set "baseuri" "cic:/matita/CoRN-Decl/ftc/MoreFunctions".
+include "CoRN.ma".
+
(* $Id: MoreFunctions.v,v 1.5 2004/04/20 22:38:50 hinderer Exp $ *)
(*#* printing FNorm %\ensuremath{\|\cdot\|_{\infty}}% *)
-(* INCLUDE
-MoreIntervals
-*)
+include "ftc/MoreIntervals.ma".
(* UNEXPORTED
Opaque Min Max.
*)
(* UNEXPORTED
-Section Basic_Results.
+Section Basic_Results
*)
(*#* *More about Functions
**Continuity
*)
-inline cic:/CoRN/ftc/MoreFunctions/I.var.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/Basic_Results/I.var".
(*#*
Trivial stuff.
*)
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_imp_inc.con.
+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.
+alias id "cI" = "cic:/CoRN/ftc/MoreFunctions/Basic_Results/cI.var".
-inline cic:/CoRN/ftc/MoreFunctions/F.var.
+alias id "F" = "cic:/CoRN/ftc/MoreFunctions/Basic_Results/F.var".
-inline cic:/CoRN/ftc/MoreFunctions/contF.var.
+alias id "contF" = "cic:/CoRN/ftc/MoreFunctions/Basic_Results/contF.var".
-inline cic:/CoRN/ftc/MoreFunctions/continuous_compact.con.
+inline "cic:/CoRN/ftc/MoreFunctions/continuous_compact.con".
(* begin show *)
-inline cic:/CoRN/ftc/MoreFunctions/Hinc.var.
+alias id "Hinc" = "cic:/CoRN/ftc/MoreFunctions/Basic_Results/Hinc.var".
(* end show *)
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_I_imp_tb_image.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_I_imp_tb_image.con".
-inline cic:/CoRN/ftc/MoreFunctions/FNorm.con.
+inline "cic:/CoRN/ftc/MoreFunctions/FNorm.con".
-inline cic:/CoRN/ftc/MoreFunctions/FNorm_bnd_AbsIR.con.
+inline "cic:/CoRN/ftc/MoreFunctions/FNorm_bnd_AbsIR.con".
(* UNEXPORTED
-End Basic_Results.
+End Basic_Results
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section Other_Results.
+Section Other_Results
*)
(*#*
The usual stuff.
*)
-inline cic:/CoRN/ftc/MoreFunctions/I.var.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/Other_Results/I.var".
-inline cic:/CoRN/ftc/MoreFunctions/F.var.
+alias id "F" = "cic:/CoRN/ftc/MoreFunctions/Other_Results/F.var".
-inline cic:/CoRN/ftc/MoreFunctions/G.var.
+alias id "G" = "cic:/CoRN/ftc/MoreFunctions/Other_Results/G.var".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_wd.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_wd.con".
(* begin show *)
-inline cic:/CoRN/ftc/MoreFunctions/contF.var.
+alias id "contF" = "cic:/CoRN/ftc/MoreFunctions/Other_Results/contF.var".
-inline cic:/CoRN/ftc/MoreFunctions/contG.var.
+alias id "contG" = "cic:/CoRN/ftc/MoreFunctions/Other_Results/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/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_const.con".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_id.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_id.con".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_plus.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_plus.con".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_inv.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_inv.con".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_minus.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_minus.con".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_mult.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_mult.con".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_nth.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_nth.con".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_scal.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_scal.con".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_abs.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_abs.con".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_recip.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_recip.con".
(* UNEXPORTED
-End Other_Results.
+End Other_Results
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section Corollaries.
+Section Corollaries
*)
-inline cic:/CoRN/ftc/MoreFunctions/I.var.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/Corollaries/I.var".
-inline cic:/CoRN/ftc/MoreFunctions/cI.var.
+alias id "cI" = "cic:/CoRN/ftc/MoreFunctions/Corollaries/cI.var".
-inline cic:/CoRN/ftc/MoreFunctions/F.var.
+alias id "F" = "cic:/CoRN/ftc/MoreFunctions/Corollaries/F.var".
-inline cic:/CoRN/ftc/MoreFunctions/G.var.
+alias id "G" = "cic:/CoRN/ftc/MoreFunctions/Corollaries/G.var".
-inline cic:/CoRN/ftc/MoreFunctions/contF.var.
+alias id "contF" = "cic:/CoRN/ftc/MoreFunctions/Corollaries/contF.var".
-inline cic:/CoRN/ftc/MoreFunctions/contG.var.
+alias id "contG" = "cic:/CoRN/ftc/MoreFunctions/Corollaries/contG.var".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_div.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_div.con".
