Modular Aspects of Rewrite-Based Specifications
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Modular Aspects of Rewrite-Based Specifications Bernhard Gramlich. N ˚ 3330 D´ecembre 1997 ` THEME 2. ISSN 0249-6399. apport de recherche.
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(64) !#"%$ &'(*)+(, - ( !./. ! =¸L¸#" ¬ a p bGY] J a \ p b p xfn\ p a J MSYNOw;[ZYZ`cb=aY\^W¢}c¤rzOU / /. }_} ¢P_¤X;b=eehd;NOYWn]¢h}8¤;\^W;M e J \= e. dJu j7I¢!E7J(G"¤JL jMGB¬L d q zOU q L. ij gb ea p bGYZ] J a \ p b p xYZW;x;dn`ahYZ\_W\_W W'bG[\^]_\^d;eh[ZgiD\ p a J M YZNOw;[ZY`8b=ahYZ\^W,¢}c¤rzOU ¢z_¤X_b=eehd;NOYZW;] ¢}c¤n\_W;M e J \= e / / dJ\?j7I¢ EHJ(G"¤JLkjFGB¬L ( DzOU Dq L jig)bea p b=YZ] J a D\ p b p xYZW;xnd;`ahYZ\^W\^W I J MYNOw;[ZYZ`cb=aY\^W;e"¢P^¤rzOU ¢}c¤b=W;x¢z^¤ zOU ¢}c¤f\_jiiYZ\_d;e[g J \^[Zx u MWn`M_Lb=[Z[la J;p MMw p \_wlM p ahYZMeQb p M Mmid;YZ_b=[ZMWia8 M¨iaM x;M#'WnM0bSa p b=W;e±\ p NbGahYZ\_W J YZ` J MW'bGj;[ZMe ah\kb=jneha p b=`a p \_N( J YZahM SË[¥b8g_M p e ^j;d;a ¯_MXMw;e Ja M)YZWi\ p NbGahYZ\_W`\^W;`M p W;YZW;]wl\_aMWiahY¥b=["¢é J YZahMc¤ SË[¥b8g_M p `\^[[¥bGw;ehMe Z O³ D ¸ ) =, n3E ] n3H 5(M <!` . 3 )[H / n3H <H/. 67/ .y(+ ( !.: 3 !%H 5 Y !. ^ ' !. / ¬ I¢ E%E K7E%H" ¤ q I!¢ E%EQK = J ¤ _ (,/ ( 6 .A- .( +. &! r. j7I¢E E%E.¤ E 2 ¢< ·¤#J,PPPJ ¢<Y;¤4 D z 2\ 'JPP,P(J\Y4 EH GG j7I¢H E%H¤ ¢<·¤Dz GGGG Dz H 2\ ,J,PPPYJ\Y4J H 2L=J,PPPYJ(L/4q" x I L Dz 2\ ,J,PPPYJ\Y4 }~ ~ } J ¢ ¢<Y ¤h¤ GG = GG ¬zs H 2L=J,PP,P(J(L/4 q"x I L=J K=¢L>J F ¤ zaU GGI M \^j;ehM p _Mta J bGa b=jl\=_M YZe"M[Z[FxnM#'W;Mxf_eYWn`M n3H YZebGehed;NMxah\Sj'M d;WnYmid;MX[gS`\_[Z[¥b=w;eYZW;]'n\ p YZ[Z[Zd;eha p b=aYZ\_W^`\_WnehYZx;M p YW v ¨;bGNw;[ZM k9 a J MQxnM p YZ_b=aYZ\_W 8f¢ ¢ ¢ J i¤h¤ J ¢ ¢ J i¤h¤ J ¢ Z ¤h¤ q x I 8¢ ¢ J 6¤#J L¢ ¢ J i¤h¤#J O ¢,Z ¤¤iq x F 8f¢ O ¢ Z ¤ J O ¢,XZ ¤ J O ¢,Z ¤¤ O O ¢¥ J M p M l tZ b p Mb=eehd;NOMxa\kj'M j;[¥b=`#¯O`\_W;ea·bGW6ae·¤X u M p M_Lb=jneha p b=`aYZ\_W YZa J giYMX[xnec¬ 8¢ ^J ¢ Q¤ J ¢ Z ¤h¤ z 8f¢^J ¢ Q¤#J = ¢ Z ¤h¤Dq x F 8f¢ = ¢ Z ¤ J = ¢ Z ¤ J = ¢,XZ ¤h¤ P =. Â(+³ ( « ! .A .A=( +n!