id "G" = "cic:/CoRN/ftc/DerivativeOps/Local_Results/G.var".
108 alias id "G'" = "cic:/CoRN/ftc/DerivativeOps/Local_Results/G'.var".
110 alias id "derF" = "cic:/CoRN/ftc/DerivativeOps/Local_Results/derF.var".
112 alias id "derG" = "cic:/CoRN/ftc/DerivativeOps/Local_Results/derG.var".
114 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_plus.con" as lemma.
116 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_inv.con" as lemma.
118 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_mult.con" as lemma.
121 As was the case for continuity, the rule for the reciprocal function has a side condition.
126 alias id "Fbnd" = "cic:/CoRN/ftc/DerivativeOps/Local_Results/Fbnd.var".
130 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_recip.con" as lemma.
137 Hint Immediate derivative_imp_inc derivative_imp_inc': included.
141 Hint Resolve Derivative_I_const Derivative_I_id Derivative_I_plus
142 Derivative_I_inv Derivative_I_mult Derivative_I_recip: derivate.
149 alias id "a" = "cic:/CoRN/ftc/DerivativeOps/Corolaries/a.var".
151 alias id "b" = "cic:/CoRN/ftc/DerivativeOps/Corolaries/b.var".
153 alias id "Hab'" = "cic:/CoRN/ftc/DerivativeOps/Corolaries/Hab'.var".
157 inline procedural "cic:/CoRN/ftc/DerivativeOps/Corolaries/Hab.con" "Corolaries__" as definition.
159 inline procedural "cic:/CoRN/ftc/DerivativeOps/Corolaries/I.con" "Corolaries__" as definition.
163 alias id "F" = "cic:/CoRN/ftc/DerivativeOps/Corolaries/F.var".
165 alias id "F'" = "cic:/CoRN/ftc/DerivativeOps/Corolaries/F'.var".
167 alias id "G" = "cic:/CoRN/ftc/DerivativeOps/Corolaries/G.var".
169 alias id "G'" = "cic:/CoRN/ftc/DerivativeOps/Corolaries/G'.var".
171 alias id "derF" = "cic:/CoRN/ftc/DerivativeOps/Corolaries/derF.var".
173 alias id "derG" = "cic:/CoRN/ftc/DerivativeOps/Corolaries/derG.var".
176 From this lemmas the rules for the other algebraic operations follow directly.
179 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_minus.con" as lemma.
181 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_scal.con" as lemma.
183 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_nth.con" as lemma.
185 alias id "Gbnd" = "cic:/CoRN/ftc/DerivativeOps/Corolaries/Gbnd.var".
187 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_div.con" as lemma.
194 Hint Resolve Derivative_I_minus Derivative_I_nth Derivative_I_scal
195 Derivative_I_div: derivate.
199 Section Derivative_Sums
202 (*#* The derivation rules for families of functions are easily proved by
203 induction using the constant and addition rules.
206 alias id "a" = "cic:/CoRN/ftc/DerivativeOps/Derivative_Sums/a.var".
208 alias id "b" = "cic:/CoRN/ftc/DerivativeOps/Derivative_Sums/b.var".
210 alias id "Hab" = "cic:/CoRN/ftc/DerivativeOps/Derivative_Sums/Hab.var".
212 alias id "Hab'" = "cic:/CoRN/ftc/DerivativeOps/Derivative_Sums/Hab'.var".
216 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_Sums/I.con" "Derivative_Sums__" as definition.
220 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_Sum0.con" as lemma.
222 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_Sumx.con" as lemma.
224 inline procedural "cic:/CoRN/ftc/DerivativeOps/Derivative_I_Sum.con" as lemma.
231 Hint Resolve Derivative_I_Sum0 Derivative_I_Sum Derivative_I_Sumx: derivate.