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[9] Underfull \hbox (badness 10000) in paragraph at lines 578--585 [] Underfull \hbox (badness 10000) in paragraph at lines 586--597 [] Underfull \hbox (badness 10000) in paragraph at lines 606--611 [] \lgwproofline=\count95 Underfull \hbox (badness 10000) in paragraph at lines 612--823 [] [10] [11] Underfull \hbox (badness 10000) in paragraph at lines 1253--1256 [] [12] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 1273. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 1273. Underfull \hbox (badness 10000) in paragraph at lines 1277--1281 [] Underfull \hbox (badness 10000) in paragraph at lines 1282--1305 [] Underfull \hbox (badness 10000) in paragraph at lines 1306--1317 [] \lgwproofline=\count96 Underfull \hbox (badness 10000) in paragraph at lines 1320--1495 [] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 1502. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 1502. Underfull \hbox (badness 10000) in paragraph at lines 1504--1508 [] [13] Underfull \hbox (badness 10000) in paragraph at lines 1509--1530 [] Underfull \hbox (badness 10000) in paragraph at lines 1531--1554 [] \lgwproofline=\count97 Underfull \hbox (badness 10000) in paragraph at lines 1557--1917 [] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 1926. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 1926. Underfull \hbox (badness 10000) in paragraph at lines 1930--1931 [] Underfull \hbox (badness 10000) in paragraph at lines 1932--1955 [] Underfull \hbox (badness 10000) in paragraph at lines 1956--1971 [] \lgwproofline=\count98 Underfull \hbox (badness 10000) in paragraph at lines 1974--2145 [] [14] Overfull \hbox (9.98274pt too wide) in paragraph at lines 2146--2148 []\OT1/cmr/m/n/10 Thus having pro-ved $\U/euf/m/n/10 [\OT1/cmr/m/n/10 M Proposi tion 3\U/eur/m/n/10 :\OT1/cmr/m/n/10 2\U/euf/m/n/10 (\OT1/cmr/m/n/10 b\U/euf/m/ n/10 )]$ \OT1/cmr/m/n/10 we pro-ceed to $\U/euf/m/n/10 [\OT1/cmr/m/n/10 M Propo sition 3\U/eur/m/n/10 :\OT1/cmr/m/n/10 2\U/euf/m/n/10 (\OT1/cmr/m/n/10 c\U/euf/ m/n/10 )]$| [] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 2152. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 2152. Underfull \hbox (badness 10000) in paragraph at lines 2154--2158 [] Underfull \hbox (badness 10000) in paragraph at lines 2159--2180 [] Underfull \hbox (badness 10000) in paragraph at lines 2181--2200 [] \lgwproofline=\count99 Underfull \hbox (badness 10000) in paragraph at lines 2203--2456 [] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 2460. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 2460. Overfull \hbox (37.87791pt too wide) in paragraph at lines 2464--2466 \OT1/cmr/m/n/10 As no-ted above we will make use of $\U/euf/m/n/10 [\OT1/cmr/m/ n/10 M Tautology B\U/euf/m/n/10 ]$ \OT1/cmr/m/n/10 in our proof of $\U/euf/m/n/ 10 [\OT1/cmr/m/n/10 M Proposition 3\U/eur/m/n/10 :\OT1/cmr/m/n/10 2\U/euf/m/n/1 0 (\OT1/cmr/m/n/10 c\U/euf/m/n/10 )]$\OT1/cmr/m/n/10 ,| [] Underfull \hbox (badness 10000) in paragraph at lines 2464--2466 [] Underfull \hbox (badness 10000) in paragraph at lines 2467--2490 [] Underfull \hbox (badness 10000) in paragraph at lines 2491--2512 [] \lgwproofline=\count100 Underfull \hbox (badness 10000) in paragraph at lines 2515--2673 [] [15] Overfull \hbox (43.03835pt too wide) in paragraph at lines 2674--2675 []\OT1/cmr/m/n/10 Thus having pro-ved $\U/euf/m/n/10 [\OT1/cmr/m/n/10 M Proposi tion 3\U/eur/m/n/10 :\OT1/cmr/m/n/10 2\U/euf/m/n/10 (\OT1/cmr/m/n/10 c\U/euf/m/ n/10 )]$ \OT1/cmr/m/n/10 we now pro-ceed with $\U/euf/m/n/10 [\OT1/cmr/m/n/10 M Proposition 3\U/eur/m/n/10 :\OT1/cmr/m/n/10 2\U/euf/m/n/10 (\OT1/cmr/m/n/10 d\ U/euf/m/n/10 )]$\OT1/cmr/m/n/10 . | [] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 2677. