27.12/28.20 YES 27.12/28.20 27.12/28.20 Problem 1: 27.12/28.20 27.12/28.20 (VAR A B V1 V2 X Y) 27.12/28.20 (THEORY 27.12/28.20 (AC mult plus union)) 27.12/28.20 (RULES 27.12/28.20 0(z) -> z 27.12/28.20 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.20 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.20 U103(tt) -> tt 27.12/28.20 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.20 U111(tt) -> tt 27.12/28.20 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.20 U121(tt) -> tt 27.12/28.20 U13(tt) -> tt 27.12/28.20 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.20 U132(tt) -> tt 27.12/28.20 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.20 U142(tt) -> tt 27.12/28.20 U151(tt) -> tt 27.12/28.20 U161(tt) -> tt 27.12/28.20 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.20 U172(tt) -> z 27.12/28.20 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.20 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.20 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.20 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.20 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.20 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.20 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.20 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.20 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.20 U202(tt,X) -> X 27.12/28.20 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.20 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.20 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.20 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.20 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.20 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.20 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.20 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.20 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.20 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.20 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.20 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.20 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.20 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.20 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.20 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.20 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.20 U242(tt,X) -> X 27.12/28.20 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.20 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.20 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.20 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.20 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.20 U26(tt) -> tt 27.12/28.20 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.20 U262(tt,X) -> X 27.12/28.20 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.20 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.20 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.20 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.20 U31(tt) -> tt 27.12/28.20 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.20 U42(tt) -> tt 27.12/28.20 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.20 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.20 U53(tt) -> tt 27.12/28.20 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.20 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.20 U63(tt) -> tt 27.12/28.20 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.20 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.20 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.20 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.20 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.20 U76(tt) -> tt 27.12/28.20 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.20 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.20 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.20 