-inline cic:/CoRN/ftc/MoreFunctions/FNorm_wd.con.
+inline "cic:/CoRN/ftc/MoreFunctions/FNorm_wd.con".
(* UNEXPORTED
-End Corollaries.
+End Corollaries
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section Sums.
+Section Sums
*)
-inline cic:/CoRN/ftc/MoreFunctions/I.var.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/Sums/I.var".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_Sumx.con.
+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.
+alias id "f" = "cic:/CoRN/ftc/MoreFunctions/Sums/f.var".
-inline cic:/CoRN/ftc/MoreFunctions/contF.var.
+alias id "contF" = "cic:/CoRN/ftc/MoreFunctions/Sums/contF.var".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_Sum0.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_Sum0.con".
-inline cic:/CoRN/ftc/MoreFunctions/Continuous_Sum.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Continuous_Sum.con".
(* UNEXPORTED
-End Sums.
+End Sums
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section Basic_Properties.
+Section Basic_Properties
*)
(*#* **Derivative
%\end{convention}%
*)
-inline cic:/CoRN/ftc/MoreFunctions/I.var.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/Basic_Properties/I.var".
-inline cic:/CoRN/ftc/MoreFunctions/pI.var.
+alias id "pI" = "cic:/CoRN/ftc/MoreFunctions/Basic_Properties/pI.var".
-inline cic:/CoRN/ftc/MoreFunctions/F.var.
+alias id "F" = "cic:/CoRN/ftc/MoreFunctions/Basic_Properties/F.var".
-inline cic:/CoRN/ftc/MoreFunctions/G.var.
+alias id "G" = "cic:/CoRN/ftc/MoreFunctions/Basic_Properties/G.var".
-inline cic:/CoRN/ftc/MoreFunctions/H.var.
+alias id "H" = "cic:/CoRN/ftc/MoreFunctions/Basic_Properties/H.var".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_wdl.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_wdl.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_wdr.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_wdr.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_unique.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_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".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_imp_Continuous'.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_imp_Continuous'.con".
(* UNEXPORTED
-End Basic_Properties.
+End Basic_Properties
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section More_Results.
+Section More_Results
*)
-inline cic:/CoRN/ftc/MoreFunctions/I.var.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/More_Results/I.var".
-inline cic:/CoRN/ftc/MoreFunctions/pI.var.
+alias id "pI" = "cic:/CoRN/ftc/MoreFunctions/More_Results/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.
+alias id "F" = "cic:/CoRN/ftc/MoreFunctions/More_Results/F.var".
-inline cic:/CoRN/ftc/MoreFunctions/F'.var.
+alias id "F'" = "cic:/CoRN/ftc/MoreFunctions/More_Results/F'.var".
-inline cic:/CoRN/ftc/MoreFunctions/G.var.
+alias id "G" = "cic:/CoRN/ftc/MoreFunctions/More_Results/G.var".
-inline cic:/CoRN/ftc/MoreFunctions/G'.var.
+alias id "G'" = "cic:/CoRN/ftc/MoreFunctions/More_Results/G'.var".
-inline cic:/CoRN/ftc/MoreFunctions/derF.var.
+alias id "derF" = "cic:/CoRN/ftc/MoreFunctions/More_Results/derF.var".
-inline cic:/CoRN/ftc/MoreFunctions/derG.var.
+alias id "derG" = "cic:/CoRN/ftc/MoreFunctions/More_Results/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/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_const.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_id.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_id.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_plus.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_plus.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_inv.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_inv.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_minus.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_minus.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_mult.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_mult.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_scal.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_scal.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_nth.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_nth.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_recip.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_recip.con".
(* UNEXPORTED
-End More_Results.
+End More_Results
*)
(* UNEXPORTED
-Section More_Corollaries.
+Section More_Corollaries
*)
-inline cic:/CoRN/ftc/MoreFunctions/I.var.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/More_Corollaries/I.var".
-inline cic:/CoRN/ftc/MoreFunctions/pI.var.
+alias id "pI" = "cic:/CoRN/ftc/MoreFunctions/More_Corollaries/pI.var".
-inline cic:/CoRN/ftc/MoreFunctions/F.var.
+alias id "F" = "cic:/CoRN/ftc/MoreFunctions/More_Corollaries/F.var".
-inline cic:/CoRN/ftc/MoreFunctions/F'.var.
+alias id "F'" = "cic:/CoRN/ftc/MoreFunctions/More_Corollaries/F'.var".
-inline cic:/CoRN/ftc/MoreFunctions/G.var.
+alias id "G" = "cic:/CoRN/ftc/MoreFunctions/More_Corollaries/G.var".
