1. Meaning Of ARYTM_2func 1 (with Origins) func dedekind_cuts Element of bool bool RAT+ equals ARYTM_2def 1 {b1 where b1 is Element of bool RAT+ for b2 being Element of RAT+ st b2 in b1 http://merak.pb.bialystok.pl/cgi-bin/mmlquery/meaning?entry=ARYTM_2:func 1

2. The Mizar Abstract Of ARYTM_2 end; theorem ARYTM_23 not ex y being set st {},y in REAL+; definition let x be Element of dedekind_cuts; func GLUED x Element of REAL+ means http://www.cs.ualberta.ca/~piotr/Mizar/mirror/http/JFM/Addenda/arytm_2.abs.html

Journal of Formalized Mathematics Addenda , 1998 University of Bialystok Association of Mizar Users The abstract of the Mizar article: Non-Negative Real Numbers. Part I by Andrzej Trybulec Received March 7, 1998 MML identifier: ARYTM_2 Mizar article MML identifier index environ vocabulary ARYTM_3, BOOLE, ORDINAL2, ORDINAL1, ARYTM_2, HAHNBAN; notation TARSKI ; constructors ; clusters ; requirements BOOLE SUBSET ; begin reserve r,s,t,x',y',z',p,q for Element of RAT+ ; definition func Subset of bool RAT+ Subset of RAT+ : r in A implies (for s st s r holds s in in RAT+ ; end; definition cluster func REAL+ equals :: ARYTM_2:def 2 RAT+ Element of REAL+ ; theorem :: ARYTM_2:1 RAT+ c= REAL+ ; theorem :: ARYTM_2:2 omega c= REAL+ ; definition cluster REAL+ func Element of means :: ARYTM_2:def 3 ex r st x in RAT+ otherwise it x; end; theorem :: ARYTM_2:3 not ex y being set st [ ,y in REAL+ ; definition let x be Element of func Element of REAL+ means :: ARYTM_2:def 4 ex r st it in x iff s r if ex r st for s holds s in x iff s r otherwise it x; end; definition let x,y be

3. Article ARYTM_2, MML Version 4.97.1001 ARYTM_2funcnot 1 ANTONYM b sub 2 /sub b sub 1 /sub otherwise it = a sub 1 /sub ; end; ARYTM_2def 4 theorem for B1 being Element of dedekind_cuts for B2 being http://mmlquery.mizar.org/mizarmode/gab/arytm_2.gab.raw

4. Non Negative Real Numbers. Part I By Andrzej Trybulec definition let x be Element of dedekind_cuts; func GLUED x Element of definition let A,B be Element of dedekind_cuts; func A + B - Element of http://markun.cs.shinshu-u.ac.jp/mizar/abstr/arytm_2.abs

< r if ex r st for s holds s in x iff s < r otherwise it = x; end; definition let x,y be Element of REAL+; pred x < x for x <=' y; theorem :: ARYTM_2:10 x <=' y implies ex z st x + z = y; theorem :: ARYTM_2:11 ex z st x + z = y or y + z = x; theorem :: ARYTM_2:12 x + y = x + z implies y = z; theorem :: ARYTM_2:13 x *' (y *' z) = x *' y *' z; theorem :: ARYTM_2:14 x *' (y + z) = (x *' y) + (x *' z); theorem :: ARYTM_2:15 x <=' x; theorem :: ARYTM_2:20 x = y + z implies z

5. Non Negative Real Numbers. Part I By Andrzej Trybulec definition func dedekind_cuts Subset of bool RAT+ equals definition let x be Element of dedekind_cuts; func GLUED x - Element of REAL+ means http://www.wakasato.org/mizar/s6.4.02m3.66.807/share/abstr/arytm_2.abs