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.20 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.20 U86(tt) -> tt 27.12/28.20 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.20 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.20 U93(tt) -> tt 27.12/28.20 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.20 isBag(empty) -> tt 27.12/28.20 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.20 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.20 isBagKind(empty) -> tt 27.12/28.20 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.20 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.20 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.20 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.20 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.20 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.20 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.20 isBin(z) -> tt 27.12/28.20 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.20 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.20 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.20 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.20 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.20 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.20 isBinKind(z) -> tt 27.12/28.20 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.20 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.20 mult(z,X) -> U171(isBin(X),X) 27.12/28.20 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.20 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.20 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.20 plus(z,X) -> U201(isBin(X),X) 27.12/28.20 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.20 prod(empty) -> 1(z) 27.12/28.20 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.20 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.20 sum(empty) -> 0(z) 27.12/28.20 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.20 union(empty,X) -> X 27.12/28.20 union(X,empty) -> X 27.12/28.20 ) 27.12/28.20 27.12/28.20 Problem 1: 27.12/28.20 27.12/28.20 Dependency Pairs Processor: 27.12/28.20 -> FAxioms: 27.12/28.20 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.12/28.20 MULT(x6,x7) = MULT(x7,x6) 27.12/28.20 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.12/28.20 PLUS(x6,x7) = PLUS(x7,x6) 27.12/28.20 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8)) 27.12/28.20 UNION(x6,x7) = UNION(x7,x6) 27.12/28.20 -> Pairs: 27.12/28.20 U101#(tt,V1) -> U102#(isBagKind(V1),V1) 27.12/28.20 U101#(tt,V1) -> ISBAGKIND(V1) 27.12/28.20 U102#(tt,V1) -> U103#(isBag(V1)) 27.12/28.20 U102#(tt,V1) -> ISBAG(V1) 27.12/28.20 U11#(tt,V1) -> U12#(isBinKind(V1),V1) 27.12/28.20 U11#(tt,V1) -> ISBINKIND(V1) 27.12/28.20 U12#(tt,V1) -> U13#(isBin(V1)) 27.12/28.20 U12#(tt,V1) -> ISBIN(V1) 27.12/28.20 U131#(tt,V2) -> U132#(isBinKind(V2)) 27.12/28.20 U131#(tt,V2) -> ISBINKIND(V2) 27.12/28.20 U141#(tt,V2) -> U142#(isBinKind(V2)) 27.12/28.20 U141#(tt,V2) -> ISBINKIND(V2) 27.12/28.20 U171#(tt,X) -> U172#(isBinKind(X)) 27.12/28.20 U171#(tt,X) -> ISBINKIND(X) 27.12/28.20 U181#(tt,X,Y) -> U182#(isBinKind(X),X,Y) 27.12/28.20 U181#(tt,X,Y) -> ISBINKIND(X) 27.12/28.20 U182#(tt,X,Y) -> U183#(isBin(Y),X,Y) 27.12/28.20 U182#(tt,X,Y) -> ISBIN(Y) 27.12/28.20 U183#(tt,X,Y) -> U184#(isBinKind(Y),X,Y) 27.12/28.20 U183#(tt,X,Y) -> ISBINKIND(Y) 27.12/28.20 U184#(tt,X,Y) -> 0#(mult(X,Y)) 27.12/28.20 U184#(tt,X,Y) -> MULT(X,Y) 27.12/28.20 U191#(tt,X,Y) -> U192#(isBinKind(X),X,Y) 27.12/28.20 U191#(tt,X,Y) -> ISBINKIND(X) 27.12/28.20 U192#(tt,X,Y) -> U193#(isBin(Y),X,Y) 27.12/28.20 U192#(tt,X,Y) -> ISBIN(Y) 27.12/28.20 U193#(tt,X,Y) -> U194#(isBinKind(Y),X,Y) 27.12/28.20 U193#(tt,X,Y) -> ISBINKIND(Y) 27.12/28.20 U194#(tt,X,Y) -> 0#(mult(X,Y)) 27.12/28.20 U194#(tt,X,Y) -> MULT(X,Y) 27.12/28.20 U194#(tt,X,Y) -> PLUS(0(mult(X,Y)),Y) 