X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FCoRN-Decl%2Fftc%2FMoreFunctions.ma;fp=helm%2Fsoftware%2Fmatita%2Fcontribs%2FCoRN-Decl%2Fftc%2FMoreFunctions.ma;h=8f0aeeda64233f21d1fa18b6ece10fd397ee6968;hb=3199f2a8428b5d19a3a803c1b03d9f90e4120f74;hp=0000000000000000000000000000000000000000;hpb=438a29376390222b94c1fe9772917c3aad50d42e;p=helm.git diff --git a/helm/software/matita/contribs/CoRN-Decl/ftc/MoreFunctions.ma b/helm/software/matita/contribs/CoRN-Decl/ftc/MoreFunctions.ma new file mode 100644 index 000000000..8f0aeeda6 --- /dev/null +++ b/helm/software/matita/contribs/CoRN-Decl/ftc/MoreFunctions.ma @@ -0,0 +1,650 @@ +(**************************************************************************) +(* ___ *) +(* ||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/ftc/MoreFunctions". + +(* $Id: MoreFunctions.v,v 1.5 2004/04/20 22:38:50 hinderer Exp $ *) + +(*#* printing FNorm %\ensuremath{\|\cdot\|_{\infty}}% *) + +(* INCLUDE +MoreIntervals +*) + +(* UNEXPORTED +Opaque Min Max. +*) + +(* UNEXPORTED +Section Basic_Results. +*) + +(*#* *More about Functions + +Here we state all the main results about properties of functions that +we already proved for compact intervals in the more general setting of +arbitrary intervals. + +%\begin{convention}% Let [I:interval] and [F,F',G,G'] be partial functions. +%\end{convention}% + +**Continuity +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +(*#* +Trivial stuff. +*) + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_imp_inc.con. + +(*#* +%\begin{convention}% Assume that [I] is compact and [F] is continuous in [I]. +%\end{convention}% +*) + +inline cic:/CoRN/ftc/MoreFunctions/cI.var. + +inline cic:/CoRN/ftc/MoreFunctions/F.var. + +inline cic:/CoRN/ftc/MoreFunctions/contF.var. + +inline cic:/CoRN/ftc/MoreFunctions/continuous_compact.con. + +(* begin show *) + +inline cic:/CoRN/ftc/MoreFunctions/Hinc.var. + +(* end show *) + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_I_imp_tb_image.con. + +inline cic:/CoRN/ftc/MoreFunctions/FNorm.con. + +inline cic:/CoRN/ftc/MoreFunctions/FNorm_bnd_AbsIR.con. + +(* UNEXPORTED +End Basic_Results. +*) + +(* UNEXPORTED +Hint Resolve Continuous_imp_inc: included. +*) + +(* UNEXPORTED +Section Other_Results. +*) + +(*#* +The usual stuff. +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +inline cic:/CoRN/ftc/MoreFunctions/F.var. + +inline cic:/CoRN/ftc/MoreFunctions/G.var. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_wd.con. + +(* begin show *) + +inline cic:/CoRN/ftc/MoreFunctions/contF.var. + +inline cic:/CoRN/ftc/MoreFunctions/contG.var. + +(* end show *) + +inline cic:/CoRN/ftc/MoreFunctions/included_imp_Continuous.con. + +inline cic:/CoRN/ftc/MoreFunctions/Included_imp_Continuous.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_const.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_id.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_plus.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_inv.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_minus.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_mult.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_nth.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_scal.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_abs.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_recip.con. + +(* UNEXPORTED +End Other_Results. +*) + +(* UNEXPORTED +Hint Resolve continuous_compact Continuous_const Continuous_id + Continuous_plus Continuous_inv Continuous_minus Continuous_mult + Continuous_scal Continuous_nth Continuous_recip Continuous_abs: continuous. +*) + +(* UNEXPORTED +Hint Immediate included_imp_Continuous Included_imp_Continuous: continuous. +*) + +(* UNEXPORTED +Section Corollaries. +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +inline cic:/CoRN/ftc/MoreFunctions/cI.var. + +inline cic:/CoRN/ftc/MoreFunctions/F.var. + +inline cic:/CoRN/ftc/MoreFunctions/G.var. + +inline cic:/CoRN/ftc/MoreFunctions/contF.var. + +inline cic:/CoRN/ftc/MoreFunctions/contG.var. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_div.con. + +inline cic:/CoRN/ftc/MoreFunctions/FNorm_wd.con. + +(* UNEXPORTED +End Corollaries. +*) + +(* UNEXPORTED +Hint Resolve Continuous_div: continuous. +*) + +(* UNEXPORTED +Section Sums. +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_Sumx.con. + +(*#* +%\begin{convention}% Assume [f] is a sequence of continuous functions. +%\end{convention}% +*) + +inline cic:/CoRN/ftc/MoreFunctions/f.var. + +inline cic:/CoRN/ftc/MoreFunctions/contF.var. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_Sum0.con. + +inline cic:/CoRN/ftc/MoreFunctions/Continuous_Sum.con. + +(* UNEXPORTED +End Sums. +*) + +(* UNEXPORTED +Hint Resolve Continuous_Sum0 Continuous_Sumx Continuous_Sum: continuous. +*) + +(* UNEXPORTED +Section Basic_Properties. +*) + +(*#* **Derivative + +Derivative is not that much different. + +%\begin{convention}% From this point on we assume [I] to be proper. +%\end{convention}% +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +inline cic:/CoRN/ftc/MoreFunctions/pI.var. + +inline cic:/CoRN/ftc/MoreFunctions/F.var. + +inline cic:/CoRN/ftc/MoreFunctions/G.var. + +inline cic:/CoRN/ftc/MoreFunctions/H.var. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_wdl.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_wdr.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_unique.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_imp_inc.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_imp_inc'.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_imp_Continuous.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_imp_Continuous'.con. + +(* UNEXPORTED +End Basic_Properties. +*) + +(* UNEXPORTED +Hint Immediate Derivative_imp_inc Derivative_imp_inc': included. +*) + +(* UNEXPORTED +Hint Immediate Derivative_imp_Continuous Derivative_imp_Continuous': + continuous. +*) + +(* UNEXPORTED +Section More_Results. +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +inline cic:/CoRN/ftc/MoreFunctions/pI.var. + +(*#* +%\begin{convention}% Assume that [F'] and [G'] are derivatives of [F] and [G], respectively, in [I]. +%\end{convention}% +*) + +inline cic:/CoRN/ftc/MoreFunctions/F.var. + +inline cic:/CoRN/ftc/MoreFunctions/F'.var. + +inline cic:/CoRN/ftc/MoreFunctions/G.var. + +inline cic:/CoRN/ftc/MoreFunctions/G'.var. + +inline cic:/CoRN/ftc/MoreFunctions/derF.var. + +inline cic:/CoRN/ftc/MoreFunctions/derG.var. + +inline cic:/CoRN/ftc/MoreFunctions/included_imp_Derivative.con. + +inline cic:/CoRN/ftc/MoreFunctions/Included_imp_Derivative.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_const.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_id.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_plus.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_inv.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_minus.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_mult.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_scal.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_nth.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_recip.con. + +(* UNEXPORTED +End More_Results. +*) + +(* UNEXPORTED +Section More_Corollaries. +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +inline cic:/CoRN/ftc/MoreFunctions/pI.var. + +inline cic:/CoRN/ftc/MoreFunctions/F.var. + +inline cic:/CoRN/ftc/MoreFunctions/F'.var. + +inline cic:/CoRN/ftc/MoreFunctions/G.var. + +inline cic:/CoRN/ftc/MoreFunctions/G'.var. + +inline cic:/CoRN/ftc/MoreFunctions/derF.var. + +inline cic:/CoRN/ftc/MoreFunctions/derG.var. + +(* begin show *) + +inline cic:/CoRN/ftc/MoreFunctions/Gbnd.var. + +(* end show *) + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_div.con. + +(* UNEXPORTED +End More_Corollaries. +*) + +(* UNEXPORTED +Section More_Sums. +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +inline cic:/CoRN/ftc/MoreFunctions/pI.var. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_Sumx.con. + +(* begin show *) + +inline cic:/CoRN/ftc/MoreFunctions/f.var. + +inline cic:/CoRN/ftc/MoreFunctions/f'.var. + +inline cic:/CoRN/ftc/MoreFunctions/derF.var. + +(* end show *) + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_Sum0.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_Sum.con. + +(* UNEXPORTED +End More_Sums. +*) + +(* UNEXPORTED +Section Diffble_Basic_Properties. +*) + +(*#* **Differentiability + +Mutatis mutandis for differentiability. +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +inline cic:/CoRN/ftc/MoreFunctions/pI.var. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_imp_inc.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_imp_Diffble.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_wd.con. + +inline cic:/CoRN/ftc/MoreFunctions/F.var. + +inline cic:/CoRN/ftc/MoreFunctions/G.var. + +inline cic:/CoRN/ftc/MoreFunctions/diffF.var. + +inline cic:/CoRN/ftc/MoreFunctions/diffG.var. + +(*#* +%\begin{convention}% Assume [F] and [G] are differentiable in [I]. +%\end{convention}% +*) + +inline cic:/CoRN/ftc/MoreFunctions/included_imp_Diffble.con. + +inline cic:/CoRN/ftc/MoreFunctions/Included_imp_Diffble.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_const.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_id.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_plus.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_inv.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_minus.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_mult.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_nth.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_scal.