< (E ] n !.H ^5(( .? <!/` . !.[ : > )?5( h/ * %' D 5 <n >E -^ <. ./: "Y(&! '.7 ! .i*# +.[!< n !H. _? Y/'.X 67 # ! )/. + d q x F Oz U Oz U 5 d q x F d q x I Oz U ífí æ ööö#. 3. d. ¢%d8¤?q x F _¢ ·¤ : ¢%d8?¤ qx F _¢ ·¤ : %¢ d8¤ q &% ) %
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(66) }8P. $ , #. Y t ! . ! =¸L¸#" ¬. ¢ b^¤YZeeha p b=YZ] J a±\ p §b p xj6gx;M#lW;YZahYZ\_W\= ¢éj'¤YZe"w p \=_MxjigYZW;x;dn`ahYZ\_W\_W7z pL¢dc¤ bGW;x`cbGehMbGW'b=[ZgiehYZe d;eYWn]a J M)W;\_W F`\^[Z[bGw;ehYZW;]w p \_wlM p a g\=rn3E)b=Wnx ¢Db_¤X I J Mw p \i\="\=¢¥`8¤ YZeQbG[e\ jigYWnx;d;`aY\^W«\_W( z pL¢dc¤)bGW;x«`cbGehMbGW'b=[ZgiehYZe M¨iw;[Z\_YZaYWn]a J b=an3H YZe)d;WnYmid;MX[g`\_[Z[¥b=w;eYZW;] bGW;x`\^[[¥bGw;ehYZW;]k`\^Wi'd;MXW6a8 \^ahM)a J b=at YZa J \^d;a `\_[Z[¥b=wnehYZW;]k`\^Wi'd;MWn`M\G n3H bGWYZW;W;M p n3HfFeahMw `\^d;[ZxjlMa p b=Wneh[¥b=aMx YZWiah\H b2ehaMw dnehYZW;]ka J M p d;[ZM = %¢ Ll ¤ q I J YZeYZex;dnMa\a J M Fb=`Xaa J b=a YZW/b a\_w J YZahM_i J YaMaM p N 2L=J,PPPYJ(L/4La J M wl\_eehYZj;YZ[ZYa g\="b)`X\_[Z[¥b=w;eMahA\ LONYZ] J ajlMQMX[YZNOYWlb=ahMXxjig bW;MX¨6a)¢éw'\^eheYjn[gOW;\_W F`\^[Z[bGw;ehYZW;]ir¤ n H ehaMw. Â! ³ =¸L¸#«" I J. )0B
(67) 3 ) n <h, * !. % .A .[H./A n X ¢ L'r ¤ qeL _8, * !. % .: ¬ MhSËY¥Fwlb p a =\=a J MMXm6dnY_bG[ZMW;`MtYeÂa p YZiY¥b=[ I J MtSË\_W;[Z=g^FY cwlb p aYZebS`\_W;eMmid;MW;`XMQ\G?2Z},@i I J MX\ p MXN @4h W¡b=[ZahM p W'b=aM¢¥x;Y p M`akb=W;xMcb=eg;¤ w p \i\G¢¥\= a J M7S \^W;[Zg^FY nw'b p a·¤ YZe)\^j;a·bGYZW;Mxj6g`\_W;eYZx;M p YZW;]a J M YZWiahM p w p MXa·b=aYZ\_ W \= ahM p NeS\= I¢ E K /JG"¤tYZW«a J MahM