27.12/28.20 U201#(tt,X) -> U202#(isBinKind(X),X) 27.12/28.20 U201#(tt,X) -> ISBINKIND(X) 27.12/28.20 U21#(tt,V1,V2) -> U22#(isBagKind(V1),V1,V2) 27.12/28.20 U21#(tt,V1,V2) -> ISBAGKIND(V1) 27.12/28.20 U211#(tt,X,Y) -> U212#(isBinKind(X),X,Y) 27.12/28.20 U211#(tt,X,Y) -> ISBINKIND(X) 27.12/28.20 U212#(tt,X,Y) -> U213#(isBin(Y),X,Y) 27.12/28.20 U212#(tt,X,Y) -> ISBIN(Y) 27.12/28.20 U213#(tt,X,Y) -> U214#(isBinKind(Y),X,Y) 27.12/28.20 U213#(tt,X,Y) -> ISBINKIND(Y) 27.12/28.20 U214#(tt,X,Y) -> 0#(plus(X,Y)) 27.12/28.20 U214#(tt,X,Y) -> PLUS(X,Y) 27.12/28.20 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.20 U22#(tt,V1,V2) -> ISBAGKIND(V2) 27.12/28.20 U221#(tt,X,Y) -> U222#(isBinKind(X),X,Y) 27.12/28.20 U221#(tt,X,Y) -> ISBINKIND(X) 27.12/28.20 U222#(tt,X,Y) -> U223#(isBin(Y),X,Y) 27.12/28.20 U222#(tt,X,Y) -> ISBIN(Y) 27.12/28.20 U223#(tt,X,Y) -> U224#(isBinKind(Y),X,Y) 27.12/28.20 U223#(tt,X,Y) -> ISBINKIND(Y) 27.12/28.20 U224#(tt,X,Y) -> PLUS(X,Y) 27.12/28.20 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.20 U23#(tt,V1,V2) -> ISBAGKIND(V2) 27.12/28.20 U231#(tt,X,Y) -> U232#(isBinKind(X),X,Y) 27.12/28.20 U231#(tt,X,Y) -> ISBINKIND(X) 27.12/28.20 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.12/28.20 U232#(tt,X,Y) -> ISBIN(Y) 27.12/28.20 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.12/28.20 U233#(tt,X,Y) -> ISBINKIND(Y) 27.12/28.20 U234#(tt,X,Y) -> 0#(plus(plus(X,Y),1(z))) 27.12/28.20 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.12/28.20 U234#(tt,X,Y) -> PLUS(X,Y) 27.12/28.20 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.20 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.20 U241#(tt,X) -> U242#(isBinKind(X),X) 27.12/28.20 U241#(tt,X) -> ISBINKIND(X) 27.12/28.20 U25#(tt,V2) -> U26#(isBag(V2)) 27.12/28.20 U25#(tt,V2) -> ISBAG(V2) 27.12/28.20 U251#(tt,A,B) -> U252#(isBagKind(A),A,B) 27.12/28.20 U251#(tt,A,B) -> ISBAGKIND(A) 27.12/28.20 U252#(tt,A,B) -> U253#(isBag(B),A,B) 27.12/28.20 U252#(tt,A,B) -> ISBAG(B) 27.12/28.20 U253#(tt,A,B) -> U254#(isBagKind(B),A,B) 27.12/28.20 U253#(tt,A,B) -> ISBAGKIND(B) 27.12/28.20 U254#(tt,A,B) -> MULT(prod(A),prod(B)) 27.12/28.20 U254#(tt,A,B) -> PROD(A) 27.12/28.20 U254#(tt,A,B) -> PROD(B) 27.12/28.20 U261#(tt,X) -> U262#(isBinKind(X),X) 27.12/28.20 U261#(tt,X) -> ISBINKIND(X) 27.12/28.20 U271#(tt,A,B) -> U272#(isBagKind(A),A,B) 27.12/28.20 U271#(tt,A,B) -> ISBAGKIND(A) 27.12/28.20 U272#(tt,A,B) -> U273#(isBag(B),A,B) 27.12/28.20 U272#(tt,A,B) -> ISBAG(B) 27.12/28.20 U273#(tt,A,B) -> U274#(isBagKind(B),A,B) 27.12/28.20 U273#(tt,A,B) -> ISBAGKIND(B) 27.12/28.20 U274#(tt,A,B) -> PLUS(sum(A),sum(B)) 27.12/28.20 U274#(tt,A,B) -> SUM(A) 27.12/28.20 U274#(tt,A,B) -> SUM(B) 27.12/28.20 U41#(tt,V2) -> U42#(isBagKind(V2)) 27.12/28.20 U41#(tt,V2) -> ISBAGKIND(V2) 27.12/28.20 U51#(tt,V1) -> U52#(isBinKind(V1),V1) 27.12/28.20 U51#(tt,V1) -> ISBINKIND(V1) 27.12/28.20 U52#(tt,V1) -> U53#(isBin(V1)) 27.12/28.20 U52#(tt,V1) -> ISBIN(V1) 27.12/28.20 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.20 U61#(tt,V1) -> ISBINKIND(V1) 27.12/28.20 U62#(tt,V1) -> U63#(isBin(V1)) 27.12/28.20 U62#(tt,V1) -> ISBIN(V1) 27.12/28.20 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.20 U71#(tt,V1,V2) -> ISBINKIND(V1) 27.12/28.20 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.20 U72#(tt,V1,V2) -> ISBINKIND(V2) 27.12/28.20 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.20 U73#(tt,V1,V2) -> ISBINKIND(V2) 27.12/28.20 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.20 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.20 U75#(tt,V2) -> U76#(isBin(V2)) 27.12/28.20 U75#(tt,V2) -> ISBIN(V2) 27.12/28.20 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.20 U81#(tt,V1,V2) -> ISBINKIND(V1) 27.12/28.20 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.20 U82#(tt,V1,V2) -> ISBINKIND(V2) 27.12/28.20 