< r if ex r st for s holds s in x iff s < r otherwise it = x; end; definition let x,y be Element of REAL+; pred x <=' y; theorem :: ARYTM_2:10 x <=' y implies ex z st x + z = y; theorem :: ARYTM_2:11 ex z st x + z = y or y + z = x; theorem :: ARYTM_2:12 x + y = x + z implies y = z; theorem :: ARYTM_2:13 x *' (y *' z) = x *' y *' z; theorem :: ARYTM_2:14 x *' (y + z) = (x *' y) + (x *' z); theorem :: ARYTM_2:15 x <=' x; theorem :: ARYTM_2:20 x = y + z implies z

6. MML Query Rendering Of ATP Proof Steps Astep 62 for x1 being set st x1 is Element of dedekind_cuts for x2 being set st x2 is Element of dedekind_cuts holds x1 + x2 = a_2_0_arytm_2(x1,x2) http://lipa.ms.mff.cuni.cz/~urban/xmlmml/html_bytst/_by/arytm_2/1977_43.html

MML Query rendering of ATP proof steps axiom: dt_c2_57 A:step 41 is Element of axiom: dt_c3_57 A:step 42 is Element of axiom: A:step 62 for x being set st x is Element of for x being set st x is Element of holds x x a_2_0_arytm_2(x ,x conjecture: A:step 63 in axiom: A:step 64 in inference: assume_negation( A:step 65 not in inference: fof_simplification( A:step 80 not in inference: split_conjunct( A:step 215 is Element of inference: split_conjunct( A:step 216 is Element of inference: fof_nnf( A:step 322 for x being set st x is Element of for x being set st x is Element of holds x x a_2_0_arytm_2(x ,x inference: variable_rename( A:step 323 for x being set st x is Element of for x being set st x is Element of holds x x a_2_0_arytm_2(x ,x inference: shift_quantors( A:step 324 for x , x being set st x is Element of x x a_2_0_arytm_2(x ,x holds x is not Element of inference: split_conjunct( A:step 325 x x a_2_0_arytm_2(x ,x x is Element of implies x is not Element of inference: split_conjunct( A:step 326 not in inference: split_conjunct( A:step 327 in inference: spm( A:step 522 not in is Element of implies is not Element of inference: rw( A:step 526 not in TRUE implies is not Element of inference: rw( A:step 527 not in TRUE implies contradiction inference: cn( A:step 528 in inference: sr( A:step 529 contradiction A:step 530 contradiction

7. MML Query func a1 + a2 Element of dedekind_cuts equals for a1, a2 being Element of dedekind_cuts holds a1 + a2 = a2 + a1; end;; ARYTM_2func 6 definition http://vega.wi.pb.edu.pl/mmlquery/fillin.php?query=list of constr where commutat

8. ARYTM_2 Semantic Presentation Show TPTP Formulae Showing IDV dedekind_cuts = { A where A is Subset of RAT+ for r being Element of RAT+ st r in . ( x in RAT+ implies ex b1 being Element of dedekind_cuts ex r being http://www.cs.miami.edu/~tptp/MizarTPTP/Articles/arytm_2.html

:: ARYTM_2 semantic presentation :: Showing IDV graph ... (Click the Palm Trees again to close it) definition func Subset-Family of RAT+ equals :: ARYTM_2:def 1 A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s r holds s in A s being Element of RAT+ st s in A r s RAT+ coherence A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s r holds s in A s being Element of RAT+ st s in A r s RAT+ is Subset-Family of RAT+ proof end; end; deftheorem defines ARYTM_2:def 1 : A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s r holds s in A s being Element of RAT+ st s in A r s RAT+ registration cluster non empty coherence not is empty proof end; end; definition func REAL+ set equals :: ARYTM_2:def 2 RAT+ s where s is Element of RAT+ s t where t is Element of RAT+ t coherence RAT+ s where s is Element of RAT+ s t where t is Element of RAT+ t is set end; deftheorem defines REAL+ ARYTM_2:def 2 : REAL+ RAT+ s where s is Element of RAT+ s t where t is Element of RAT+ t set IR A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s r holds s in A s being Element of RAT+ st s in A r s set RA s where s is Element of RAT+ s t where t is Element of RAT+ t : for x y being set holds not x y in A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s r holds s in A s being Element of RAT+ st s in A r s proof end;

