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A Simple Deduction System for First-Order Logic with Equality, Free Constructors and Induction

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(1)A Simple Deduction System for First-Order Logic with Equality, Free Constructors and Induction Jean Goubault-Larrecq. To cite this version: Jean Goubault-Larrecq. A Simple Deduction System for First-Order Logic with Equality, Free Constructors and Induction. [Research Report] RR-3653, INRIA. 1999. �inria-00073019�. HAL Id: inria-00073019 https://hal.inria.fr/inria-00073019 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(57) G   >  H´ G F  v  v >  ´ G F   TVc] H J k J †ŒX k J™T`spJKMrhX_]ˆJS›]L‘b^#Sbs[ WVJL^ k WVd‰aUSUT\¥bJL c :‰Ÿ õS c H X k ]L‘p†ŒX k ^_Sfd}K‰^‚²pTVK‰^_W)c‡JLi›aUJLSb[ %J :H Q :  Q QW:NªX_† k aUWVJ:^_rbrUW\T`[^_]‡TVX#SbcLK‘ :NK%aŽcm]!ghJ P X k ^ªceXZW\ZJLs c‡dfce]‡JLK >   F /v  v > ƒ  F  † T\SR] H J¥ k cm]«[^ZceJZ/‘ :HMTVc—SUX_]™aUSUT\¥Ž^#gUW\JZ‘¹T\SR] H J‰c‡JL[X#Sbs[L^#c‡J  >  ´ G F /v  v > 0´ G F   T`c¨^MKMXZce]¨Y#JSbJ k ^_W;aUSUT\¥bJ k X_O† :@HZŸ ¬=J[^_S¸spJ¥bSUJ €ML $^:H &«^Z\c P»TVS¯] H J‰¥ k cm]%[L^#c‡J#‘Š^#Sbs¯^#ƒc  >  ´ G F „v  v > ­´ G F   T\S¯] H J}c‡JL[ XZSbs [^ZceJZ‘;gUap]«’oJ0’¨T`c H ]‡Xªr k X‚ZJ k JLc‡aUW\]ˆc—T\SbsUJrhJSbspJLS›]‡WVdX_†o^rb^ k ]ˆTV[aUWV^ k ^#W\YZX k T˜] H KŸ XªspX] H T`c‘)’oJ0c H ^#W\W J ²prUWVX k J™’^“dpcX_†Br k XfsUab[ TVSUY‰^_WVW¹TVsUJKMrŽX#]‡JLSZ]—aUSUT\¥bJ k cIX_† ^MY#TV#JSªcedpce]‡JLK :HZŸ¬­J™¥ k ce]X#gbc‡J k #JZ´  4 y^w76 #– O: G , >   F  v  v >   F  4"– #  “– {\/G\ 4– / y##– |"/õ#– "[!3  >– • e@” :  !"91 3 ½  ž5 21M/ spXZ K /8  >  v  v >  :9‚1U#– M  %– {## ƒ >  H´ G F  v  v >  ´ G F    ; € Á2Á <y ·UX k JL#J k ; d &4‘¹%y $&$Lš¹‘UceTVSb[ J ­aUSUT\¥bJc > ) ’¨T\] H F ) ‘U’oJ™Kabcm] H ^“#kJ $ > ) & G F ) Ÿ ¬=J‰¥ k cm][WV^#T\K ] H ^‚3] > ) T`c:TVS.spX#K 0 Ÿ õSbspJLJLs¹‘^#cˆc‡aUKMJ‰X#S¯] H J}[ XZSZ] k ^ k d] H ^_] $ > &/G > Ÿ IH JLS > ) G F)  Ÿ õSRrb^ k ]ˆTV[aUWV^ k ‘ F ) T`c™^}‚^ k T`^_gUWV3J B;‘^_SŽs¹´ $Ø^ &b $ B & G > ) Ÿ fT\SŽ[ J : T`c™ceXZW\ZJLs¹‘ ) F ) G  > ) ) ‘ H JSb[J BG  > Ÿ X k JLX‚#J k ‘ $Ø^ &TVKMrUW\TVJLc¨] H ^_N] B?SspX#K ‘bc‡X] H ^‚N] B}TVcc‡X#KMJ >1=#‘)%y $/>;$Cš; ‘ >QG  &4Ÿ ap]¨] H JQ S : [ XZS›]ˆ^_TV) Sbc] H J:JLi›ab^_]‡TVX#SbEc > )  F ) ^#Sb s >1=U F =#‘U] H ^‚]T`c‘b] H J«JiZaŽ^‚]‡TVX#SŽc > ) >=:^#Sb s >1=U F =_Ÿ IH J k J†ŒX k J >1=«T`c¨SUX_]c‡X#WV#Js}TVQ S :‰‘U[ XZSZ] k ^ZspT`[]‡TVSUY0] H J™†±^#[]I] H ^‚] :@T`c¨ceXZW\ZJLs¹Ÿ fXŽ´ $±g &IspXZ K ­TVcJ ²U^#[ ]‡WVdJLi›ab^#W;]‡X  > „v  v >  ZŸ ¬=J:SUX‚’¢[ W`^_TVK ] H ^‚)] $ > &EG F †ŒX k JZJ k ?d &4‘B\y $n&$Aš;Ÿ fTVSb[\J ¯aUSUT\¥bJLc >  F ) ´ $Ø[ & $ > ) &G F ) Ÿ fTVSb[ J :8TVc c‡X#WV#Js¹‘“†Œ $ F ) & E  > )  v  v) >   G