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.20 U83#(tt,V1,V2) -> ISBINKIND(V2) 27.12/28.20 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.20 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.20 U85#(tt,V2) -> U86#(isBin(V2)) 27.12/28.20 U85#(tt,V2) -> ISBIN(V2) 27.12/28.20 U91#(tt,V1) -> U92#(isBagKind(V1),V1) 27.12/28.20 U91#(tt,V1) -> ISBAGKIND(V1) 27.12/28.20 U92#(tt,V1) -> U93#(isBag(V1)) 27.12/28.20 U92#(tt,V1) -> ISBAG(V1) 27.12/28.20 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.20 ISBAG(union(V1,V2)) -> ISBAGKIND(V1) 27.12/28.20 ISBAG(singl(V1)) -> U11#(isBinKind(V1),V1) 27.12/28.20 ISBAG(singl(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBAGKIND(union(V1,V2)) -> U41#(isBagKind(V1),V2) 27.12/28.20 ISBAGKIND(union(V1,V2)) -> ISBAGKIND(V1) 27.12/28.20 ISBAGKIND(singl(V1)) -> U31#(isBinKind(V1)) 27.12/28.20 ISBAGKIND(singl(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.20 ISBIN(0(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.20 ISBIN(mult(V1,V2)) -> ISBINKIND(V1) 27.12/28.20 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.20 ISBIN(plus(V1,V2)) -> ISBINKIND(V1) 27.12/28.20 ISBIN(prod(V1)) -> U91#(isBagKind(V1),V1) 27.12/28.20 ISBIN(prod(V1)) -> ISBAGKIND(V1) 27.12/28.20 ISBIN(sum(V1)) -> U101#(isBagKind(V1),V1) 27.12/28.20 ISBIN(sum(V1)) -> ISBAGKIND(V1) 27.12/28.20 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.20 ISBIN(1(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(0(V1)) -> U111#(isBinKind(V1)) 27.12/28.20 ISBINKIND(0(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(mult(V1,V2)) -> U131#(isBinKind(V1),V2) 27.12/28.20 ISBINKIND(mult(V1,V2)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(plus(V1,V2)) -> U141#(isBinKind(V1),V2) 27.12/28.20 ISBINKIND(plus(V1,V2)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(prod(V1)) -> U151#(isBagKind(V1)) 27.12/28.20 ISBINKIND(prod(V1)) -> ISBAGKIND(V1) 27.12/28.20 ISBINKIND(sum(V1)) -> U161#(isBagKind(V1)) 27.12/28.20 ISBINKIND(sum(V1)) -> ISBAGKIND(V1) 27.12/28.20 ISBINKIND(1(V1)) -> U121#(isBinKind(V1)) 27.12/28.20 ISBINKIND(1(V1)) -> ISBINKIND(V1) 27.12/28.20 MULT(0(X),Y) -> U181#(isBin(X),X,Y) 27.12/28.20 MULT(0(X),Y) -> ISBIN(X) 27.12/28.20 MULT(mult(0(X),Y),x6) -> U181#(isBin(X),X,Y) 27.12/28.20 MULT(mult(0(X),Y),x6) -> ISBIN(X) 27.12/28.20 MULT(mult(0(X),Y),x6) -> MULT(U181(isBin(X),X,Y),x6) 27.12/28.20 MULT(mult(1(X),Y),x6) -> U191#(isBin(X),X,Y) 27.12/28.20 MULT(mult(1(X),Y),x6) -> ISBIN(X) 27.12/28.20 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.12/28.20 MULT(mult(z,X),x6) -> U171#(isBin(X),X) 27.12/28.20 MULT(mult(z,X),x6) -> ISBIN(X) 27.12/28.20 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.12/28.20 MULT(1(X),Y) -> U191#(isBin(X),X,Y) 27.12/28.20 MULT(1(X),Y) -> ISBIN(X) 27.12/28.20 MULT(z,X) -> U171#(isBin(X),X) 27.12/28.20 MULT(z,X) -> ISBIN(X) 27.12/28.20 PLUS(0(X),0(Y)) -> U211#(isBin(X),X,Y) 27.12/28.20 PLUS(0(X),0(Y)) -> ISBIN(X) 27.12/28.20 PLUS(0(X),1(Y)) -> U221#(isBin(X),X,Y) 27.12/28.20 PLUS(0(X),1(Y)) -> ISBIN(X) 27.12/28.20 PLUS(plus(0(X),0(Y)),x6) -> U211#(isBin(X),X,Y) 27.12/28.20 PLUS(plus(0(X),0(Y)),x6) -> ISBIN(X) 27.12/28.20 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.12/28.20 PLUS(plus(0(X),1(Y)),x6) -> U221#(isBin(X),X,Y) 27.12/28.20 PLUS(plus(0(X),1(Y)),x6) -> ISBIN(X) 27.12/28.20 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.12/28.20 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.12/28.20 PLUS(plus(1(X),1(Y)),x6) -> ISBIN(X) 27.12/28.20 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.12/28.20 PLUS(plus(z,X),x6) -> U201#(isBin(X),X) 27.12/28.20 PLUS(plus(z,X),x6) -> ISBIN(X) 27.12/28.20 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.12/28.20 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.12/28.20 PLUS(1(X),1(Y)) -> ISBIN(X) 27.12/28.20 PLUS(z,X) -> U201#(isBin(X),X) 27.12/28.20 PLUS(z,X) -> ISBIN(X) 27.12/28.20 