Automatic Verification of Sufficient Completeness for Conditional Constrained Term Rewriting Systems
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Automatic Verification of Sufficient Completeness for Conditional Constrained Term Rewriting Systems Adel Bouhoula — Florent Jacquemard. N° 5863 Mars 2006. N 0249-6399. ISRN INRIA/RR--5863--FR+ENG. Thème SYM. apport de recherche.
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(42) tM ^ xI(Cruz{¸ºtvruz®tvx^z®gÂÁYrdxegª8i bxY( !rvz{;9tvz®Å9» im z{Itvxicª ic2r6t © t ¯ x v r q p * h b w x } y r x ª Á ¢ i ~ v r i Pos(t) bxYrvz±tz{xYtvx!psi/z}r xstvi/iKgtvxegtviWi rviwqFtp xt|ª8Å2Á!Xdz}egwgiMy®icrØviWxqruc~y±t3~gxª%z{gv*icgz{ y}AÁz®gJ¹ tvt|icvh °&z{tve z}r s&< ,2t3» b3xYz±rvª8z±ticz{ÅxYic vp!p Å2z{ z®twgz}r3y®iDsxsic/gx~ticr%¯2wItp h!t[s]xYrutxÁ ^© iKi!z®t ~rvi Á var (t) !;
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(44) J ( ( z®r #Èz±tihCg\@g z®g . I;9ª«& xh-Å2 vz} , twgy®icrtxtvi/thCr/CÁ Jr3~rv~yçÀ8°%iCz}si/Itvz{ª«p¯rv~gwrutvz{tv~gtvz®xr3°&z{tve²tegi/z®t th!x#xÅ2gegz}z}rvwghÇy®icirc´q Á tvJicrvrvz{~gxYw*rutvtz{xtv~sti/tz{xYhC r/σAÁ J z®r3I½ '+2» ,v»½ª«x3C½ ti/z}© r<h !¸«t°%z±i/ªy®y}tvÂegrvi!xvstvxYiWh! wgz{z¯ icxtvª z®Åσi&rux~Iwt6r tz®z±t~srtvz®xy{y!t°&egiegz}Å26e!vhCz}wgy®icrrKÅ2xª z}twgy®ic r tegi¯/xqgxhCz®Ðxª σ xYYt6z®rCxYgy{p³x~atvi/hCr/Á i~rvibxYrutuÈg´Ðgxt62tz{xYЪ«x#rv~gwrutvz{tv~gtvz®xr#ggy®z}/2tz{xY¿ /xh!^xIruz{tvz®x8Ádegi hMxIr t&Yi/gicy[xh!h!x.z®r t6/i xªrvxh!imtvicvhCr t , . . . , t z}rDsicgxtic#wIp mgi(t , . . . , t ) Á Î £c¤/¥2¦ O¤c £ Î ¥ ¤ Ê ¥ ¶£¯¦ Ï1§(º¦ £ Ê £ P © i·rrv~gh!i¢z®Åic ²x^r tz{Ity}~i À°&egz}6e z}rC·Èz±tirvit!xª gicgz®c2tvimrupqhw^xYy®r°&z±te#viW~g6rvz{ÅYi xqxYy{iWz®Itvi/git62tvz®xCz® T (C) Á<dpIz®cy®y{pYÀ L xYYt6Lz®rØtvegi rupqItYtz®Kic|I~y®z{t p ¸UrupqIttvz} sz®rvic|I~y{z{t p ÂÀrvxh!i3ruz®h!gy{z{È^c2tvz®x.x6sicvz®g y®z{¬Yi»c¼¾½I¼ØMy®i´sz}xYgegz}K^2tve.x6sicvz®gK 5®c@ 6RÀ .