A Simple Deduction System for First-Order Logic with Equality, Free Constructors and Induction
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(56) > k JLK^_TVSbc¨ceXZW\ZJLsªTVS: k H k k k k k k cT\Sb[J:SUXM JLc ^ TV^#gUW\Jc^ JJZJ [ J^]Js¹- !:ºSUJL#J T\S[ JL^#cJLc KMX k JXZJ k !:»spJL[ k J^#cJLc¨ce] k TV[ ]WVdTVS ] H J[^ZceJ}X_¨] H J/.( v!u  w10 ^_Sb s .( v^u  2 00k abW\Jc H J k :J >ghJL[ XZKMJLc%cX#WV#Js¹^_Sbs I : IspJ[ k JL^#cJLc:T\S^#W\W H ¨ k k X_] J aUWVJLcL ^Z[ H¸k abW\Jªr k JLcJ k #JLc:] H J cJ ]4cX#^_WVWIaUSUT\¥bJ k cLAT\S] H J cJSbcJ] H ^] H JLSUJZJ k : Q : A] H J cJ ]MX_ aUSUT\¥bJ k cMX#b : T`cMJ ²U^Z[]WVd¯] H ^]X_k : X k JLX#J k A^_S¦T kk JLspab[T\gbW\Jcedpce]JLK : N X#SUJ]X¯ H T`[ H SbX k aUWVJ ^_rbrUW\TVJLc N K%abce]GghJMX_o] H J0X k K P¹X k > F v v > F )^_SsK%acm]«ghJMcX#WV#Js¹ IH JcaUgbce]T\]ap]T\XZS
(57) G > H´ G F v v > ´ G F TVc] H J k J X k JT`spJKMrhX_]JS]Lb^#Sbs[ WVJL^ k WVdaUSUT\¥bJL c : õS c H X k ]LpX k ^_Sfd}K^²pTVK^_W)cJLiaUJLSb[ %J :H Q : Q QW:NªX_ k aUWVJ:^_rbrUW\T`[^_]TVX#SbcLK :NK%acm]!ghJ P X k ^ªceXZW\ZJLs cdfce]JLK > F /v v > F T\SR] H J¥ k cm]«[^ZceJZ/ :HMTVcSUX_]aUSUT\¥^#gUW\JZ¹T\SR] H JcJL[X#Sbs[L^#cJ > ´ G F /v v > 0´ G F T`c¨^MKMXZce]¨Y#JSbJ k ^_W;aUSUT\¥bJ k X_O :@HZ ¬=J[^_S¸spJ¥bSUJ ML $^:H &«^Z\c P»TVS¯] H J¥ k cm]%[L^#cJ#^#Sbs¯^#c > ´ G F v v > ´ G F T\S¯] H J}cJL[ XZSbs [^ZceJZ;gUap]«oJ0¨T`c H ]Xªr k XZJ k JLcaUW\]cT\SbsUJrhJSbspJLS]WVdX_o^rb^ k ]TV[aUWV^ k ^#W\YZX k T] H K XªspX] H T`c)oJ0c H ^#W\W J ²prUWVX k J^dpcX_Br k XfsUab[ TVSUY^_WVW¹TVsUJKMrX#]JLSZ]aUSUT\¥bJ k cIX_ ^MY#TV#JSªcedpce]JLK :HZ¬J¥ k ce]X#gbcJ k #JZ´ 4 y^w76 # O: G , > F v v > F 4" # {\/G\ 4 / y## |"/õ# "[!3 > e@ : !"91 3 ½ 5 21M/ spXZ K /8 > v v > :91U# M % {## > H´ G F v v > ´ G F ; Á2Á <y ·UX k JL#J k ; d &4¹%y $&$L¹UceTVSb[ J aUSUT\¥bJc > ) ¨T\] H F ) UoJKabcm] H ^#kJ $ > ) & G F ) ¬=J¥ k cm][WV^#T\K ] H ^3] > ) T`c:TVS.spX#K 0 õSbspJLJLs¹^#ccaUKMJX#S¯] H J}[ XZSZ] k ^ k d] H ^_] $ > &/G > IH JLS > ) G F) õSRrb^ k ]TV[aUWV^ k F ) T`c^}^ k T`^_gUWV3J B;^_Ss¹´ $Ø^ &b $ B & G > ) fT\S[ J : T`cceXZW\ZJLs¹ ) F ) G > ) ) H JSb[J BG > X k JLX#J k $Ø^ &TVKMrUW\TVJLc¨] H ^_N] B?SspX#K bcX] H ^N] B}TVccX#KMJ >1=#)%y $/>;$C; >QG &4 ap]¨] H JQ S : [ XZS]^_TV) Sbc] H J:JLiab^_]TVX#SbEc > ) F ) ^#Sb s >1=U F =#U] H ^]T`cb] H J«JiZa^]TVX#Sc > ) >=:^#Sb s >1=U F =_ IH J k JX k J >1=«T`c¨SUX_]cX#WV#Js}TVQ S :U[ XZSZ] k ^ZspT`[]TVSUY0] H J±^#[]I] H ^] :@T`c¨ceXZW\ZJLs¹ fX´ $±g &IspXZ K TVcJ ²U^#[ ]WVdJLiab^#W;]X > v v > Z ¬=J:SUX¢[ W`^_TVK ] H ^)] $ > &EG F X k JZJ k ?d &4B\y $n&$A; fTVSb[\J ¯aUSUT\¥bJLc > F ) ´ $Ø[ & $ > ) &G F ) fTVSb[ J :8TVc cX#WV#Js¹ $ F ) & E > ) v v) > G
(58) U d $±g &ZT] X#WVWVXc)] H ^] $ F ) & EsUX#) K GT p õSrb^ k ]TV[aUW`^ k gfd $±[ &+ $ > ) &!G F ) pX k JZJ k ;d &¹%y $&$ j IH J JLKMKM^0] H JSªXZW\WVXco k XZK $g &^_Sbs $±[ & ? O TV#JS^}cX#WV#JLsRcedpce]JK3X# JLiab^_]TVX#Sb)c : G > F v v > F ¹[^_WV\W /| ¦spX#K :»X_ :»] H J cJ ] > v v > Z . . . . . . . . . . .
(59) y#e21Uo | LZ c/ ê_ |#y ! V õ#| m:98#!"/y/ } $ À 1+ & : v > B Q : B´HG> v B> 91U:. À + 0 ! \ Ø / _ | "# õ#| : "$ ©#| : 21M 1"KS1b |M #[e !3 > '9{kK !"91 21MspXZ K : G $±sUX#K : >` &/[ B+ ; ÁÁ2<y J @ ] : G : v >?B}^_Sb5 s : G , : B}´ G > v B>)k pT\Sb[%J : T`c¨ceXZW\ZJLs¹ >CG B; k H H ¬=J%¥ ce]Gc X°] ^_)] : T`ccX#WV#Js¹G¬ k T]J : ^#c > F v v > F ; H J k J%J^#[ H > ) y$x&@$ j T`cªcX#WV#Js£TVT S : Xc H X3] H ^_5 ] : T`cceXZW\ZJLs¹!SUX_]TV[JR] H ^] JL^Z[ H JLiab^_]TVX#S(X# : T`cX#:] H JRX k K > ) B}´ G> F ) BH´ G> X k B>) iab^]T\XZSbc > ) BH´ G> F ) B}´ G> ´AceTVSb[ J > ) T`c¨ceXZW\ZJLsTV5 S : > ) T\Srb^ k ]T`[ aUW`^ k spXfJcoSbX_]IXf[L[ a k TVS ~ >?B; H JLSb[ E J > ) G B;w fXJL^Z[ H cab[ H Jiab^]T\XZS%T`cX_] H JX k " K > F