PROD(union(A,B)) -> U251#(isBag(A),A,B) 27.12/28.20 PROD(union(A,B)) -> ISBAG(A) 27.12/28.20 PROD(singl(X)) -> U241#(isBin(X),X) 27.12/28.20 PROD(singl(X)) -> ISBIN(X) 27.12/28.20 SUM(union(A,B)) -> U271#(isBag(A),A,B) 27.12/28.20 SUM(union(A,B)) -> ISBAG(A) 27.12/28.20 SUM(empty) -> 0#(z) 27.12/28.20 SUM(singl(X)) -> U261#(isBin(X),X) 27.12/28.20 SUM(singl(X)) -> ISBIN(X) 27.12/28.20 UNION(union(empty,X),x6) -> UNION(X,x6) 27.12/28.20 UNION(union(X,empty),x6) -> UNION(X,x6) 27.12/28.20 -> EAxioms: 27.12/28.20 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) = mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) = plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) = union(x7,x6) 27.12/28.20 -> Rules: 27.12/28.20 0(z) -> z 27.12/28.20 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.20 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.20 U103(tt) -> tt 27.12/28.20 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.20 U111(tt) -> tt 27.12/28.20 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.20 U121(tt) -> tt 27.12/28.20 U13(tt) -> tt 27.12/28.20 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.20 U132(tt) -> tt 27.12/28.20 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.20 U142(tt) -> tt 27.12/28.20 U151(tt) -> tt 27.12/28.20 U161(tt) -> tt 27.12/28.20 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.20 U172(tt) -> z 27.12/28.20 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.20 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.20 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.20 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.20 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.20 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.20 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.20 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.20 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.20 U202(tt,X) -> X 27.12/28.20 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.20 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.20 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.20 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.20 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.20 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.20 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.20 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.20 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.20 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.20 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.20 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.20 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.20 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.20 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.20 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.20 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.20 U242(tt,X) -> X 27.12/28.20 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.20 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.20 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.20 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.20 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.20 U26(tt) -> tt 27.12/28.20 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.20 U262(tt,X) -> X 27.12/28.20 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.20 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.20 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.20 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.20 U31(tt) -> tt 27.12/28.20 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.20 U42(tt) -> tt 27.12/28.20 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.20 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.20 U53(tt) -> tt 27.12/28.20 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.20 