≈. h!ich*wbi/6ruegz® x:L tvx·#Èg. ´s6≈ic¯. vic~gy}my®g~^i L ⊆ T (C) ¸ºy{z®.¬≺i!ª«.xY z{ruti!°%i/y®yruxYutictvi/hCrÂÁ "# ( , ( xY·tvegiy®g~^i L i %xqxy®ic/xhwgz{^2tvz®x^rDxªØtvxYh!rKxª<tegi ª«xYvh P (t , . . . , t ) °&egicvi P ∈ L t , . . . , t ∈ T (C, © Xi¢)hC Á µp;p#i´qxtvqicÅYi/;Itvtz®egxi.[Àsg.gy®icz®h!c2stz{t xYp¢¶xYxh*ªKrvwg~gz®wrutvtvz{z®tvx~s.tz{z®xYrz®Ir*tvª«i/vxYgh ittici/.hCr r%tvtx¯~giYxYÁ rutv6z®Yt6r z® rutv6z®eItvª«x°%¯°%µpYÀ< trveg~i/6e#iª«tvxYevi t si/Èz}gri*z{ICtvicrvxvy®~svti/z{tvxYic##ª«xYr%KtCv~giYxYÁ<drutveg6i3z®Yrui/t tcxYªrDrvx·y®~s¸ºtv/z®xx^rurtvx~ª8ttegxi3Âxru~^wr tr tz±tz{~sItvt z®x z}r%σgYi/gvxYx~gtvic#sz®wqgp Mª«xKy{y8*Á tvJicvhCxYr&r z®tcz{It c sol (c) c z}r (
(45) ( ). I;9cσ& »* z± ,ª sol+K(c) 6= ∅ ¸º^ (
(46) ( I;9& »3xtvegicv°&z}rviWÂÁ Å2z}J wgy®icr3(°&z{tve t» Á
(47) 9» x tvi!tteJcK2t z}tregi¢¢Yy{z®rvgrvic~gh!Ksttvi/z®xh ²tte∈2t T t(F,z}r3Xy®z{g)iWtx z}i/r3tveggicxt*°&icz{tvrue²tvz®¢tv/z®ÅxiÀrutvru6z®z®/Iit cqÀ7p°&gegxYz}6 eRy{hCz®gµicp·vz{rut epvhCiM&<µ ,rup xYh!wb ii & i/´qviWrvrvicz®·tveixYrutv6z®YtWÀgª«xYKz®r t6/i f (x, x) JcK z}rrvi/hCItz®cy®y{p#ic|I~gz®Åµy{icItKtvx f (x, x ) Jc ∧ x ≈ x K Á J » 1. 1. n. n. S. Λ. p. p. p. 1. 1. n. 1. n. 1. n. 0. 0. n.
(48)
(49) » » !"#%$'& » » » ()( "*'+ ,
(50) - &."* ( , » 0 + /-12 . . z}°&rDegic.vi ic|I~z}r%tvz®!x/xs2 =~g^txYtvz®Dx..xªxy{z{z{tvicicYticy®#ric|I~ tvz®xz}r&s y®→z{tvi/t6w^yçi/Á t © °icimi/È.^t #°xM/xtqi/Åich!grcz{icÁ I© ti tx!xrvi/^i ruz}sy}i/ ~^ru 9iW&F rØ t(/eg»i/(3hCxrvªi/y®tvÅegici r&ª«YxYrvh tvi/ΓhC⇒r%xL ru )z® ( Γtv ~g ,iD» i/+C´It 9i/&F g(/ic»Cwqr&pM!tvegixYgrutvic6sz}z®/gictvitvrvi/pqLh*hwbÁ xy}r = → Às!tveixYggiWtvz®Åi ∧ ⇒ Á<degz}rØ°µpÀI°%iK/¢siÈ^gi. Î Ï «¤ Î ¦ (<§ Î £c¤/¥2¦ Ê Ï¶¥ ÊIÍ ¥ º¤ P J )'+ ,
(51) - &X ) ( , » + v» # , »2 & »mz}r* )x^-r +t ,
(52) z{g iWCy}~rvi ρ xª7tegi ª«txYi/vhh!r Γ ⇒ l →¸U/ry®JcKy®ic·ru~icrv6[e ÁDtvy®eiªótt &Γz}·r!¯z{Y/eIxt2Re ~tz{rvxYz®giWxªmic|Ii ~y{z®gtviWz®xr u·=eµvÅYÀi3ctvey®iy{iWÄrh!tveii rvxvtcÀO xz®ªKrmtegi.xYv~gruy®tvi6ÀØz®tvYegtWi Á © egic¢ltegi3xYrsz{tvz®x Γ z}ri/h!st )pYÀ'ρ+ ,z}
