X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FCoRN-Decl%2Ftransc%2FRealPowers.ma;fp=matita%2Fcontribs%2FCoRN-Decl%2Ftransc%2FRealPowers.ma;h=fd081416a06d63a6b8fadb81dd41cb59cf548e21;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1 diff --git a/matita/contribs/CoRN-Decl/transc/RealPowers.ma b/matita/contribs/CoRN-Decl/transc/RealPowers.ma new file mode 100644 index 000000000..fd081416a --- /dev/null +++ b/matita/contribs/CoRN-Decl/transc/RealPowers.ma @@ -0,0 +1,197 @@ +(**************************************************************************) +(* ___ *) +(* ||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/transc/RealPowers". + +include "CoRN.ma". + +(* $Id: RealPowers.v,v 1.5 2004/04/23 10:01:08 lcf Exp $ *) + +(*#* printing [!] %\ensuremath{\hat{\ }}% #^# *) + +(*#* printing {!} %\ensuremath{\hat{\ }}% #^# *) + +include "transc/Exponential.ma". + +(* UNEXPORTED +Opaque Expon. +*) + +(*#* *Arbitrary Real Powers + +**Powers of Real Numbers + +We now define +$x^y=e^{y\times\log(x)}$#xy=ey*log(x)#, whenever +[x [>] 0], inspired by the rules for manipulating these expressions. +*) + +inline "cic:/CoRN/transc/RealPowers/power.con". + +(* NOTATION +Notation "x [!] y [//] Hy" := (power x y Hy) (at level 20). +*) + +(*#* +This definition yields a well defined, strongly extensional function +which extends the algebraic exponentiation to an integer power and +still has all the good properties of that operation; when [x [=] e] it +coincides with the exponential function. +*) + +inline "cic:/CoRN/transc/RealPowers/power_wd.con". + +inline "cic:/CoRN/transc/RealPowers/power_strext.con". + +inline "cic:/CoRN/transc/RealPowers/power_plus.con". + +inline "cic:/CoRN/transc/RealPowers/power_inv.con". + +(* UNEXPORTED +Hint Resolve power_wd power_plus power_inv: algebra. +*) + +inline "cic:/CoRN/transc/RealPowers/power_minus.con". + +inline "cic:/CoRN/transc/RealPowers/power_nat.con". + +(* UNEXPORTED +Hint Resolve power_minus power_nat: algebra. +*) + +inline "cic:/CoRN/transc/RealPowers/power_zero.con". + +inline "cic:/CoRN/transc/RealPowers/power_one.con". + +(* UNEXPORTED +Hint Resolve power_zero power_one: algebra. +*) + +(* UNEXPORTED +Opaque nexp_op. +*) + +inline "cic:/CoRN/transc/RealPowers/power_int.con". + +(* UNEXPORTED +Hint Resolve power_int: algebra. +*) + +inline "cic:/CoRN/transc/RealPowers/Exp_power.con". + +inline "cic:/CoRN/transc/RealPowers/mult_power.con". + +inline "cic:/CoRN/transc/RealPowers/recip_power.con". + +(* UNEXPORTED +Hint Resolve Exp_power mult_power recip_power: algebra. +*) + +inline "cic:/CoRN/transc/RealPowers/div_power.con". + +(* UNEXPORTED +Hint Resolve div_power: algebra. +*) + +inline "cic:/CoRN/transc/RealPowers/power_ap_zero.con". + +inline "cic:/CoRN/transc/RealPowers/power_mult.con". + +inline "cic:/CoRN/transc/RealPowers/power_pos.con". + +(* UNEXPORTED +Hint Resolve power_mult: algebra. +*) + +inline "cic:/CoRN/transc/RealPowers/power_recip.con". + +(* UNEXPORTED +Hint Resolve power_recip: algebra. +*) + +inline "cic:/CoRN/transc/RealPowers/power_div.con". + +(* UNEXPORTED +Hint Resolve power_div: algebra. +*) + +(* UNEXPORTED +Section Power_Function +*) + +(*#* **Power Function + +This operation on real numbers gives birth to an analogous operation +on partial functions which preserves continuity. + +%\begin{convention}% Let [F, G : PartIR]. +%\end{convention}% +*) + +alias id "J" = "cic:/CoRN/transc/RealPowers/Power_Function/J.var". + +alias id "F" = "cic:/CoRN/transc/RealPowers/Power_Function/F.var". + +alias id "G" = "cic:/CoRN/transc/RealPowers/Power_Function/G.var". + +inline "cic:/CoRN/transc/RealPowers/FPower.con". + +inline "cic:/CoRN/transc/RealPowers/FPower_domain.con". + +inline "cic:/CoRN/transc/RealPowers/Continuous_power.con". + +(* UNEXPORTED +End Power_Function +*) + +(* NOTATION +Notation "F {!} G" := (FPower F G) (at level 20). +*) + +(* UNEXPORTED +Section More_on_Power_Function +*) + +(* UNEXPORTED +Opaque Expon Logarithm. +*) + +(*#* From global continuity we can obviously get local continuity: *) + +inline "cic:/CoRN/transc/RealPowers/continuous_I_power.con". + +(*#* The rule for differentiation is a must. *) + +(* UNEXPORTED +Transparent Logarithm. +*) + +(* UNEXPORTED +Opaque Logarithm. +*) + +inline "cic:/CoRN/transc/RealPowers/Derivative_power.con". + +inline "cic:/CoRN/transc/RealPowers/Diffble_power.con". + +(* UNEXPORTED +End More_on_Power_Function +*) + +(* UNEXPORTED +Hint Resolve Derivative_power: derivate. +*) +