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.20 U63(tt) -> tt 27.12/28.20 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.20 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.20 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.20 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.20 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.20 U76(tt) -> tt 27.12/28.20 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.20 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.20 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.20 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.20 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.20 U86(tt) -> tt 27.12/28.20 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.20 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.20 U93(tt) -> tt 27.12/28.20 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.20 isBag(empty) -> tt 27.12/28.20 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.20 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.20 isBagKind(empty) -> tt 27.12/28.20 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.20 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.20 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.20 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.20 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.20 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.20 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.20 isBin(z) -> tt 27.12/28.20 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.20 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.20 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.20 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.20 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.20 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.20 isBinKind(z) -> tt 27.12/28.20 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.20 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.20 mult(z,X) -> U171(isBin(X),X) 27.12/28.20 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.20 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.20 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.20 plus(z,X) -> U201(isBin(X),X) 27.12/28.20 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.20 prod(empty) -> 1(z) 27.12/28.20 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.20 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.20 sum(empty) -> 0(z) 27.12/28.20 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.20 union(empty,X) -> X 27.12/28.20 union(X,empty) -> X 27.12/28.20 -> SRules: 27.12/28.20 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.12/28.20 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.12/28.20 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.12/28.20 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.12/28.20 UNION(union(x6,x7),x8) -> UNION(x6,x7) 27.12/28.20 UNION(x6,union(x7,x8)) -> UNION(x7,x8) 27.12/28.20 27.12/28.20 Problem 1: 27.12/28.20 27.12/28.20 SCC Processor: 27.12/28.20 -> FAxioms: 27.12/28.20 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.12/28.20 MULT(x6,x7) = MULT(x7,x6) 27.12/28.20 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.12/28.20 PLUS(x6,x7) = PLUS(x7,x6) 27.12/28.20 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8)) 27.12/28.20 UNION(x6,x7) = UNION(x7,x6) 27.12/28.20 -> Pairs: 27.12/28.20 U101#(tt,V1) -> U102#(isBagKind(V1),V1) 27.12/28.20 U101#(tt,V1) -> ISBAGKIND(V1) 27.12/28.20 U102#(tt,V1) -> U103#(isBag(V1)) 27.12/28.20 U102#(tt,V1) -> ISBAG(V1) 27.12/28.20 U11#(tt,V1) -> U12#(isBinKind(V1),V1) 27.12/28.20 U11#(tt,V1) -> ISBINKIND(V1) 27.12/28.20 U12#(tt,V1) -> U13#(isBin(V1)) 27.12/28.20 U12#(tt,V1) -> ISBIN(V1) 27.12/28.20 U131#(tt,V2) -> U132#(isBinKind(V2)) 27.12/28.20 U131#(tt,V2) -> ISBINKIND(V2) 27.12/28.20 U141#(tt,V2) -> U142#(isBinKind(V2)) 