(58) UŸ d $±g &‘ZT˜] †ŒX#WVWVX‚’c)] H ^‚] †Œ $ F ) & EsUX#) K GT p Ÿ õSrb^ k ]ˆTV[aUW`^ k ‘ gfd $±[ &+‘ $ > ) &!G F ) ‘p†ŒX k JZJ k ;d &ˆ‘¹%y $&$ j Ÿ IH J  JLKMKM^0] H JSª†ŒXZW\WVX‚’co† k XZK $Œg &^_Sbs $±[ &Ÿ ? O TV#JS­^}c‡X#WV#JLsRcedpce]‡JK3X#† JLi›ab^_]‡TVX#Sb)c : G  >   F  v  v >   F  ‘¹[^_WV\W /|–  ¦spX#K :»X_† :»] H J c‡J ]  > „v  v >  ZŸ Œ. Œ. ‡. Š. ‹. ‡. ‡. ‡. ‹. ‡. ‡.

(59)    y#–•e21U–o”ˆ   |   LZ •c/ ꔈ_• |#y­• ! V–   õ–#|  m”:–98#!"/y/ } $ À 1+ & : v >  B Q :ƒ B´HG>  v B–> 9‚1U–:. À + 0 • ! \– Ø• / Œˆ” _• | "#  “–  õ–#| :   "$ ˆ  ‚–   ©–#| :  21M 1"K‚–S1b– ˆ|M– –#[e” !3  >– •'9{kK  !"91 21MŠspXZ K :  G $±sUX#K :    >` &/[  B+  ; € ÁÁ2<y  J @ ] :  G  : v >?B}^_Sb5 s :  G , :ƒ B}´ G >  v B–>)kŸ pT\Sb[%J :  T`c¨ceXZW\ZJLs¹‘ >CG  B;Ÿ k H H  ¬=J%¥ ce]Gc X‚’°] ^_)] : T`c—c‡X#WV#Js¹ŸG¬ k T˜]ˆJ : ^#c >   F  v  v >   F  ‘;’ H J k J%J^#[ H > ) ‘y$x&@$ j ‘ T`cªc‡X#WV#Js£TVT S :  Ÿ Xc H X‚’3] H ^_5 ] :  T`cceXZW\ZJLs¹‘!SUX_]ˆTV[JR] H ^‚] JL^Z[ H JLi›ab^_]‡TVX#S(X#† :  T`cX#†:] H JR†ŒX k K > )  B}´ G>   F )  BH´ G>  X k B>)Ÿ i›ab^‚]ˆT\XZSbc > )  BH´ G>   F )  B}´ G>  ´AceTVSb[ J > ) T`c¨ceXZW\ZJLs‰TV5 S :  ‘ > ) T\Srb^ k ]‡T`[ aUW`^ k spXfJcoSbX_]IXf[L[ a k TVS ~ >?B;‘ H JLSb[ E J > ) G  B;w Ÿ fX™JL^Z[ H c‡ab[ H Ji›ab^‚]ˆT\XZS%T`cX_†Ž] H J†ŒX k " K >  F  BH´ G>  Ÿ ¬­J¨[WV^#T\K¢] H ^‚] > TVc ] H JLSc‡X#WV#JLsT\7 S :  Ÿ  J ] :  gŽkJ : KMTVS›aŽc ] H JGJLi›ab^‚]ˆT\XZJS > )  F ) ) &Ÿ fT\SŽ) [ kJ :  TVcceXZW\ZJLs¹‘f’J H ^“ZJ#´ ) $±^ & > ) TVcISbX_]† k JLJGTVM S :  ‘ $±g & > T`cSUX_]I† k JJGT\S F ‘U^#Sbs $Ø[ & > T`cSUX_]I† k JJGT\ S >ZBhXŸ õSªrb^ k ]‡T`[ abWV^ k ´ $±s & > ) G  >)Ÿ IH JLB S :  G :   B­) H´ G">  v $ > )  F ) &#)  