(53) r& /- y{y®&fic )# ) ,( + , » + v» # ) , » ,& »/#Á + Jrvit R , xª/= x s z{tvz®x^y7cxYr tz{ic7Àqicrv[Á /xruJÐtv6ti/z®gh ic[Àsv~gy®icvric°&z}r&vz{/tviWy{r<y®ict.x wqp3~gy®i ( » ¸«icrv[Á ( ÀI gi/g» xtv2 ic!v» wq#p » ( ( » Á z{ª ª«xYrvxh!i t JdK −−→ s JdK t| = lσ bxYrvz±tz{xY p tJdKru~wr tz±t~stvz®x sσJdKÀ s = t[rσ] Àstρeg≡i rvΓ~gw⇒rutvz{ltv→~stz{xYr JcKσ ∈z}rDRru~^6e¢te2t d ∧ ¬cσ z}r~gr2tvz}ruÈ^wgy®i3 uσ ↓ vσ tª«xY ruy{z{y tvz®uÅi=¸ºvviWru∈8ÁΓtvÀ[6°&^egrui/z{tvi z®ÅuσiµÂ/↓y{xIru~vσvi!r xt6ª −−→rmÁ ª«x
(54) x∃w,tiCtveui−−ru→ich!wItv←z}−C−szvOi/i/ ^−iC−→wbit ¸º°%viWi/rui/[¶Á /−x−→gÂmz±tsz{xYi/gr3xtviWrKÄteg/i!xvru9i tv`6i´sz®Iz®Åtir z®/;xvruictv°&6vz®z{tvIi!tz}r%~gy{ÈgiW´sr/icÁdeiC¢Åµz®y{sz}sicz{^t icpxsª%icYtvt&eiª«xxh ^sz±tÁ z{xY¯z}r si/ÈgiW¯°&vt tvegi#rupsrutvi/h R °&egicviWrtveiz{Iti/gvi/ttvz®xxªtegi J1/xrutv6z®gictvicvh s JcK z}. r v» +! 9 ;& »wqRp R z{ªtegi/i!z®r rvxh!i t JcK ru~6e²te2t s JcK −−→ t JcK ÁÃKtvegicv°&z}rui s JcK z}Yr3rK/Cy{rvy®~gicwstvRicvh 7v» +!z® ru;9t& »/^À[xi 3xª< !Ry®i ªó'tm e ^ .&Grvz} s i xMÁ ªJ !/~gxy®i rutvx6ª z®gÁHiWJÇtvicvxYh rutvt6JcKz®gicz®r#Øtvic» v h &<= v» +!z®rm 9 ½;9 & »Cwq-p + Rv» +z{ª! 9z{t ;& »xI¸«t6icz®rv[r Á t JcK I½ '+ 7» +! ;9& »µÂØz±ª tσ z®rØviWs~/z{wgy®i ¸«icrv[Áz®vics~^z®wgy{iµÂª«xR c i Å c i v p ® z v W i s ~ ® z g w ® y D i u r Y x { y g ~ v t ® z x xª x~ggz{gmª«xY Ádegi si/Ègz{tvz®xrDxª 9» ,'
(55) À «½ -+ ) » ' » «½ -+ )!\ » U½I» - » vimtvei r t6σgc#xic2r 5{W@ 6RÁ t Î £c¤/¥ §¤ Î ¥!£ Ê § §¦s¤ Î P © irrv~gh!i&ª«vxYh x2°³xYCtve^2ticÅicvpMx^sz±tz{xYybxYr tz{ic!i/°&z±tiKrupsrutvich z}r xªtveiª«xYvh R = R ]R °&egi/i R /xItz{^r/xgz±tz{xYysxYrutv6z®gic3~gy®icrxªtvegi%ª«xh Γ ⇒ f (` , . . . , ` ) → rRJcK rv~x6eCgy®tpeÁ 2t f ∈ D À ` , . . . , ` ∈ T (C, X ) R xYItz®r/xrutv6z®giW!vic°&vz{tviD~gy®icrØ°&z±te¢xYr tv~^tvxYØrvpIhw^xYy®rª«vxYh . . . . . p. R. p. ∗ R. R. ∗ R. R. + R. ∗ R. . R. .