-inline cic:/CoRN/ftc/MoreFunctions/G'.var.
+alias id "G'" = "cic:/CoRN/ftc/MoreFunctions/More_Corollaries/G'.var".
-inline cic:/CoRN/ftc/MoreFunctions/derF.var.
+alias id "derF" = "cic:/CoRN/ftc/MoreFunctions/More_Corollaries/derF.var".
-inline cic:/CoRN/ftc/MoreFunctions/derG.var.
+alias id "derG" = "cic:/CoRN/ftc/MoreFunctions/More_Corollaries/derG.var".
(* begin show *)
-inline cic:/CoRN/ftc/MoreFunctions/Gbnd.var.
+alias id "Gbnd" = "cic:/CoRN/ftc/MoreFunctions/More_Corollaries/Gbnd.var".
(* end show *)
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_div.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_div.con".
(* UNEXPORTED
-End More_Corollaries.
+End More_Corollaries
*)
(* UNEXPORTED
-Section More_Sums.
+Section More_Sums
*)
-inline cic:/CoRN/ftc/MoreFunctions/I.var.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/More_Sums/I.var".
-inline cic:/CoRN/ftc/MoreFunctions/pI.var.
+alias id "pI" = "cic:/CoRN/ftc/MoreFunctions/More_Sums/pI.var".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_Sumx.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_Sumx.con".
(* begin show *)
-inline cic:/CoRN/ftc/MoreFunctions/f.var.
+alias id "f" = "cic:/CoRN/ftc/MoreFunctions/More_Sums/f.var".
-inline cic:/CoRN/ftc/MoreFunctions/f'.var.
+alias id "f'" = "cic:/CoRN/ftc/MoreFunctions/More_Sums/f'.var".
-inline cic:/CoRN/ftc/MoreFunctions/derF.var.
+alias id "derF" = "cic:/CoRN/ftc/MoreFunctions/More_Sums/derF.var".
(* end show *)
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_Sum0.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_Sum0.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_Sum.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_Sum.con".
(* UNEXPORTED
-End More_Sums.
+End More_Sums
*)
(* UNEXPORTED
-Section Diffble_Basic_Properties.
+Section Diffble_Basic_Properties
*)
(*#* **Differentiability
Mutatis mutandis for differentiability.
*)
-inline cic:/CoRN/ftc/MoreFunctions/I.var.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Basic_Properties/I.var".
-inline cic:/CoRN/ftc/MoreFunctions/pI.var.
+alias id "pI" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Basic_Properties/pI.var".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_imp_inc.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_imp_inc.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_imp_Diffble.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_imp_Diffble.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_wd.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_wd.con".
-inline cic:/CoRN/ftc/MoreFunctions/F.var.
+alias id "F" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Basic_Properties/F.var".
-inline cic:/CoRN/ftc/MoreFunctions/G.var.
+alias id "G" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Basic_Properties/G.var".
-inline cic:/CoRN/ftc/MoreFunctions/diffF.var.
+alias id "diffF" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Basic_Properties/diffF.var".
-inline cic:/CoRN/ftc/MoreFunctions/diffG.var.
+alias id "diffG" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Basic_Properties/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/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_const.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_id.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_id.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_plus.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_plus.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_inv.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_inv.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_minus.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_minus.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_mult.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_mult.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_nth.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_nth.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_scal.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_scal.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_recip.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_recip.con".
(* UNEXPORTED
-End Diffble_Basic_Properties.
+End Diffble_Basic_Properties
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section Diffble_Corollaries.
+Section Diffble_Corollaries
*)
-inline cic:/CoRN/ftc/MoreFunctions/I.var.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Corollaries/I.var".
-inline cic:/CoRN/ftc/MoreFunctions/pI.var.
+alias id "pI" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Corollaries/pI.var".
-inline cic:/CoRN/ftc/MoreFunctions/F.var.
+alias id "F" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Corollaries/F.var".
-inline cic:/CoRN/ftc/MoreFunctions/G.var.
+alias id "G" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Corollaries/G.var".
-inline cic:/CoRN/ftc/MoreFunctions/diffF.var.
+alias id "diffF" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Corollaries/diffF.var".
-inline cic:/CoRN/ftc/MoreFunctions/diffG.var.
+alias id "diffG" = "cic:/CoRN/ftc/MoreFunctions/Diffble_Corollaries/diffG.var".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_div.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_div.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_Sum0.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_Sum0.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_Sumx.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_Sumx.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_Sum.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_Sum.con".
(* UNEXPORTED
-End Diffble_Corollaries.
+End Diffble_Corollaries
*)
(* UNEXPORTED
-Section Nth_Derivative.