9. Template Item.constructors For ARYTM_2dfs 8 func DEDEKIND_CUT a1 Element of dedekind_cuts means (RAT+ \/ dedekind_cuts) \ {{b2 where b2 is Element of RAT+ b2 b1} where b1 is Element of RAT+ http://megrez.mizar.org/cgi-bin/mmlquery/meaning?filledfilename=item.constructor

10. The Mizar Article ARYTM_2 definition let x; func DEDEKIND_CUT x Element of dedekind_cuts means Def3 ex proof thus x in RAT+ implies ex IT being Element of dedekind_cuts, http://mizar.uwb.edu.pl/JFM/Addenda/arytm_2.miz.html

The Mizar article: Non-Negative Real Numbers. Part I by Andrzej Trybulec Received March 7, 1998 Association of Mizar Users MML identifier: MML identifier index Back to top

11. MML Query, Version 1.4.01, MML 4.95.999 dedekind_cuts ; = ARYTM_2funcnot 1 = ARYTM_2func 1 ; ARYTM_2dfs 2 = ARYTM_2func 2 = REAL+ ; ARYTM_2def 2 = ARYTM_2func 2 = REAL+ http://212.33.66.35/cgi-bin/mmlquery/fire?input=ARYTM_3:func 5 occur ordered by

12. WoW Forums - Post Search dedekind_cuts Not my proof don t want to take credit - I just like it because it starts very basic. The wiki article just shows you the outlin. http://forums.worldofwarcraft.com/search.html;jsessionid=448146F95BB3CA94706691A

13. Dedekind Cuts Wisdom Archive. Body Mind and Soul. Faith and Belief. God and Religion. Law of Attraction. Life and Beyond. Love and Happiness. Peace of Mind http://www.experiencefestival.com/dedekind_cuts

Articles Archives Start page News Contact Community General Newsletter Contact information Site map Most recommended Search the site Archive Photo Archive Video Archive Articles Archive More ... Wisdom Archive Body Mind and Soul Faith and Belief God and Religion ... Yoga Positions Site map 2 Site map Dedekind cuts A Wisdom Archive on Dedekind cuts Dedekind cuts A selection of articles related to Dedekind cuts More material related to Dedekind Cuts can be found here: Index of Articles related to Dedekind Cuts Dedekind cuts Page 2 ARTICLES RELATED TO Dedekind cuts Dedekind cuts: Encyclopedia - Mathematical analysis Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. These topics are often studied in the context of real numbers, complex numbers, and their functions. However, they can also be defined and studied in any space of mathematical objects that is equipped with a definition of "nearness" (a topological space) or "distance" (a metric space). Mathematical analysis ... Including: Mathematical analysis - History Mathematical analysis - Subdivisions Read more here: Dedekind cuts: Encyclopedia - Upper bound In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. Formally, given a partially ordered set (P, ≤), an element u of P is an upper bound of a subset S of P, if s ≤ u, for all elements s of S. Using ≥ instead of ≤ leads to ...

14. SUGSI-C004 Translate this page func dedekind_cuts - Subset-Family of RAT+ equals ARYTM_2def 1 cluster dedekind_cuts - non empty; end;. definition func REAL+ equals http://moodle.int-univ.com/mod/resource/view.php?id=607

15. Nabble - Mizar - PATE Deadline Extended To April 11, 2007 RAT+ \/ dedekind_cuts \ {{ s s t} t {}}; but making the definition in GRAPH_5 a redefinition of ARYTM_2def 2 looks like a horror. http://www.nabble.com/PATE-deadline-extended-to-April-11,-2007-td9785527.html