(56). . . . R.
(57). . . D. C. 1. 1. D. n. n. C. C. )' .078
(58) .&5:9²& * 30 /xxYrur tvt6z®gz{icictvi/°&i/iCz±ti6rvpsh!r thCi/h6Árm ^Cicxvh!~gz{tgg²xYi´gY6e8t Àri/g°%i/icy®rvyi/Irtz® tvz®x5 ¯6çÀxtvª%egictvegpMi¢vrviit ~xrvicª%Ytvvx xY~gYi/gictvicvtvhCir3z{^z®vi/ich!g~i/Iz®twgy®y{iMy®p*wq p²i/y®i/Å2z®Åi/I t rvitDxª8tvicvhCrcÀgwqph!icrxªgx Uti/hMz®ybi/y®Yi/h!icYtª«xYy{y®x2°&z{gxss~tvz®xv~y{iWr/Á (#½ ,\ » ;= ] ! , » (/» Q gx Uti/hMz®y®r » xYr tz{ic³tvi/i6h!hC u » v» u ( &< , » » T (F,GX )= (Q, -+0∆) * , ] » -(/+ » ∆ ( ] ) vxs( s ~ ,!tvz® x¼ & » ( » t := 9» v» c ¼ W ¼ ¼ f ( u , . . . , u ) JcK f∈F t c u u ∈Q degiCx çtvicvh!z®y}r3iCy®°%µpsr3xYrvz®sicviWh!xss~gy®x·Å2z®wgy®iMi/^h!z{ÁC ;vtvz}~y®cÀ[°%irrv~gh!ite2t*tvegi!ti/h z}r&y{z®giWWÁ f (u , . . . , u ) ¥ ¥2¦s¤ (º¦ ¦ / xÊ rutv6z®gÊ ic¢Ê tvicvhCÎ r ÀsxY -Ê xP mª«z{xÅYDi/rvegxvt&/x°&egrutvi/6 z®giWz®²r&/tvy{iWi/iYª«xhhMhC/xI tvGi/´ItW=Àsz®(Q, rKsi∆)ÈicÀ#tegwIi¢p gxss~tvz®x¶i/y}2tz{xY²xY . . !! #"$ &% x y' x 1y. '. . ' x y ' x 1y '. x ny. 1. x y. ' x ny. n. )(. . . `G,x. . `x. +*. `. 8. G. t[x] Jx: x uy ∧ dK `x t[f (x1 , . . . , xn )] Jx1 : x u1 y ∧ . . . ∧ xn : x un y ∧ c ∧ dσK. {z ªtegi/i3i´sz}r t6r u := f ( u , . . . , u ) JcK ∈ ∆ rv~6e#tvet f (u , . . . , u ) = uσ ¸º°i3rrv~gh!ite2t&tegi3Å2z®wgy®icr%xª sx#gxtmxqc~gmz®tveixYrutv6z®gicti/h  À{ÁcÁ/Á/À i ª«icrveÅ2z®wgy{iWr/Ámdegi u ,...,u i`i/´sz{ÅYitrvz±tcz{ÅYim#tv6rvz±tz{ÅYim/y{xIru~viWrxªtegi3i/y}2tvz®xt[x]` Jx:iuicrv^∧iWdKtz{ÅYi/y®p¢sxi/xtviW¢xwqp ` ` Á / » &F ½ ! ½I» ( » (» ½ -+ »9 ( icgi/62tic ;=] ) ( , » + »»½ M G & u u ¼ó)»c¼ ( ! G x Jx: u K ` t JcKt » v» c ( (
(59) ( . I;9& »W¼ (
(60) , ½ , » ' Y 9» ,' L(G, d
(61) egi3w^x2ÅYiDh!i/h*wbi/6rvegz{.xYr tz{It6r t: u Àq°&z{tve u ∈ Q Àqviz{Iti/gvi/tviWwqp 8 sol (t: u ) = {σ
(62)
(63) tσ ∈ L(G, u )} Á y}xgtvYiM~teYi2t3Á °%iMc~^ruiCrv~6e¯/xrutv6z®ItKª«x z®rut^itvxviWr tvz}t3¢tvi/htvx·#z®Åic¯rvxvtmxYmIp·Yz{ÅYi/¯vic~gy}tvi/i x y. 1. x 1y. 1. x ny. n. n. 1. x y. n. ∗. !! #"$ -, .. x y. ∗. x y'. +. . x y. x y. x y. x y. x y.
