:: Basic Properties of Extended Real Numbers :: by Noboru Endou , Katsumi Wasaki and Yasunari Shidama :: :: Received September 7, 2000 :: Copyright (c) 2000-2019 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies NUMBERS, SUPINF_1, REAL_1, RELAT_1, XXREAL_0, ARYTM_3, COMPLEX1, CARD_1, ARYTM_1, FINSEQ_1, CARD_3, FUNCT_1, SUBSET_1, SUPINF_2, NAT_1, FUNCT_2, FUNCOP_1, XBOOLE_0, TARSKI, ORDINAL4, PARTFUN1, RFINSEQ, XCMPLX_0, MESFUNC1; notations TARSKI, XBOOLE_0, SUBSET_1, CARD_1, ORDINAL1, NUMBERS, XXREAL_0, XXREAL_3, XCMPLX_0, XREAL_0, COMPLEX1, REAL_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, BINOP_1, FUNCT_2, DOMAIN_1, RFINSEQ, RFUNCT_3, RVSUM_1, FUNCOP_1, FINSEQ_1, FINSEQ_4, NAT_1, NAT_D, SUPINF_1, SUPINF_2, MEASURE6; constructors BINOP_1, DOMAIN_1, REAL_1, NAT_1, FINSEQ_4, FINSOP_1, RVSUM_1, SUPINF_2, RFINSEQ, RFUNCT_3, MEASURE6, BINARITH, BINOP_2, SUPINF_1, CLASSES1, NAT_D, RELSET_1, XXREAL_3, FINSEQ_1, FUNCT_1, FUNCT_2, INT_1, FUNCOP_1, COMPLEX1, ARYTM_2; registrations XBOOLE_0, FUNCT_1, ORDINAL1, RELSET_1, FUNCT_2, NUMBERS, XXREAL_0, XREAL_0, NAT_1, INT_1, FINSEQ_1, VALUED_0, CARD_1, XXREAL_3; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; begin :: Operations "x * y","x / y", "|.x.|" in R_eal numbers definition let x,y be R_eal; redefine func x * y -> R_eal; end; theorem :: EXTREAL1:1 for x,y being ExtReal holds for a,b being Real holds x = a & y = b implies x * y = a * b; definition let x,y be ExtReal; redefine func x / y -> R_eal; end; theorem :: EXTREAL1:2 for x,y being ExtReal holds for a,b being Real st x = a & y = b holds x / y = a / b; definition let x be ExtReal; func |. x .| -> R_eal equals :: EXTREAL1:def 1 x if 0 <= x otherwise -x; projectivity; end; registration let x be Real, a be Complex; identify |.x.| with |.a.| when x = a; end; theorem :: EXTREAL1:3 for x being R_eal st 0 <= x holds |. x .| = x; theorem :: EXTREAL1:4 for x being R_eal st x < 0 holds |. x .| = -x; registration let x be R_eal; cluster |.x.| -> non negative; end; ::\$CT 2 begin :: Sum of Finite Sequence of Extended Real Numbers :: from CONVFUN1, 2010.02.13, A.T. definition let F be FinSequence of ExtREAL; func Sum(F) -> R_eal means :: EXTREAL1:def 2 ex f being sequence of ExtREAL st it = f.(len F) & f.0 = 0. & for i being Nat st i < len F holds f.(i+1)= f.i+F.(i+1); end; theorem :: EXTREAL1:7 Sum(<*> ExtREAL) = 0.; theorem :: EXTREAL1:8 for a being R_eal holds Sum<*a*> = a; theorem :: EXTREAL1:9 for a,b being R_eal holds Sum<*a,b*> = a+b; theorem :: EXTREAL1:10 for F,G being FinSequence of ExtREAL st not -infty in rng F & not -infty in rng G holds Sum(F^G) = Sum(F)+Sum(G); theorem :: EXTREAL1:11 for F,G being FinSequence of ExtREAL, s being Permutation of dom F st G = F*s & not -infty in rng F holds Sum(F) = Sum(G); reserve x,y,w,z for ExtReal, a for Real; begin :: Basic properties of absolute value for extended real numbers theorem :: EXTREAL1:12 x = a implies |.x.| = |.a qua Complex.|; theorem :: EXTREAL1:13 |.x.| = x or |.x.| = -x; theorem :: EXTREAL1:14 0 <= |.x.|; theorem :: EXTREAL1:15 x <> 0 implies 0 < |.x.|; theorem :: EXTREAL1:16 x = 0 iff |.x.| = 0; theorem :: EXTREAL1:17 |.x.| = -x & x <> 0 implies x < 0; theorem :: EXTREAL1:18 x <= 0 implies |.x.| = -x; theorem :: EXTREAL1:19 |.x * y.| = |.x.| * |.y.|; theorem :: EXTREAL1:20 -|.x.| <= x & x <= |.x.|; theorem :: EXTREAL1:21 |.x.| < y implies -y < x & x < y; theorem :: EXTREAL1:22 -y < x & x < y implies 0 < y & |.x.| < y; theorem :: EXTREAL1:23 -y <= x & x <= y iff |.x.| <= y; theorem :: EXTREAL1:24 |.x + y.| <= |.x.| + |.y.|; theorem :: EXTREAL1:25 -infty < x & x < +infty & x <> 0 implies |.x.|*|. 1. / x .| = 1.; theorem :: EXTREAL1:26 x = +infty or x = -infty implies |.x.|*|. 1./x .| = 0; theorem :: EXTREAL1:27 x <> 0 implies |. 1./x .| = 1./|.x.|; theorem :: EXTREAL1:28 not((x=-infty or x=+infty) & (y=-infty or y=+infty)) & y<>0 implies |. x/y.| = |.x.|/|.y.|; theorem :: EXTREAL1:29 |.x.| = |.-x.|; theorem :: EXTREAL1:30 |.+infty.| = +infty & |.-infty.| = +infty; theorem :: EXTREAL1:31 x is Real or y is Real implies |.x.|-|.y.| <= |.x-y.|; theorem :: EXTREAL1:32 |.x-y.| <= |.x.| + |.y.|; ::\$CT theorem :: EXTREAL1:34 |.x.| <= z & |.y.| <= w implies |.x+y.| <= z + w; theorem :: EXTREAL1:35 x is Real or y is Real implies |.|.x.|-|.y.|.| <= |.x-y.|; theorem :: EXTREAL1:36 0 <= x * y implies |.x+y.| = |.x.|+|.y.|; begin theorem :: EXTREAL1:37 x <> +infty & y <> +infty & not ( x = +infty & y = +infty or x = -infty & y = -infty ) implies min(x,y) = (x + y - |.x - y.|) / 2; theorem :: EXTREAL1:38 x <> -infty & y <> -infty & not ( x = +infty & y = +infty or x = -infty & y = -infty ) implies max(x,y) = (x + y + |.x - y.|) / 2; definition let x,y be R_eal; redefine func max(x,y) -> R_eal; redefine func min(x,y) -> R_eal; end; theorem :: EXTREAL1:39 min(x,y) + max(x,y)= x + y; begin ::Addenda :: missing, 2007.07.20, A.T. theorem :: EXTREAL1:40 |.x.| = +infty implies x = +infty or x = -infty; theorem :: EXTREAL1:41 |.x.| < +infty iff x in REAL;