p N bG[]^Mj p bZI¢!E7J(G"¤ J YZ` J YZW6aM p w p Mae M^M p g 8 j E jigYZahehMX[®#b=W;x = b=eSYxnMWi= ahYZa gd;Wn`ahYZ\_Wf YZW;`Mb=WigYZWi'W;YZaM¢_n X = ¢L'¤q L=¤D x;M p YZ_b=aY\^WkNkd;eha`\^Wia·b=YZWYZWi'W;YZahMX[gNbGWi g nk ehaMw;e8;W;\^WiFaM p NOYZW'b=aY\^W)\= n X = ¢ L'r¤ qeL \_d;[Zx YZNwn[gW;\^WiFaM p NOYZW'b=aY\^Wk\== n ;ehYZNOw;[Zg)jigb=wnw;[Zg6YZW;] ' eh[ZYZ] J ah[ZgN\ p Meh\_w J YZehahYZ`cbGahMxBYZWiahM p w p MXa·b=aYZ\_W¡ J Y` J g6YZM[Zx;ebw p \_wlM p x;M` p Mcb=eM\ p b=WigB¢¥w p \= wlM p ¤ p Mx;d;`XahYZ\_W£ehahMXw¡YZ W n X ¢ L'¤ q LOYeb=e \^[Z[\= ek¢Db=ehed;NYZW;]OahM p NYZW'bGahYZ\_W\= n¤¬ I b=¯^M bGeeMNbGW6aYZ`kx;\_Nb=YZWD\ p E K = = i¬ z I¢ EHJ(G"¤ ;&c YZa J +j'MYZW;]a J MehMXak\=W'bGahdi p b=[ Wid;NkjlM p e8 I J YZeg6YZM[Zx;ekbM[Z[D\_d;Wnx;Mx«N\^W;\_a\_W;M!¢ E K = =¤Dhb=[Z]_MXj p b¢ *J¤Sjig«x;M#'W;YZW;] 8 ¢<¢ +J =#¤ JP,PP(Jc_¢ J ¤h* ¤ z ¢ 8_¢ :JP,PP(Ju #¤ J = PP,P ¤ \ p b=Wig 8 j E"b=Wnx ¢h<¢ ,J L¤h*¤ z _¢ ,J }8¤;b=Wnxa·bG¯iYWn ] ?ah\j'M a J MQM[Z[D\_d;W;xnMx[ZM¨iYZ`\_] p bGw J YZ``\^Nkj;YZW'b=aYZ\_Wk\= q9= x@ b=W;
(68) x ¢éa J Md;ehdlb=[i\ p x;M p YZW;]t\_W)Wlb=ahd p b=[^Wid;Nkj'M p e·¤Xi R[ZMcb p [Zgi_¢ *J¤ Yef`\_NOw'b=aYZj;[ZM YZa J nDX = ¢ L'r¤ qeL 6 `##;M_ ]' 2ËP @#4± J MW;`M n X = ¢ L'r¤ qeL Ye aM p NOYZW'b=aYWn]'. ³i¸l=³ ) =, n E ] n H 5(Z 7 __`# !. ^ . !.(3 ):
(69) n E <.[./"Y(+ ( !.F . n H <h/. 67/ .A&! ' !.k .[H(&! ' !.k(. /,.] ,. n E X n H _8, * !. % .: ! =¸L¸#" ¬ n\ p bw p \i\=Qjig«`\^Wia p bGx;YZ`ahYZ\^W b=eehd;NOMAn E bGW;xan H b p M]^Y^MW¡b=ebGj'\=^Med;` J a J b=a nZv z n E Xn H YZe W;\^Wi ahM p NYZW'b=aYZW;]' Mt`\_WnehYZx;M p b`\_d;WiaM p M¨;bGNw;[ZM\G"NYZW;YZNb=[ p bGW;¯ ¬d qax d+ qyx{d qax PPP J M p M)) [ \' ]'8MNb8gb=eehd;NOM8d j7 I ¢!kE ¤ z I ¢!E E KAE H ¤\ p bG[Z[ Y ;{tQ Y YZW;`M2( YZeNYZW;YZNb8[ bGW;xkn E YZeWn\_Wi `\_[Z[¥b=wnehYZW;]M `\^W;`[Zd;x;M a J b=ab=[Z[>d b p M ah\_wj;[¥bG`·¯¢é\_a J M p YZeM_lYF M_iYb=[Z[>d M p M a\_w J YaM_iYZx;MWiahYgiYZW;])b=jneha p b=`aYZ\_W\Ga J Mw p YZW;`YZw'b=[lehdnj;ahM p Ne ¢¥Y M_®ia J MY pp Mw;[¥bG`MNMXW6ajigkeh\^NM p Me J _b p YbGj;[ZMc¤fYZW3 \_dn[xgiYZM[Zxb=WYW 'W;YZahM n H x;M p YZ_bGahYZ\_W`\_Wia p b=xnY`XahYZW;])aM p NOYWlb=ahYZ\^W)\= n H ¤ \=)b=wnw;[Zg6YZW;]TLC MXNNb}#9 bGW;xBM¨iw;[Z\_YZahYZW;]a J MOFb=`Xaka J p bGa3 `\^Wia·b=YZW;eYZWi'W;YZaM[Zg«Nb=Wig\_d;aM p n3E ehahMXw;eM \_j;a#b=YZWb=WYZWilW;YZahMk¢<n3EQ4 X = ¢%lL ¤rqeL ¤D x;M YZ_b=ahYZ\^W ¢ k ¤¬ ¢d8 ¤ q ¢%d&c ¤rq ¢d= ¤ q PP,P P ¦dnajig3LC MNONb},3 a J YZe YZNw;[ZYZMe"W;\^Wi ahM p NYZW'b=aYZ\_Wk\==n3=E J MW;`Mb`\_Wia p b=x;YZ`aYZ\_W. ñ¥ëfíñ¥ø.
(70)
(71) !#"%$ &'(*)+(, - ( !./. }cz. 6 W v ¨;b=Nwn[M 9a J M)ehgieahMN 3n E z7 8f¢%L=J ¢%Ll¤ J(N_¤ q 8f¢%N[JN[JN^¤ YZetW;\_Wi `\^[[¥bGw;ehYZW;]kb=W;xln3H z. ¢S¤ J ¢%L=JlL ¤ q h J O ¢%Ll¤rq L )YZe Mcb=eYZ[gOe J \= Wa\jlM)j'\^a J d;W;YZmid;M[ZgO`\_[Z[¥b=w;eYZW;] O bGW;x`\_[Z[¥b=w;eYZW;]`X\_Wi'dnMWiac u MW;`M^'jig I J MX\ p MXN },S@ M `\^W;`[Zd;x;M)aM p NOYZW'b=aY\^Wk\==n3E X n3H $&%L ('³AB .X' h ^ <!` . 3 ) ¢%L=JLl¤ q. n3E z 6. f¢ J ¢L'¤#JN^¤ qJ8f¢%NJ(NJ(N_¤ 8 . :. n3Hz. ¢%L=JLl¤>q H E I. O O. ¢L'¤rqe ¢L'r¤ qeL O. ¢)¤ N. L M. $ t , * !. % .h & #! & ! <!` !.[>/. !.^:h D, ']> 5 , >.. A -. .T* ! H !.# / !. 3 ( Y 5( M ' !# , :
(72) .kX % , (] '((.[7' <^. "u !.[ h .[F. &! ' !.H(.?/. /Z L O ¢L'¤ qs ! (/, 95 < :%5#' ] D, '] ^ . (.% + h ;b { F../"Y(+!< ( ' . % n3E _8 ! !. +!.* "u !.( : M e J b=[Z[le J \=BW;\=Ba J b=ax p \^w;w;YZW;])a J M`\^[Z[bGw;ehYZW;] `\^Wi'd;MXW;`M `\_Wnx;YZahYZ\_W\ p n H YZW I J M\ p MN