(64) . + » & !&F &F v» ! » +. ¥ ¶¦ ( ¥ ¥2¦ . ¶¦g¥ Y·ÎhMhC GÎ (R ) = (Q P (RagxY),C∆i/ÅYi/(Rp¶x))^r t°&ez{z®g6e;iW¶icgi/°&i/62z±tti.icrrvtvpser iti/y®h gR~^À%iM°i·xª/³x~/x rutvR~RgtCx²hCxYyruª«tvx6hCz®gr/icÁC Itvt~ i/i z{xtvªz®Å#icy{tvpYÀsi/tviCegz}YrD/h!xh!rutvm~ª«xtz{ xY[Às°&egz}6e#Yi/gicy{z_E/iWrtvegi3xYgi xª 5 @6RÀ/xviWrubxrØtxMtvegi /xh!gy®i/h!i/Ittvz®x/xh!gy®itvz®x çics~^z®wgy{i!ti/hCr/À[°&ei/iCi/Åicvprv~gwrvit xª%x çtvicvh!z®y}r!¸óª«x tvegi/xh!gy®i/h!i/Ittvz®x^Âz}r i/gicrvi/Itic#wIptvegi3h!xYruRt&icgi/6yO~ggz{ÈicDxªz±t6r&i/y®i/h!i/It6r/Á tegi#N8ruitvtz®L(R trv~gws)tvicw^vi#hCr*tvexi.ªrvtviegt!i#y{xi/Iªótt6ez®gz{Ärvtvz®esi.i¢rvx~gªwstvtegi/i#h!~gr/xxªKtvruegtv6i.z®y{gi/ªóictM²e^~gy®ic¶r rvxz}ªsicRrMxª&teg;ixgxi¢^r ŵtvz}z{gwiWy{Äi x~gy{iWª«r*xY*xª icYR6eÄruxYu t y{i/t (R ) wbitveiDrui/txªtegi&gxY Uti/h!z{y®rxªtegiª«xh mgi(t , . . . , t ) °&egi/i { t , . . . , t } z}r S∈S hC2´sz®hCyqru~gw^rui/Qtxª L(R rv~6e te2t i~gz±È^wgy®iÁdegirvitxªstv6À ^ruz{tvz®xr ∆ (R ) /xItz{^r[i/À ÅYi/p t := )~6e²tve^2t t , . . . ,°& t z{tve;gxÈy®i v r f ( u , . . . , u ) J¬cK eµÅY i!viWrubWictvz®Åi!ruxYut6r S , . . .Á , S À tfz}rm∈tvegFi mgi xªØtegirvSit 8 , ..u. ,
(65)
(66) S u→ ∈S Q u (R, . .). , u u hC∈2Qt6eicr (Rf (u) ,u. . ,. ., .u. ,)u À c≡ eθ ¦ ( tvxY%rvpqh*wbxy}rxIt6z®gz{ 0 : Int ¢tvei ~g^ pgicÊ sOiWicN[ri/rvtxIntwbi rv!~/rv/xicvrtruxYª«xYrvz{pqIh*tvicwbxicy}r rp,^s#: YIntrvrv~g→h!imIntMÀrvit #C y®xitªRxYr =tv~^{s(p(x)) Á → x} degiMxYr tz{ic.tvi/i6h!hC G (R ) /xrutv~ticYrKw^x2ÅYi erKtvevici*gxY →Uti/x,h!p(s(x)) z{y®r s(x) À p(x) s~^xtvz®x¸ótveiM~gy®y}ic2r tv tv8icmz}r30sicgx:=ticwq0p 0 wbi/y®x2°Çs(x)w^iW/~:=ruitves(z®r g0xY) çtvi/h!z{^yxgy®p·p(x)icgi/62:=ticr 0p(ÂÀ0) tvegiMª«xy®y®x2°&z{g#gx&3 p(x) := p p(x) s(x) := s s(x) degi gxqxªxªtvegi ª«xYy{y®x2°&z{y®i/h!hCC/#wbi ª«x~g^#z®0 5 6çÁ / » &F ½ ! ½Y» S ¼ ( » (» H R Y 7v» +! 9 ;& » 9» ( T (C) ¼ L(G, u ) u © + iD )ru e^ y® yg+ x^ruz}si/wbi/y®x2°¶tegigxhCyqª«xhÇYhMhC G (R ) Yrvrvxsz}2tic3tx R °%icy®yg x^r tz{giW3ti/h » » z±ª À À tª«xYJcK Dy®y i ∈ [1..n] Á c = x : u ∧ . . .∧x : u ∧d {x , . . . , x } = var (t) u ∈ Q (R ) sort(u ) = sort(x ) !#"$ %