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_recip.con. + +(* UNEXPORTED +End Diffble_Basic_Properties. +*) + +(* UNEXPORTED +Hint Immediate Diffble_imp_inc: included. +*) + +(* UNEXPORTED +Section Diffble_Corollaries. +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +inline cic:/CoRN/ftc/MoreFunctions/pI.var. + +inline cic:/CoRN/ftc/MoreFunctions/F.var. + +inline cic:/CoRN/ftc/MoreFunctions/G.var. + +inline cic:/CoRN/ftc/MoreFunctions/diffF.var. + +inline cic:/CoRN/ftc/MoreFunctions/diffG.var. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_div.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_Sum0.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_Sumx.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_Sum.con. + +(* UNEXPORTED +End Diffble_Corollaries. +*) + +(* UNEXPORTED +Section Nth_Derivative. +*) + +(*#* **Nth Derivative + +Higher order derivatives pose more interesting problems. It turns out that it really becomes necessary to generalize our [n_deriv] operator to any interval. +*) + +inline cic:/CoRN/ftc/MoreFunctions/I.var. + +inline cic:/CoRN/ftc/MoreFunctions/pI.var. + +(* UNEXPORTED +Section Definitions. +*) + +(*#* +%\begin{convention}% Let [n:nat], [F:PartIR] and assume that [F] is n-times differentiable in [I]. +%\end{convention}% +*) + +inline cic:/CoRN/ftc/MoreFunctions/n.var. + +inline cic:/CoRN/ftc/MoreFunctions/F.var. + +inline cic:/CoRN/ftc/MoreFunctions/diffF.var. + +inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_fun.con. + +inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_char + (* begin hide *).con. + +(* end hide *) + +inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_strext.con. + +inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_wd.con. + +inline cic:/CoRN/ftc/MoreFunctions/N_Deriv.con. + +(* UNEXPORTED +End Definitions. +*) + +(* UNEXPORTED +Section Basic_Results. +*) + +(*#* +All the usual results hold. +*) + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_n_wd.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_wdr.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_wdl.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_unique.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_n_imp_Diffble.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_imp_Diffble.con. + +inline cic:/CoRN/ftc/MoreFunctions/le_imp_Diffble_n.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_n_imp_le.con. + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_n_imp_inc.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_imp_Diffble_n.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_imp_inc.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_imp_inc'.con. + +inline cic:/CoRN/ftc/MoreFunctions/included_imp_Derivative_n.con. + +inline cic:/CoRN/ftc/MoreFunctions/included_imp_Diffble_n.con. + +inline cic:/CoRN/ftc/MoreFunctions/Included_imp_Derivative_n.con. + +inline cic:/CoRN/ftc/MoreFunctions/Included_imp_Diffble_n.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_plus.con. + +(* UNEXPORTED +End Basic_Results. +*) + +(* UNEXPORTED +Section More_Results. +*) + +(*#* +Some new results hold, too: +*) + +inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_Feq.con. + +inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_lemma.con. + +inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_S.con. + +inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_plus.con. + +(*#* +Some useful characterization results. +*) + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_O.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_Sn.con. + +(* UNEXPORTED +End More_Results. +*) + +(* UNEXPORTED +Section Derivating_Diffble. +*) + +(*#* +As a special case we get a differentiation operator%\ldots%#...# +*) + +inline cic:/CoRN/ftc/MoreFunctions/F.var. + +(* begin show *) + +inline cic:/CoRN/ftc/MoreFunctions/diffF.var. + +(* end show *) + +inline cic:/CoRN/ftc/MoreFunctions/Diffble_imp_Diffble_n.con. + +inline cic:/CoRN/ftc/MoreFunctions/Deriv.con. + +(* UNEXPORTED +End Derivating_Diffble. +*) + +(* UNEXPORTED +Section Corollaries. +*) + +(*#* +%\ldots%#...# for which the expected property also holds. +*) + +inline cic:/CoRN/ftc/MoreFunctions/Deriv_lemma.con. + +(*#* +Some more interesting properties. +*) + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_1.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_chain.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_imp_Continuous.con. + +inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_imp_Continuous'.con. + +(* UNEXPORTED +End Corollaries. +*) + +(* UNEXPORTED +End Nth_Derivative. +*) + +(* UNEXPORTED +Hint Resolve Derivative_const Derivative_id Derivative_plus Derivative_inv + Derivative_minus Derivative_mult Derivative_scal Derivative_nth + Derivative_recip Derivative_div Derivative_Sumx Derivative_Sum0 + Derivative_Sum: derivate. +*) + +(* UNEXPORTED +Hint Immediate Derivative_n_imp_inc Derivative_n_imp_inc' Diffble_n_imp_inc: + included. +*) + +(* UNEXPORTED +Hint Resolve Deriv_lemma N_Deriv_lemma: derivate. +*) + +(* UNEXPORTED +Hint Immediate Derivative_n_imp_Continuous Derivative_n_imp_Continuous': + continuous. +*) +