+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.
+alias id "I" = "cic:/CoRN/ftc/MoreFunctions/Nth_Derivative/I.var".
-inline cic:/CoRN/ftc/MoreFunctions/pI.var.
+alias id "pI" = "cic:/CoRN/ftc/MoreFunctions/Nth_Derivative/pI.var".
(* UNEXPORTED
-Section Definitions.
+Section Definitions
*)
(*#*
%\end{convention}%
*)
-inline cic:/CoRN/ftc/MoreFunctions/n.var.
+alias id "n" = "cic:/CoRN/ftc/MoreFunctions/Nth_Derivative/Definitions/n.var".
-inline cic:/CoRN/ftc/MoreFunctions/F.var.
+alias id "F" = "cic:/CoRN/ftc/MoreFunctions/Nth_Derivative/Definitions/F.var".
-inline cic:/CoRN/ftc/MoreFunctions/diffF.var.
+alias id "diffF" = "cic:/CoRN/ftc/MoreFunctions/Nth_Derivative/Definitions/diffF.var".
-inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_fun.con.
+inline "cic:/CoRN/ftc/MoreFunctions/N_Deriv_fun.con".
-inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_char
- (* begin hide *).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_strext.con".
-inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_wd.con.
+inline "cic:/CoRN/ftc/MoreFunctions/N_Deriv_wd.con".
-inline cic:/CoRN/ftc/MoreFunctions/N_Deriv.con.
+inline "cic:/CoRN/ftc/MoreFunctions/N_Deriv.con".
(* UNEXPORTED
-End Definitions.
+End Definitions
*)
(* UNEXPORTED
-Section Basic_Results.
+Section Basic_Results
*)
(*#*
All the usual results hold.
*)
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_n_wd.con.
+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_wdr.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_wdl.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_n_wdl.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_unique.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/Diffble_n_imp_Diffble.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_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/le_imp_Diffble_n.con".
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_n_imp_le.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/Diffble_n_imp_inc.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_imp_Diffble_n.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/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_Derivative_n.con".
-inline cic:/CoRN/ftc/MoreFunctions/included_imp_Diffble_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_Derivative_n.con".
-inline cic:/CoRN/ftc/MoreFunctions/Included_imp_Diffble_n.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Included_imp_Diffble_n.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_plus.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_n_plus.con".
(* UNEXPORTED
-End Basic_Results.
+End Basic_Results
*)
(* UNEXPORTED
-Section More_Results.
+Section More_Results
*)
(*#*
Some new results hold, too:
*)
-inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_Feq.con.
+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_lemma.con".
-inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_S.con.
+inline "cic:/CoRN/ftc/MoreFunctions/N_Deriv_S.con".
-inline cic:/CoRN/ftc/MoreFunctions/N_Deriv_plus.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_O.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_Sn.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_n_Sn.con".
(* UNEXPORTED
-End More_Results.
+End More_Results
*)
(* UNEXPORTED
-Section Derivating_Diffble.
+Section Derivating_Diffble
*)
(*#*
As a special case we get a differentiation operator%\ldots%#...#
*)
-inline cic:/CoRN/ftc/MoreFunctions/F.var.
+alias id "F" = "cic:/CoRN/ftc/MoreFunctions/Nth_Derivative/Derivating_Diffble/F.var".
(* begin show *)
-inline cic:/CoRN/ftc/MoreFunctions/diffF.var.
+alias id "diffF" = "cic:/CoRN/ftc/MoreFunctions/Nth_Derivative/Derivating_Diffble/diffF.var".
(* end show *)
-inline cic:/CoRN/ftc/MoreFunctions/Diffble_imp_Diffble_n.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Diffble_imp_Diffble_n.con".
-inline cic:/CoRN/ftc/MoreFunctions/Deriv.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Deriv.con".
(* UNEXPORTED
-End Derivating_Diffble.
+End Derivating_Diffble
*)
(* UNEXPORTED
-Section Corollaries.
+Section Corollaries
*)
(*#*
%\ldots%#...# for which the expected property also holds.
*)
-inline cic:/CoRN/ftc/MoreFunctions/Deriv_lemma.con.
+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_1.con".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_chain.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".
-inline cic:/CoRN/ftc/MoreFunctions/Derivative_n_imp_Continuous'.con.
+inline "cic:/CoRN/ftc/MoreFunctions/Derivative_n_imp_Continuous'.con".
(* UNEXPORTED
-End Corollaries.
+End Corollaries
*)
(* UNEXPORTED
-End Nth_Derivative.
+End Nth_Derivative
*)
(* UNEXPORTED