(67) &(')*,+.-/%&#&001 .5 * &5 4 .5#.
(68)
(69) ©gi&x2Åqruz{ey{ÁyIgx2°¶si/Ègiª«xhCy®y{pmtvei%vxYwgy{ich °iiØz®Itvi/icrutviW3z® tvez®r^ic z{tri/y}2tz{xYr[°&z{tveMz{^s~tz{ÅYiØtegi/xYvich . C. NF. NF. C. NF. C. C. C. C. C. C. S. C. NF. C. x 1y. 1. x. 1. C. 1. x ny. 1. n. n. x 1y. x y. NF. x 1y. n y NF. n. x ny. C. x y. NF. x ny. 1. C. n. 1. C. n. l→r JeK∈RC , t=lθ. . C. NF. Int x y. x. y. x. y. x y. x y. . C. x. y. x y. x. y. x. y. x. y. x y. x. y. x. y. %. x y ∈QNF (RC ). C. x y. NF. 1 x 1y. n x ny. 1. C. C. n. x iy. NF. i. C. i. » /-12 R ( ( 9 » & = )%$'& » » &,& t ∈ T (F) »9v»*»Z ( ( s , T (C) ( ! K G t −−→ s ¼ dsi/egÈigz{gtvz®xxsiW~srv~gz{i¢°%tegi.i3gvxi#tvz®gxvxY^xxIªruruz®~sg¯/z{¿z{icYjqtKic/tvxz®h!x³gy®iti/ª«xYg!iWrv6rexicª86¬qtvez{i ¯srvi~sÈic#z®i/Iª«~gtxtvz®h!x·gy®ruipqti/hgw^iWrvxYry®rcz}Á r!wrvicÄxYa ic|I~gz®Å2y®i/It )- 9 ( = ;)& f ∈ D (rv~sz®i/Itvy®p#/xh!gy®iti # R &,& t , . . . , t ∈ T (C) »9v»» Z ( ( ( ! ! ¼ /. !! #"$ . . ∗ R. !! #"$ . s ∈ T (C). . f (t1 , . . . , tn ). + −− → R. 1. '. n. s. " % /-12 R ( ( 9 » &<=] )%$'& » » » \ »9 = + » * » + ( = ;)& f ∈ D ( ( 9 » &<=] )%$'& » » # ? ¼ %pz®s~^tvz®xxY¢tegi3q~gh*wbi/&xªxs//~gviciWrxªrvpqh*wbxy}rxª D z®·Mz®Åi/¢tvicvh t ∈ T (F) Á. R. ¼. 2. Ï §¤ Ê ¤ ÊIÎ ¥ Ê ¶£ P J¹/y®~rvi z}r + » +! 9
(70) ,\ » » v» xª ¸ºgi/gxtvic¢wqp ÂØz±ªz±t%z®r%Å2y®z®z{.qph!xssicy xª R À¯z±t3z}r3 ,-+! 9
(71) ,\ » » v» C xª R ¸Usi/gxtviWwqp R |= R C Âmz ¶ª«xY y{yRrv~g|=wruCtvz{tv~gtvz®x σ YvxY~gsz®gª«xY C À Ádegz}rDsi/Ègz{tvz®xz}ri/i/6y®zFEcicCtxxYr tz{ic¢/y®~rvicr&r%ª«xYy{y®x2°DrcÁ R |= Cσ ( , » +b &F (/» JcK ( z{^s~tz{ÅYiMtvegicxi/h R + » - » +;9= R |= C JcK H &,& ( !; (
(72) ,
(73) ( σ ∈ sol (c) Ø» ! \ » R |=CCσ ¼ "! ÖÙµßUÖãßçݾá&ÚWÕ¾Ý Ö ÞKâWÔ6äRãäUݾÔâ ü®ÔÜ7ÚWÜUÔ/ÞµÙµÓ äUݾÔâ3ÜUÙµÕ¾Ö ßuæµü®ÔÜ7äUéµÖ<ßRã6ëÖ<Ô6üܺÖuã6Þ2ã6ìµÝ¾Õ¾Ý äóí . Ind. !! #"$ .