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 2677. Underfull \hbox (badness 10000) in paragraph at lines 2679--2684 [] Underfull \hbox (badness 10000) in paragraph at lines 2687--2710 [] Underfull \hbox (badness 10000) in paragraph at lines 2711--2732 [] \lgwproofline=\count101 Underfull \hbox (badness 10000) in paragraph at lines 2733--2942 [] Overfull \hbox (39.64948pt too wide) in paragraph at lines 2943--2945 []\OT1/cmr/m/n/10 Thus having pro-ved $\U/euf/m/n/10 [\OT1/cmr/m/n/10 M Proposi tion 3\U/eur/m/n/10 :\OT1/cmr/m/n/10 2\U/euf/m/n/10 (\OT1/cmr/m/n/10 d\U/euf/m/ n/10 )]$ \OT1/cmr/m/n/10 we now pro-ceed with $\U/euf/m/n/10 [\OT1/cmr/m/n/10 M Proposition 3\U/eur/m/n/10 :\OT1/cmr/m/n/10 2\U/euf/m/n/10 (\OT1/cmr/m/n/10 f\ U/euf/m/n/10 )]$| [] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 2947. 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Underfull \hbox (badness 10000) in paragraph at lines 4230--4256 [] Overfull \hbox (12.56895pt too wide) in paragraph at lines 4257--4289 $\U/euf/m/n/10 [\OT1/cmr/m/n/10 S[][]M Proposition 3\U/eur/m/n/10 :\OT1/cmr/m/n /10 2\U/euf/m/n/10 (\OT1/cmr/m/n/10 g\U/euf/m/n/10 ) (\OT1/cmr/m/n/10 ii\U/euf/ m/n/10 )\OT1/cmr/m/n/10 : [][]: \U/euf/m/n/10 ([] [] [] [] ([] [] [])[] [] [] [ ] [] [] ([] [] [])[])]$ | [] Underfull \hbox (badness 10000) in paragraph at lines 4257--4289 [] \lgwproofline=\count106 Underfull \hbox (badness 10000) in paragraph at lines 4405--4405 []$[] [] [] [] \U/euf/m/n/10 ([] [] [])[] [] [] Underfull \hbox (badness 10000) in paragraph at lines 4405--4405 \U/euf/m/n/10 ([] [] [])[] [] ([] [] [])[] [] [] Underfull \hbox (badness 10000) in paragraph at lines 4455--4455 []$\U/euf/m/n/10 ([] [] [])[] [] ([] [] [])[] [] [] Underfull \hbox (badness 10000) in paragraph at lines 4704--4704 []$[] [] [] [] \U/euf/m/n/10 ([] [] [])[] [] [] Underfull \hbox (badness 10000) in paragraph at lines 4704--4704 \U/euf/m/n/10 ([] [] [])[] [] ([] [] [])[] [] [] Underfull \hbox (badness 10000) in paragraph at lines 4810--4810 []$\U/euf/m/n/10 ([] [] [])[] [] ([] [] [])[] [] [] Underfull \hbox (badness 5817) in paragraph at lines 4935--4935 []$[][]\OT1/cmr/m/n/10 : \U/euf/m/n/10 ([] [] [] [] ([] [] [])[] [] [] Underfull \hbox (badness 10000) in paragraph at lines 4292--4990 [] [19] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 4994. 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Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 5353. Overfull \hbox (23.8277pt too wide) in paragraph at lines 5355--5358 \OT1/cmr/m/n/10 We now have the ne-ces-sary me-ans to show our ob-jective, na-m ely $M Proposition 3\U/eur/m/n/10 :\OT1/cmr/m/n/10 2\U/euf/m/n/10 (\OT1/cmr/m/n /10 h\U/euf/m/n/10 )$\OT1/cmr/m/n/10 .| [] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 5361. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 5361. Underfull \hbox (badness 10000) in paragraph at lines 5363--5368 [] [20] Underfull \hbox (badness 10000) in paragraph at lines 5369--5393 [] Underfull \hbox (badness 10000) in paragraph at lines 5394--5414 [] \lgwproofline=\count108 Underfull \hbox (badness 10000) in paragraph at lines 5417--5725 [] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 5732. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `subscript' on input line 5732. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 5732. Underfull \hbox (badness 10000) in paragraph at lines 5739--5741 [] Underfull \hbox (badness 10000) in paragraph at lines 5742--5770 [] Overfull \hbox (8.64034pt too wide) in paragraph at lines 5771--5803 $\U/euf/m/n/10 [\OT1/cmr/m/n/10 S[][]M Proposition 3\U/eur/m/n/10 :\OT1/cmr/m/n /10 2\U/euf/m/n/10 (\OT1/cmr/m/n/10 h\U/euf/m/n/10 )[]\OT1/cmr/m/n/10 : [][]: \ U/euf/m/n/10 ([] [] [] [] ([] [] [])[]) \OMS/cmsy/m/n/10 ` [] [] [] [] \U/euf/m /n/10 ([] [] [])[]]$ | [] Underfull \hbox (badness 10000) in paragraph at lines 5771--5803 [] \lgwproofline=\count109 Underfull \hbox (badness 3965) in paragraph at lines 5879--5879 []$[][]\OT1/cmr/m/n/10 : \U/euf/m/n/10 ([] [] [] [] ([] [] [])[]) [] [] Underfull \hbox (badness 4279) in paragraph at lines 5990--5990 []$[][]\OT1/cmr/m/n/10 : \U/euf/m/n/10 ([] [] [] [] ([] [] [])[]) [] [] Underfull \hbox (badness 4608) in paragraph at lines 6099--6099 []$[][]\OT1/cmr/m/n/10 : \U/euf/m/n/10 ([] [] [] [] ([] [] [])[]) [] [] Underfull \hbox (badness 10000) in paragraph at lines 5806--6188 [] [21] Overfull \hbox (46.8784pt too wide) in paragraph at lines 6189--6190 []\OT1/cmr/m/n/10 We now have the ne-ces-sary me-ans to prove the in-duction st ep of $M Proposition 3\U/eur/m/n/10 :\OT1/cmr/m/n/10 2\U/euf/m/n/10 (\OT1/cmr/m /n/10 h\U/euf/m/n/10 )(\OT1/cmr/m/n/10 ii\U/euf/m/n/10 )$ | [] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 6193. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 6193. Underfull \hbox (badness 10000) in paragraph at lines 6195--6198 [] Underfull \hbox (badness 10000) in paragraph at lines 6199--6223 [] Underfull \hbox (badness 10000) in paragraph at lines 6224--6252 [] \lgwproofline=\count110 Underfull \hbox (badness 10000) in paragraph at lines 6255--6881 [] [22] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 6887. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 6887. Underfull \hbox (badness 10000) in paragraph at lines 6889--6892 [] Underfull \hbox (badness 10000) in paragraph at lines 6893--6916 [] Underfull \hbox (badness 10000) in paragraph at lines 6917--6940 [] \lgwproofline=\count111 Underfull \hbox (badness 10000) in paragraph at lines 7017--7017 []$[][]\OT1/cmr/m/n/10 : \U/euf/m/n/10 ([] [] [] [] [] [] []) [] [] Underfull \hbox (badness 10000) in paragraph at lines 6943--7261 [] [23] [24] [25] [26] [27] (./page.bbl Underfull \hbox (badness 10000) in paragraph at lines 4--8 \OT1/cmr/m/n/10 20050502/home/grue/check/GRD-2005-05-26-UTC-06-49-32-964854/, [] Underfull \hbox (badness 1152) in paragraph at lines 16--20 []\OT1/cmr/m/n/10 Klaus Grue. \OT1/cmr/m/it/10 Pe-ano arit-h-me-tic\OT1/cmr/m/ n/10 . http://www.diku.dk/[]\OT1/cmtt/m/n/10 ~\OT1/cmr/m/n/10 grue/logiweb/ [] ) [28] (./page.aux) ) Here is how much of TeX's memory you used: 2674 strings out of 95835 36161 string characters out of 1195527 118770 words of memory out of 1000001 5554 multiletter control sequences out of 10000+50000 20823 words of font info for 82 fonts, out of 500000 for 1000 14 hyphenation exceptions out of 1000 24i,10n,39p,220b,330s stack positions out of 1500i,500n,5000p,200000b,5000s Output written on page.dvi (28 pages, 168024 bytes).