(73) },@ YZe wl\_eheYZj;[ZMQw p \=iYZx;MxM)bGx;x;YZahYZ\_Wlb=[Z[Zg p Mmid;Y p M a J bGa n E YZe [ZM#a±F[ZYZW;Mcb p Â(+³ ( « ! .): == n35(^E ] kn3%'H 5(5# > <!-k` % !.[ 3 : 3 ) ?./ ? & ! n3 E !. <8 . Y'./X"Y(& ! ! '# !.&* .[ n3H _?/. 67 # d d qa ¢%8d ¤?qax F ¢_· ¤ : x F zOU zOU ¢%8d ¤? 5 d qax q x F ¢_· ¤ : F zOU ¢%8d ¤ d qax ! J =¸L¸#p " ¬(¢b^¤I bGW;x¢¥j'¤ J \_[Zxkqjig3 CLM&% NO) Np %b)~ }#&9'% !¢Db_p ¤" bG
(74) W;x ¢_¢é· j'¤ :¤ ¢é`c¤Â\_[Z[Z\= ejigkYWnx;d;`aY\^Wk\_W7p ¢%d8J¤¢ép `#· a Mkw \i\G\= C MNNb}#'9 ¢é`c¤h¤X 6 W w'b aY`Xd;[¥b l`\^[Z[bGw;ehYZW;]k`X\_Wi'dnMW;`MO\= n3B H YZe W;\_atW;MMx;MXx M M^. ³ « = _8 n, 5* ! *. <%% ."u : .( ) : ,./] n _ % * !. !.t .A.Xh n X ¢%Ll¤ q L=J ¢%Ll¤iq ! =¸L¸#p " ¬ I J M SËY®éFw'b p a L \= a J MOMmid;YZ_b=[ZMW;`MOYZe)a p YiY¥ b=[ ;\ p a J M SË\_W;[Zg^FY Lw'b p a,s bGehed;NMka J b=at n ) . =. 53. . 3. =. YZekahM NYZW'bGahYZW;]l C MaAn ¬ z = ¢L'¤Aq L 6rn ¬z = ¢L'¤kq 6 jiiYZ\_d;e[gi=n YZeahM p NYZW'b=aYZW;]'¦g¡b=ehed;NwnahYZ\_W¡bGW;x C MNNb }3M¯iW;\=a J bGa¢_n _ n ¤SYZeaM p NOYWlb=ahYZW;]l u MW;`M_ d;ehYZW;]ka J M 7 & <"Y( ^/ !. b=w;w p \6bG` J \="¦§bG` J NbGY p
(75) |tM p e J \= YZa;Z2Z}_ C MNNb=e}^ËP4h \ p aM p NOYZW'b=aY\^W\G na_O<¢ n _tn. r¤ z©<¢ na_tn ¤ _n. YZa"ehd `Me ah\e J \= a J bGaQ¢_nW_tn ¤ 7/ + <"Y( ^/ , n YF M_W ¬ q x
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(79) x x I J M p p p [¥b=aahM w \^w'M a gYZeb`\^W;ehMXm6dnMW;`M\= ¢}c7 ¤ qax 8qax # qax hqax 'b=Wnx ífí æ ööö#.
(80) }#9. $ , #. Y t ! . D¢ P_¤7qax 8qax # qax q x \=)'¢}c¤YZe"ea p bGYZ] J a \ p b p xkj6gkb=WkM8b=ehgk`8b=ehM b=W'bG[Zg6eYZe"M¨iw;[Z\_YZahYZW;]ta J MtehwlM`Y¥b=[Le J b=wlM\G a J M ¢ L'D¤ qeLb=W;x = ¢%Ll¤rq ¢DP_¤YZeb=[Zeh\SMcb=egjigbSeha#b=W;x'b p x`cbGehMbGW'b=[ZgiehYZec^j;d;a MXehehMXW6aY¥b=[Z[Zg = \^W[ZM#Da [YZW;M8b p YZa gk\G>n u MXW;`M_lM0b p M x;\^W;M_. p d;[ZMe p M[ZYZMe. ³i¸l=³ )+ =, n E ] n H 5(Z 7 __`# !. ^ . !.(3 ):
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