(74). . . Ind.
(75)
(76) » » !"#%$'& » » » ()( "*'+ ,
(77) - &."* ( , » 0 + /-12 . . §¤ ¤ ¥ ¥ « ¤ )§ £c¤/¥2¦ Ϥc¥ ¥2¦ . ¶¦s¥2£ ~stvxYh!tviW#Ê z®sÊI~^Î tvz®Ê Åimtegi/xYÎ vich gÍ x2Åqz{gª«xÎ tegi rh!Ê i ¬Iz®Êq.Ê xª<d&Kj.rv^iWz{È^P / tv z®x5 r&6Y°%riz{.si/tvÅYei/z®y®r&x^;bi/gWi/Á° degi3ggvxIxq/6ice¯s~ª«xvi xrv~gªwg5 @Y68xYz}r&y}r y}sru~gxMwz{Y¢ruiWt¢egiCxY¢gtxIegxi3ªØgwqxphCz®syb~ª«xtvz®h x[/Á!xdruegtvi/6iz®gª«xYicv¢iYÀ7tvz±i/t*i cY¯hMw^ihCc y®y{GiW·tv(Rx·sz})rÀg6e°&egz}6ie#gz}rvxq~xrvªic#xwª«y{xz®Yttvegz®i xrmicgsi/~g62z{tz{#xY¢tegxi ª gxsiWs~giDª«x6egiW6¬Iz®gMru~s/z{icYt&xYh!gy{i/tvicgicrrgiÈgiWz®.jqiWtz{xY#sÁ
(78) xtvitvettegimgvxs/ics~giKxª 5 @ 67z®rrvx~g¢ Miª«z®0~st652t6óz{ÂxYÁ %y®icy{p#c~xrvh!i gxªy®iy{tici gª«xte.tegici*rutvsz}iWtvxz®x62^tr/icÀs#°%i3x/ icgtx~gtDicgrictrvei/2ItDt&°%z®i3grueity{yz{y[tvxeg^i ruz}svi/xsegiWicsv~gii ¸Uruxicªi 5 d@6egegi/xYi/vicih>Á*N8MitK~^r&z{/y®xy®~r xty®y®vtvpi z{tr&vz®z®gy®i3x.i´gh!gy®iÁ ¦ ( Ê Q © i¢xYhMy{i/tvitegi¢rv^iWz{È^/tvz®xÄxª´sh!gy{i¯C°&z±te rvxvt Bool ª«xY %xqxy®icr/Àt °%xx^r t6Itr3xª C À true, /xsz{false tvz®x^y7: Bool ~gy{iWr 8 ¶xgi#wgz{^p;giÈgiW¶rvpIhw^xYy ≤: Int × Int → Bool z® D 0Á N[i/t R w^i#tvegi#ª«xy®y®x2°&z{g¯rui/txª Ï. . . . NF. C. . D. 0 ≤ 0 → true, 0 ≤ p(0) → false,. s(x) ≤ y p(x) ≤ y. → x ≤ p(y), → x ≤ s(y),. © i ruex2° tve^2t&tvei ª«xy®y{x2°&z®g!y}~rvi z®r&.z®s~^tvz®Åimtegi/xYvichÆxª R 8. 