Nabble.setVar("skin",null); Nabble.page = 'forum.TopicDump'; Nabble.addCssRule(document.styleSheets[0],'.nabble a:link','color:'+document.linkColor); Nabble.addCssRule(document.styleSheets[0],'.nabble a:visited','color:'+document.vlinkColor); Nabble Software Math Software Mizar Nabble.userHeader(14353); PATE deadline extended to April 11, 2007 View: Threaded Chronologically All Messages Nabble.selectOption(Nabble.get("nabble.viewSelect"),Nabble.tview); New views Rating Filter: Alert me PATE deadline extended to April 11, 2007 document.write(Nabble.ratingStars(3)); by document.write(''); Adam Naumowicz document.write(''); document.write(Nabble.formatDateLong(new Date(1175497168000))); :: Rate this Message: Reply to Author View Threaded Show Only this Message Dear All, I hope this deadline extension can help some of you to contribute papers for this workshop. Best regards, Adam Naumowicz Final call for papers, the deadline for submission has been extended to April 11, 2007. Call for Papers RDP (RTA 07 + TLCA 07) Workshop PATE Proof Assistants and Types in Education June 25 2007 http://www.rdp07.org/pate.html

16. Dedekind_cuts吧 —— 维客(wiki) Translate this page dedekind_cuts dedekind_cuts dedekind_cuts http://post.wiki.cn/post/Dedekind_cuts

var skin = 'monobook';var stylepath = 'http://post.wiki.cn/skins'; Personal tools 提交Bug

17. Abcmiz_0.abs, Type;ABCMIZ_0mode.01 Type 36,1446 Noetherian ARYTM_1func.11 45833 -;ARYTM_1func.22 - 81,1555 arytm_2.abs, dedekind_cuts;ARYTM_2func.11 dedekind_cuts 23559 REAL+;ARYTM_2func.22 REAL+ http://ftp.icm.edu.pl/packages/mizar/version/abstr/symbtags

<%;AFINSQ_1:func.3:4 ;AFINSQ_1:func.4:5 197,4421 ^;AFINSQ_1:func.0:6^ 210,4574 ^;AFINSQ_1:func.5:7^ 268,5764 <%;AFINSQ_1:func.6:8 <%;AFINSQ_1:func.7:9 <%;AFINSQ_1:func.8:10 <%;AFINSQ_1:func.9:11 <%;ALGSEQ_1:func.3:3 <^;ALTCAT_1:func.1:1 <*;AMI_1:func.22:24 <*;AMI_1:func.23:25 <=;AMISTD_1:pred.1:1 <=;ARMSTRNG:pred.3:6 <=_;ARROW:pred.1:1 <=_ 90,2175 >=_;ARROW:pred.1:2>=_ 98,2258 <_;ARROW:pred.1:3 <_ 99,2293 >_;ARROW:pred.1:4>_ 100,2327 arytm_0.abs, +;ARYTM_0:func.1:1+ 52,1292 *;ARYTM_0:func.2:2* 63,1772 opp;ARYTM_0:func.3:3opp 81,2412 inv;ARYTM_0:func.4:4inv 86,2501 [*;ARYTM_0:func.5:5[* 107,2816 arytm_1.abs, -';ARYTM_1:func.1:1-' 45,833 -;ARYTM_1:func.2:2- 81,1555 arytm_2.abs, DEDEKIND_CUTS;ARYTM_2:func.1:1DEDEKIND_CUTS 22,536 REAL+;ARYTM_2:func.2:2REAL+ 33,800 DEDEKIND_CUT;ARYTM_2:func.3:3DEDEKIND_CUT 53,1079 GLUED;ARYTM_2:func.4:4GLUED 64,1338 <=';ARYTM_2:pred.1:1 <;ARYTM_2:pred.1:2 <=';ARYTM_3:pred.3:3 <;ARYTM_3:pred.3:4 <=;BCIALG_1:pred.1:7 ;BINARI_5:func.0:3 ;BINARI_5:func.3:4 40,754 binarith.abs, Tuple;BINARITH:mode.0:1Tuple 27,979 'or';BINARITH:func.1:1'or' 47,1455 'xor';BINARITH:func.2:2'xor' 48,1503 -';BINARITH:func.3:3-' 101,2284 'not';BINARITH:func.4:4'not' 118,2622 carry;BINARITH:func.5:5carry 125,2806 Binary;BINARITH:func.6:6Binary 136,3076 Absval;BINARITH:func.7:7Absval 145,3271 +;BINARITH:func.8:8+ 153,3389 add_ovfl;BINARITH:func.9:9add_ovfl 162,3563 are_summable;BINARITH:pred.1:1are_summable 171,3761 ^;BINARITH:func.10:10^ 191,4245

