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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/ftc/NthDerivative".
19 (* $Id: NthDerivative.v,v 1.5 2004/04/20 22:38:50 hinderer Exp $ *)
26 Section Nth_Derivative.
31 We now study higher order differentiability.
33 %\begin{convention}% Throughout this section:
34 - [a, b] will be real numbers with [a [<] b];
35 - [I] will denote the compact interval [[a,b]];
36 - [F, G, H] will denote partial functions.
42 We first define what we mean by the derivative of order [n] of a function.
45 inline cic:/CoRN/ftc/NthDerivative/a.var.
47 inline cic:/CoRN/ftc/NthDerivative/b.var.
49 inline cic:/CoRN/ftc/NthDerivative/Hab'.var.
53 inline cic:/CoRN/ftc/NthDerivative/Hab.con.
55 inline cic:/CoRN/ftc/NthDerivative/I.con.
59 inline cic:/CoRN/ftc/NthDerivative/Derivative_I_n.con.
62 Unlike first order differentiability, for our definition to be
63 workable it is better to define directly what it means for a function
64 to be [n] times differentiable instead of quantifying over the
65 [Derivative_I_n] relation.
68 inline cic:/CoRN/ftc/NthDerivative/Diffble_I_n.con.
75 Implicit Arguments Derivative_I_n [a b].
79 Implicit Arguments Diffble_I_n [a b].
88 These are the expected extensionality and uniqueness results.
91 inline cic:/CoRN/ftc/NthDerivative/a.var.
93 inline cic:/CoRN/ftc/NthDerivative/b.var.
95 inline cic:/CoRN/ftc/NthDerivative/Hab'.var.
99 inline cic:/CoRN/ftc/NthDerivative/Hab.con.
101 inline cic:/CoRN/ftc/NthDerivative/I.con.
105 inline cic:/CoRN/ftc/NthDerivative/Diffble_I_n_wd.con.
107 inline cic:/CoRN/ftc/NthDerivative/Derivative_I_n_wdr.con.
109 inline cic:/CoRN/ftc/NthDerivative/Derivative_I_n_wdl.con.
111 inline cic:/CoRN/ftc/NthDerivative/Derivative_I_n_unique.con.
118 Section Basic_Results.
123 We now explore the concept of [n] times differentiability. Notice
124 that, unlike the first order case, we do not pay so much attention to
125 the relation of [n] times derivative, but focus rather on the
126 definition of [Diffble_I_n].
129 inline cic:/CoRN/ftc/NthDerivative/a.var.
131 inline cic:/CoRN/ftc/NthDerivative/b.var.
133 inline cic:/CoRN/ftc/NthDerivative/Hab'.var.
137 inline cic:/CoRN/ftc/NthDerivative/Hab.con.
139 inline cic:/CoRN/ftc/NthDerivative/I.con.
144 We begin by showing that having a higher order derivative implies being differentiable.
147 inline cic:/CoRN/ftc/NthDerivative/Diffble_I_n_imp_diffble.con.
149 inline cic:/CoRN/ftc/NthDerivative/deriv_n_imp_diffble.con.
152 If a function is [n] times differentiable then it is also [m] times differentiable for every [m] less or equal than [n].
155 inline cic:/CoRN/ftc/NthDerivative/le_imp_Diffble_I.con.
158 The next result consolidates our intuition that a function is [n]
159 times differentiable if we can build from it a chain of [n]
163 inline cic:/CoRN/ftc/NthDerivative/Diffble_I_imp_le.con.
166 As expected, an [n] times differentiable in [[a,b]] function must be
167 defined in that interval.
170 inline cic:/CoRN/ftc/NthDerivative/Diffble_I_n_imp_inc.con.
173 Also, the notions of derivative and differentiability are related as expected.
176 inline cic:/CoRN/ftc/NthDerivative/Diffble_I_n_imp_deriv_n.con.
178 inline cic:/CoRN/ftc/NthDerivative/deriv_n_imp_Diffble_I_n.con.
181 From this we can prove that if [F] has an nth order derivative in
182 [[a,b]] then both [F] and its derivative are defined in that interval.
185 inline cic:/CoRN/ftc/NthDerivative/Derivative_I_n_imp_inc.con.
187 inline cic:/CoRN/ftc/NthDerivative/Derivative_I_n_imp_inc'.con.
194 First order differentiability is just a special case.
199 inline cic:/CoRN/ftc/NthDerivative/F.var.
201 inline cic:/CoRN/ftc/NthDerivative/diffF.var.
203 inline cic:/CoRN/ftc/NthDerivative/diffFn.var.
207 inline cic:/CoRN/ftc/NthDerivative/deriv_1_deriv.con.
209 inline cic:/CoRN/ftc/NthDerivative/deriv_1_deriv'.con.
216 As usual, nth order derivability is preserved by shrinking the interval.
219 inline cic:/CoRN/ftc/NthDerivative/included_imp_deriv_n.con.
221 inline cic:/CoRN/ftc/NthDerivative/included_imp_diffble_n.con.
224 And finally we have an addition rule for the order of the derivative.
227 inline cic:/CoRN/ftc/NthDerivative/Derivative_I_n_plus.con.
234 Section More_Results.
237 inline cic:/CoRN/ftc/NthDerivative/a.var.
239 inline cic:/CoRN/ftc/NthDerivative/b.var.
241 inline cic:/CoRN/ftc/NthDerivative/Hab'.var.
245 inline cic:/CoRN/ftc/NthDerivative/Hab.con.
247 inline cic:/CoRN/ftc/NthDerivative/I.con.
251 (*#* **The Nth Derivative
253 We now define an operator that returns an nth order derivative of an
254 n-times differentiable function. This is analogous to the quantifier
255 elimination which we would get if we had defined nth differentiability
256 as an existential quantification of the nth derivative relation.
259 inline cic:/CoRN/ftc/NthDerivative/n_deriv_I.con.
262 This operator is well defined and works as expected.
265 inline cic:/CoRN/ftc/NthDerivative/n_deriv_I_wd.con.
267 inline cic:/CoRN/ftc/NthDerivative/n_deriv_lemma.con.
269 inline cic:/CoRN/ftc/NthDerivative/n_deriv_inc.con.
271 inline cic:/CoRN/ftc/NthDerivative/n_deriv_inc'.con.
274 Some basic properties of this operation.
277 inline cic:/CoRN/ftc/NthDerivative/n_Sn_deriv.con.
279 inline cic:/CoRN/ftc/NthDerivative/n_deriv_plus.con.
286 Section More_on_n_deriv.
290 Some not so basic properties of this operation (needed in rather specific situations).
293 inline cic:/CoRN/ftc/NthDerivative/n_deriv_I_wd'.con.
295 inline cic:/CoRN/ftc/NthDerivative/n_deriv_I_wd''.con.
297 inline cic:/CoRN/ftc/NthDerivative/n_deriv_I_strext'.con.