0 ≤ x = true ⇒ 0 ≤ s(x) 0 ≤ x = f alse ⇒ 0 ≤ p(x). → true, → false.. u¸ W JKggy®pIz®g tvegi²gxqg~tz{xY¿v~gy®icr¢xª G (R ) tvx¸ WÂÀD°%ixwgtz®Ðt °%x ru~wgxIy}r¸«z®s~tvz®xr ti/^ 8 0 ≤ s(x ) = 0 ≤ s(x ) = true Jx : s(x) K Ádegi È6rutKru~gwxYy7/wbi ª«~gvtvegicDz{^r t6Itvz}2tic.wqp G (R ) z®Itvx true Jx : 0 K À^#tvegz}rDiW|Y~^2tvz®xi/°&z±ticrwqp R z®Itvx!tegi3t~stvxYy{xYp true = true Ádegi*rvic/xrv~gwgYxYy[cwbi 0 ≤ s(0) = true reqvz{pqh!^gxy®tvz{egÈiWicru.z}rcz®Á<YtRtDx*z®tregbi xYtrru~sz®wgtvxYy®i y{xYw^iWp /true ~^ruiC=¸ µÂ%truez®r&rutv~rvz}z{gtvy®Mp#trveghCi3y®y}y{ic~ trviMe¸uµÂ#Àgtv°&egegi z}6rve[ic/Àsxz®¢.tegrvz}~gr&wg/xIrviyçÀqÁ z}r&xYruz}sicviW¢r#z®s~tvz®x 0 ≤ x1 = true Jx1 : x s(x)y K. NF. 1 x y. 1. 1. C. 1 x. NF. y. C. D. . 3. [¤/¥ Î ¥ Î Ï º¥W¥ Ê Ï § «¡ ( º¤ !Ë P J~z}y%gxwgy®i/h z® tegi#z®/vichMicIty%gxsicg~gvi#xªmjsictz{xYa¯z®rtx²si/tviWt &ª«° xYegy{y®i/x2tv°&egz{icC!rvy®~syØtegiz®i/ItDx~gxY;sz{z®tvz®xru.tª«x&iWrYvxxYªK~g¢ti/viWh s~z{³z®wgz®y{xYz{t pr Àtwv~^rvictvz®#x¶x#tvvegi i3RxtvRz®xic.s~x/ªz{wz®y{iYsÁ0~atvxz®Åimtvtvegegz}rMicxgi/~gh^Á xIruiYÀ°%i¢~rvi¢tegi ) ( , » + 9» t JcK (Cr tvxYgy®p¢YvxY~gics~^z®wgy{i # R » » v» n & 7» ( n > 0 Γ ⇒ -+ n ( !; (
(79) ,
(80) σ , i ∈ [1..n] ( ! ! t = l σ ¬c ∨ c σ ∨ . . . ∨ c σ ( ( ( . ;& » l -→ Jc K R + Rr |= ¼ Γ σ Jc ∧ c σ K ∨ . . . ∨ Γ σ Jc ∧ c σ K
(81) xtvimte2t&wqpsiÈ^gz±tz{xY[Àsi/ÅYi/pCr tvxYgy®p!x~g¢viWs~/z{wgy®i x^r tz{giWCti/h1z®r%°%ic¬qy®pCviWs~/z{wgy®iÁ<l·xYvicx2ÅiccÀY°&egi/ R z®.tvz}regi3gx~gxqx·ªx/ª<xqdÅegici/vxYYvi/icIh(tcÀ^i/ÁÅYi/dp.egi3rutvxYxqgÅYi/y{6p.rviz}rx~gxt&tvviW~si~Á /z{wgy®i/xrutv6z®giW.tvi/hz®rx~viWs~/z{wgy®iÁ degz}rKz}rmruegx2°& . . . D. !! #"$ i. i. . i. i'. D. 1 1. Ind. . 1 1. '. n n.
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