BH´ G> ¬J¨[WV^#T\K¢] H ^] > TVc ] H JLScX#WV#JLsT\7 S : J ] : gkJ : KMTVSac ] H JGJLiab^]T\XZJS > ) F ) ) & fT\S) [ kJ : TVcceXZW\ZJLs¹fJ H ^ZJ#´ ) $±^ & > ) TVcISbX_] k JLJGTVM S : $±g & > T`cSUX_]I k JJGT\S F U^#Sbs $Ø[ & > T`cSUX_]I k JJGT\ S >ZBhX õSªrb^ k ]T`[ abWV^ k ´ $±s & > ) G >) IH JLB S : G : B) H´ G"> v $ > ) F ) &#) BH´ G"> v B0 ) > G : BH´ G > v > ) F ) BR´ G > v B "> $±cT\Sb[J > ) G B &U^_Ss > TVcISbX_]¨ k JJT\5 S : BH´ G> $gfd $±s &¨^_SbsªcT\Sb[J > ) T`c¨SUX_]I k JJTV5 S : gfd $±^ & & ) > ) TVcoSUX_]o k JLJ!TVS F ) BH´ G> $gfd $±s &o^_SsceTVSb[= J > ) T`coSbX_] k JJT\S F ) gfd $g & & > ) T`c SbX_] k JJT\S B> $gfd $±[ & & k fXMT\SspJJ: s > ) T`c¨ceXZW\ZJLsT\Q S : iab^]T\XZ S B >)´J0c H X¢] H ^] BT`cceXZW\ZJLsTVC S : õSbspJLJLs¹ BRTVcSUX#] k JLJTV3 S :Y B¯´ G > )ceTVSb[J ~ B.G > K0X k JLX#J k A B TVcGSUX#]G k JJT\S] H J k TVY H ]e§ H ^_SsceT`spJ0X_! BJ">X k ] H Jc^_KMJ k JL^ZceXZS ce:X B TVc ceXZW\ZJLsTVQ S : GU: BH´ G> v B> ©]¨X#WVWVXc] H ^_] : TVc¨cX#WV#Js¹ IH ^_k] : ^#SbR s : H ^ZJ:] H Jc^#K0JMcJ ]cGX#oaUSUT\¥bJ k cGT`cXZgfTVX#ac;SUX#]T`[ TVSUY}] H ^]:^#Sd abSUT¥J k X_ JLT] H J k K%acm]IaUSUT\Jd > ^_Sbs BM¥ k cm] ·BTVSb^_WVWVd# >SspX#K : BRSª sUX#K : $±cT\Sb[)J : T`cIceXZW\ZJLs & >CS sUX# K : $ØceTVSb[J : T`c!cX#WV#Js & BMSspXZ K : X k JXZJ k ZX k JZJ k d^ k TV^#gUWVJ D D TVc¨TVSspXZ K : > v B+«TA^_SsX#SUWVd}T\ D T`cIT\S spXZ K : > v B+Z IH J k JLcaUW]IXZW\WVXc ? 4 y 4 6 # : G > F v v > F 3 ½ 5 {KQ 4J / yZ |"/õ ue| LZ# hc !3 > e e@ : :91U% b>& ©" 9 8#!b G4 m . À 1 + 0 õ \ ) LG4c | : ! ; Á2Á <y dTVSbspa[]TVX#SªX#S] H J[L^ k spTVSb^_WVT]mdX#BspXZ K ospX#K :w ©)] H T`c¨[^ k spTVSb^_WVT\]mdTV\c 1bf] H JLSªspXZ K 8 spXZ K :UcXMgfd JLKMKM^Mvb\yZI GO > ´ G F v v > 0H´ G F p] H ^_]TVcLI :"G A ] H J k ¨TVcJ#W\J] BªgJ0^^ k TV^#gUW\J%T\SRsUX# K