18. NiceConstructorNames < Mizar < Mizar TWiki constr_name(r3_arytm_3, = ,_). constr_name(k1_arytm_2, dedekind_cuts ,_). constr_name(k2_arytm_2, REAL+ ,_). constr_name(k3_arytm_2, DEDEKIND_CUT ,_). http://wiki.mizar.org/cgi-bin/twiki/view/Mizar/NiceConstructorNames

Skip to topic Skip to bottom Jump: Mizar Mizar Web Mizar Web Home Changes Index Search Webs Main Mizar Sandbox TWiki ... Test My links My home page edit Big Tits Videos Edit ... Printable Mizar.NiceConstructorNames r1.6 - 13 Jul 2006 - 02:18 - JosefUrban topic end Skip to actions Pretty Names for Mizar Constructors Set MML = http://mmlquery.mizar.org/mml/4.48.930/ Here is the table of constructors for MML version 4.48.930 - see http://mmlquery.mizar.org/mml/4.48.930/ . First comes the (linked) constructor, then its Mizar symbol (made Prolog-parsable), and finally the suggested nice both human and machine usable name. If '_' is there instead, no one supplied the nice name yet. Feel free to add/edit the nice names, or discuss here the naming conventions and related things. The nice names should only contain characters [a-zA-Z0-9_], and start with a downcase letter. This perl one-liner will produce a sed script for renaming the constructors into the nice names from the following table (if you save it as foo), and check for duplicate names: This perl one-liner will print the constructors (in the non-nice syntax) used in a file foo (useful for checking that all constructors have been nicely named): A reasonably automated rule for naming the selectors (strating with 'u') seems to be prepending 'the_' to it. Some hyphens need to be changed to underscore, otherwise not many clashes.

19. Constr_name( A Href=%MML%hidden.html M1 M1_hidden /a ,set,set constr_name( a href=%MML%arytm_3.html R3 r3_arytm_3 /a , = apos; ,_). constr_name( a href=%MML%arytm_2.html K1 k1_arytm_2 /a , dedekind_cuts ,_). http://kti.ms.mff.cuni.cz/cgi-bin/viewcvs.cgi/MPTP2/constr_names1.pl?rev=1.5

20. Dedekind Cuts http//medlibrary.org/medwiki/dedekind_cuts. All Wikipedia text is available under the terms of the GNU Free Documentation License. http://medlibrary.org/medwiki/Dedekind_cuts

Welcome to the MedLibrary.org Wikipedia Supplement on Dedekind cuts Please Click to Return to Front Page We subscribe to the HONcode principles. Verify here Web medlibrary.org Dedekind cuts This MedLibrary.org supplementary page on Dedekind cuts is provided directly from the open source Wikipedia as a service to our readers. Please see the note below on authorship of this content, as well as the Wikipedia usage guidelines. To search for other content from our encyclopedia supplement, please use the form below: Related Sponsors BM Pharmacy Generic pharmaceuticals at unbeatable prices. Insomnia, men's health, hair health, pain control. bmpharmacy.com Online Pharmacy ... frugalmed.com Ads by Tiva In mathematics , a Dedekind cut , named after Richard Dedekind , in a totally ordered set S is a partition of it, ( A B ), such that A is closed downwards (meaning that for all a in A x a implies that x is in A as well) and B is closed upwards, and A contains no greatest element. The cut itself is, conceptually, the "gap" defined between A and B . The original and most important cases are Dedekind cuts for rational numbers and real numbers . Dedekind used cuts to prove the completeness of the reals without using the axiom of choice (proving the existence of a complete ordered field to be independent of said axiom). See also

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