B­H´ G">  v B0 ) > G :   B­H´ G >  v > )  F )  BR´ G >  v B "> $±c‡T\Sb[J > ) G  B &‘U^_SŽs > TVcISbX_]¨† k JJ™T\5 S :   BH´ G>  $Œgfd $±s &¨^_Sbsªc‡T\Sb[J > ) T`c¨SUX_]I† k JJ™TV5 S :  gfd $±^ & &‘ ) > ) TVcoSUX_]o† k JLJ!TVS F )  BH´ G>  $Œgfd $±s &o^_SŽsceTVSb[= J > ) T`coSbX_] † k JJ—T\S F ) gfd $Œg & &‘ > ) T`c SbX_] † k JJ—T\S B> $Œgfd $±[ & & kŸ fXMT\SŽspJJ: s > ) T`c¨ceXZW\ZJLsT\Q S : Ÿ i›ab^‚]ˆT\XZ S B  >)´™’J0c H X‚’¢] H ^‚] BT`c™ceXZW\ZJLsTVC S :  Ÿ õSbspJLJLs¹‘ BRTVc™SUX#]™† 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 ^_SŽsceT`spJ0X_! † BJ">­†ŒX k ] H J‰cˆ^_KMJ k JL^ZceXZS † ce:X B TVc ceXZW\ZJLsTVQ S :  GU:ƒ BH´ G>  v B–>ŠŸ ©]¨†ŒX#WVWVX‚’c] H ^_] :  TVc¨c‡X#WV#Js¹Ÿ IH ^_k] :  ^#SbR s :  H ^“ZJ:] H J‰c‡^#K0JMc‡J ]ˆcGX#†oaUSUT\¥bJ k cGT`c—XZgf›TVX#aŽc‘;SUX#]‡T`[ TVSUY}] H ^‚]:^#S›d abSUT˜¥ŽJ k X_† JLT˜] H J k K%aŽcm]IaUSUT\†ŒJd > ^_Sbs BM¥ k cm]Ÿ ·BTVSb^_WVWVd#‘ >SspX#K :  ‘ BRSª sUX#K :  $±c‡T\Sb[)J :  T`cIceXZW\ZJLs & ‘ >CS sUX# K :  $ØceTVSb[J :  T`c!c‡X#WV#Js &‘ BMSspXZ K :  Ÿ X k JX‚ZJ k ‘Z†ŒX k JZJ k d‚^ k TV^#gUWVJ D ‘ D TVc¨TVSspXZ K :    > v B+«T˜†A^_SŽsX#SUWVd}T\† D T`cIT\S spXZ K :    > v B+ZŸ IH J k JLc‡aUW˜]I†ŒXZW\WVX‚’cŸ ?  4 y 4 6 #– : G  >   F  v  v >   F  ˆ 3 ½  ž5 – ˆ  ‚– {KQ 4J – / yZˆ– |"/õˆ– ue|   LZ#– h–c•   !3  >– e• e@” : :9‚1U–•‡%–   b>&  ©"–  9– 8#!bˆ– G4– m” . À 1 + 0  õ–   \– )   LG”4c• | :  ! ; € Á2Á <y dTVSbspaŽ[]‡TVX#SªX#S] H J™[L^ k spTVSb^_WVT˜]md‰X#†BspXZ K ospX#K :‰wŸ ©†)] H T`c¨[^ k spTVSb^_WVT\]md‰TV\c 1b‘f] H JLSªspXZ K 8 spXZ K :‰‘Uc‡XMgfd  JLKMKM^MvbŸ\yZI‘ GO >  ´ G F v  v > 0H´ G F   ‘p] H ^_]TVcLI‘ :"G AŸ  ] H J k ’¨TVc‡J#‘ŽW\J] BªgŽJ0^‰‚^ k TV^#gUW\J%T\SRsUX# K