(60) spX# K : J )] KHspJLSUX_]J:] H J%T`spJKMrhX_]JS]KMXZce]!YZJSUJ k ^#W aUSUT\¥bJ k > ´ G F v v > ´ G F X_ : ©]!T`cJWVW§©fSUX¨S] H ^_]LcTVSb[ \J H ^_SQ s =^ k J«ghX_] H KMXZce]Y#JSbJ k ^_W ] H J k JGJ²pTVce]c^ k JSb^#KMT\SUY cab[ H ] H ^_]
(61) GU H q JL[^#W\W;] H ^]!^ª# K| LTVc¨^0gUT lmJL[ ]TV#J«K^_rUrbT\SUY0 k X#K ^ k TV^#gUWVJLco]X^ k T`^_gUWVJLcX_B] H J:c^#K0J«cX k ]L J] >gJ $ B & gbcJ k #J] H ^]´ $Ø^ &&BS spXZ K H W õSbspJJs¹ BMSªspXZ K UspXZ K :8gfd:^#cceabK0rU]TVX#S#^#Sbs%sUX#K : G spXZ K H gfd}[ XZSbce] k ab[]T\XZS IH JLS´ $g & B
(62) G>w õSbspJLJLs¹Ugfd $Ø^ & B IH G B;bcX B
(63) G B IH GB G> ¬=J[ W`^_TVK,] H ^_]L%´ $Ø[ &B.G >) õSbspJLJLs¹;T\! B:G >R] H JL9 S B G Bgfd $g & ceX BªX#abWVsSUX#]ghJTVSRspXZ K [ XZS] k ^#spT`[]T\SUY] H J±^#[]¨] H ^]J H ^#JrUT`[4#Js B k XZK spX#K spX# K : ©]GX#WVW\Xc] H ^]L´ $±s & >
(64) S. spX# K õSbspJLJLs¹¹gfdspJ¥bSUT\]TVX#S BBS.sUX# K )gUap]] H JL( S >¯T`c k JLJTV( S B ¸gfd $g &´ cT\S[ % J RT`c¨T`spJKMrhX_]JS]L $Øs &oX#WVWVXc ¡W`cXb´ $±J & $ > &)G >)X õSbspJLJLs¹T\
(65) $ > &!G>)fceTVSb[Jgfd0spJ¥bSUT\]TVX# S $ B & G>)Z] H J!±^#[ ]A] H ^_] GTVc gUT lmJ[]T\ZJ X#aUW`s}TVKMrUWVd B G>) H TV[ H T`c¨T\KMrhXZcceTVgUWVJgd $±[ & ¬=JSUX@[WV^#T\K ] H ^_]L ´ $ê &?>USsUX#K IH#Z õSbspJLJLs¹BX_] H J k ¨T`ceJ >+ IH G >)ce X >+ G ># KH G >f oaU] >?G > gfd $±J &hceX >+ =G >) IH J k J X k J >QSsUX#K b[ XZSZ] k ^ZspT`[]TVSU Y $±s & k fX $ê' & H X#W`sUcL IH JLS´ $±Y &>+ H G B; õSspJJs¹gfd $±s & A >+ (G >) S] H J:X#] H J k!H ^#Sbs¹ >+ G >+ H hceJX >G ># H f[ õS k rb^ ]TV[aUWV^ k A ># H MT`c!^^ k T`^_gUWVJ#;^_Ss] H J k JX k ?J ># H Kabce]G^_W`cXgJ0^^ k T`^_gUWVJ D ¨T\] H $ D &G >U fTVSb[J $ B & G>^#Sbs %TVcgUT lmJ[]T\ZJ#H Dk GBhH ceX H $Y & X#WVW\XH cL d $ê' &^_Ss $±Y & o]X#YZJ ] J ¨T\] ] J±^#[ ]] ^_? ] : G H IoJ K^d k T\] J :2^#c : H v > ;B;I¨T] H > S spX#K :HZ 3 ½ 54. y. . . . . . . . . . . z.
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