(60) spX# K :‰Ÿ  J )] KHspJLSUX_]ˆJ:] H J%T`spJKMrhX_]ˆJS›]—KMXZce]!YZJSUJ k ^#W aUSUT\¥bJ k  >  ´ G F  v  v >  ´ G F   X_† :‰Ÿ ©]!T`c’JWVW˜§©‹fSUX‚’¨S] H ^_]L‘Žc‡TVSb[ \J H ^_SŽQ s =^ k J«ghX_] H KMXZce]Y#JSbJ k ^_W ‘ ] H J k JGJ²pTVce]ˆc^ k JSb^#KMT\SUY c‡ab[ H ] H ^_]

(61) GU H ›Ÿ 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«c‡X k ]LŸ  J] >gŽJ  $ B & Ÿ  gbc‡J k #J] H ^‚]´ $Ø^ &&BS  spXZ K H WŸ õSbspJJs¹‘ BMSªspXZ K UspXZ K :8gfd:^#cˆceabK0rU]‡TVX#S‘#^#Sbs%sUX#K : G spXZ K H gfd}[ XZSbce] k ab[]ˆT\XZSŸ IH JLS´ $Œg & B

(62) G>ŠwŸ õSbspJLJLs¹‘Ugfd $Ø^ & B IH G B;‘bc‡X B

(63) G B IH  GB  G>ŠŸ ¬=J[ W`^_TVK,] H ^_]L%´ $Ø[ &B.G  >)‡Ÿ õSbspJLJLs¹‘;T\!† B:G >R] H JL9 S B G Bgfd $Œg & ‘ceX Bª’X#abWVsSUX#]—ghJTVSRspXZ K ‘ [ XZS›] k ^#spT`[]ˆT\SUY] H J™†±^#[]¨] H ^‚]’J H ^“#J™rUT`[4‹#Js B† k XZK spX#K spX# K :‰Ÿ ©]G†ŒX#WVW\X‚’c—] H ^‚]L´ $±s & >

(64) S.  spX# K Ÿ õSbspJLJLs¹‘¹gfdspJ¥bSUT\]‡TVX#S BBS.sUX# K ‘)gUap]™] H JL( S >¯T`c—† k JLJTV( S B ¸gfd $Œg &´ c‡T\SŽ[ % J RT`c¨T`spJKMrhX_]ˆJS›]L‘ $Øs &o†ŒX#WVWVX‚’cŸ ¡W`c‡Xb´ $±J &  $ > &)G  >)X Ÿ õSbspJLJLs¹‘›T\

(65) †  $ > &!G>)‘fceTVSb[Jgfd0spJ¥bSUT\]‡TVX# S  $ B & G>)‘Z] H J!†±^#[ ]A] H ^_] GTVc gUT lmJ[]ˆT\ZJ ’X#aUW`s}TVKMrUWVd B G>)‘Ž’ H TV[ H T`c¨T\KMrhXZcˆceTVgUWVJ™g›d $±[ & Ÿ ¬=J‰SUX‚’@[WV^#T\K ] H ^_]L ´ $ê† &?>USœsUX#K IH#Z Ÿ õSbspJLJLs¹‘BX_] H J k ’¨T`ceJ >+ IH G >)‘ce X >+ G ># KH  G >f Ÿ oaU] >?G  > gfd $±J &‘hceX >+ =G  >)Ÿ IH J k J †ŒX k J >QSsUX#K ‘b[ XZSZ] k ^ZspT`[]‡TVSU Y $±s & kŸ fX $ê'† & H X#W`sUcLŸ IH JLS´ $±Y &>+ H G B;Ÿ õSŽspJJs¹‘Žgfd $±s & A‘ >+ (G >)Ÿ  S] H J:X#] H J k!H ^#Sbs¹‘ >+ G >+ H ›‘hceJX >G ># H f[Ÿ õS k rb^ ]ˆTV[aUWV^ k A‘ ># H MT`c!^‚^ k T`^_gUWVJ#‘;^_SŽs] H J k J†ŒX k ?J ># H Kabce]G^_W`c‡XgŽJ0^‚^ k T`^_gUWVJ D ‘Ž’¨T\] H  $ D &G >ŠUŸ fTVSb[J  $ B & G>^#Sbs %TVcgUT lmJ[]ˆT\ZJ#H ‘ Dk GBhH ‘ŽceX H $ŒY & †ŒX#WVW\X‚’H cLŸ d $ê'† &^_SŽs $±Y & ‘o]ˆX#YZJ ] J ’¨T\] ] J†±^#[ ]] ^_? ] : G H ‘I’oJ K‰^“dž’ k T\]‡ J :2^#c : H v > ;B;‘I’¨T˜] H > S  spX#K :HZŸ 3 ½  ž54. y. ‡. Š. ‡. Š. ‡. ‹. ‡. ‡. ‡. ‡. z.

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