27.12/28.20 U141#(tt,V2) -> ISBINKIND(V2) 27.12/28.20 U171#(tt,X) -> U172#(isBinKind(X)) 27.12/28.20 U171#(tt,X) -> ISBINKIND(X) 27.12/28.20 U181#(tt,X,Y) -> U182#(isBinKind(X),X,Y) 27.12/28.20 U181#(tt,X,Y) -> ISBINKIND(X) 27.12/28.20 U182#(tt,X,Y) -> U183#(isBin(Y),X,Y) 27.12/28.20 U182#(tt,X,Y) -> ISBIN(Y) 27.12/28.20 U183#(tt,X,Y) -> U184#(isBinKind(Y),X,Y) 27.12/28.20 U183#(tt,X,Y) -> ISBINKIND(Y) 27.12/28.20 U184#(tt,X,Y) -> 0#(mult(X,Y)) 27.12/28.20 U184#(tt,X,Y) -> MULT(X,Y) 27.12/28.20 U191#(tt,X,Y) -> U192#(isBinKind(X),X,Y) 27.12/28.20 U191#(tt,X,Y) -> ISBINKIND(X) 27.12/28.20 U192#(tt,X,Y) -> U193#(isBin(Y),X,Y) 27.12/28.20 U192#(tt,X,Y) -> ISBIN(Y) 27.12/28.20 U193#(tt,X,Y) -> U194#(isBinKind(Y),X,Y) 27.12/28.20 U193#(tt,X,Y) -> ISBINKIND(Y) 27.12/28.20 U194#(tt,X,Y) -> 0#(mult(X,Y)) 27.12/28.20 U194#(tt,X,Y) -> MULT(X,Y) 27.12/28.20 U194#(tt,X,Y) -> PLUS(0(mult(X,Y)),Y) 27.12/28.20 U201#(tt,X) -> U202#(isBinKind(X),X) 27.12/28.20 U201#(tt,X) -> ISBINKIND(X) 27.12/28.20 U21#(tt,V1,V2) -> U22#(isBagKind(V1),V1,V2) 27.12/28.20 U21#(tt,V1,V2) -> ISBAGKIND(V1) 27.12/28.20 U211#(tt,X,Y) -> U212#(isBinKind(X),X,Y) 27.12/28.20 U211#(tt,X,Y) -> ISBINKIND(X) 27.12/28.20 U212#(tt,X,Y) -> U213#(isBin(Y),X,Y) 27.12/28.20 U212#(tt,X,Y) -> ISBIN(Y) 27.12/28.20 U213#(tt,X,Y) -> U214#(isBinKind(Y),X,Y) 27.12/28.20 U213#(tt,X,Y) -> ISBINKIND(Y) 27.12/28.20 U214#(tt,X,Y) -> 0#(plus(X,Y)) 27.12/28.20 U214#(tt,X,Y) -> PLUS(X,Y) 27.12/28.20 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.20 U22#(tt,V1,V2) -> ISBAGKIND(V2) 27.12/28.20 U221#(tt,X,Y) -> U222#(isBinKind(X),X,Y) 27.12/28.20 U221#(tt,X,Y) -> ISBINKIND(X) 27.12/28.20 U222#(tt,X,Y) -> U223#(isBin(Y),X,Y) 27.12/28.20 U222#(tt,X,Y) -> ISBIN(Y) 27.12/28.20 U223#(tt,X,Y) -> U224#(isBinKind(Y),X,Y) 27.12/28.20 U223#(tt,X,Y) -> ISBINKIND(Y) 27.12/28.20 U224#(tt,X,Y) -> PLUS(X,Y) 27.12/28.20 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.20 U23#(tt,V1,V2) -> ISBAGKIND(V2) 27.12/28.20 U231#(tt,X,Y) -> U232#(isBinKind(X),X,Y) 27.12/28.20 U231#(tt,X,Y) -> ISBINKIND(X) 27.12/28.20 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.12/28.20 U232#(tt,X,Y) -> ISBIN(Y) 27.12/28.20 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.12/28.20 U233#(tt,X,Y) -> ISBINKIND(Y) 27.12/28.20 U234#(tt,X,Y) -> 0#(plus(plus(X,Y),1(z))) 27.12/28.20 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.12/28.20 U234#(tt,X,Y) -> PLUS(X,Y) 27.12/28.20 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.20 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.20 U241#(tt,X) -> U242#(isBinKind(X),X) 27.12/28.20 U241#(tt,X) -> ISBINKIND(X) 27.12/28.20 U25#(tt,V2) -> U26#(isBag(V2)) 27.12/28.20 U25#(tt,V2) -> ISBAG(V2) 27.12/28.20 U251#(tt,A,B) -> U252#(isBagKind(A),A,B) 27.12/28.20 U251#(tt,A,B) -> ISBAGKIND(A) 27.12/28.20 U252#(tt,A,B) -> U253#(isBag(B),A,B) 27.12/28.20 U252#(tt,A,B) -> ISBAG(B) 27.12/28.20 U253#(tt,A,B) -> U254#(isBagKind(B),A,B) 27.12/28.20 U253#(tt,A,B) -> ISBAGKIND(B) 27.12/28.20 U254#(tt,A,B) -> MULT(prod(A),prod(B)) 27.12/28.20 U254#(tt,A,B) -> PROD(A) 27.12/28.20 U254#(tt,A,B) -> PROD(B) 27.12/28.20 U261#(tt,X) -> U262#(isBinKind(X),X) 27.12/28.20 U261#(tt,X) -> ISBINKIND(X) 27.12/28.20 U271#(tt,A,B) -> U272#(isBagKind(A),A,B) 27.12/28.20 U271#(tt,A,B) -> ISBAGKIND(A) 27.12/28.20 U272#(tt,A,B) -> U273#(isBag(B),A,B) 27.12/28.20 U272#(tt,A,B) -> ISBAG(B) 27.12/28.20 U273#(tt,A,B) -> U274#(isBagKind(B),A,B) 27.12/28.20 U273#(tt,A,B) -> ISBAGKIND(B) 27.12/28.20 U274#(tt,A,B) -> PLUS(sum(A),sum(B)) 27.12/28.20 U274#(tt,A,B) -> SUM(A) 27.12/28.20 U274#(tt,A,B) -> SUM(B) 27.12/28.20 U41#(tt,V2) -> U42#(isBagKind(V2)) 27.12/28.20 U41#(tt,V2) -> ISBAGKIND(V2) 27.12/28.20 U51#(tt,V1) -> U52#(isBinKind(V1),V1) 27.12/28.20 U51#(tt,V1) -> ISBINKIND(V1) 27.12/28.20 U52#(tt,V1) -> U53#(isBin(V1)) 27.12/28.20 U52#(tt,V1) -> ISBIN(V1) 27.12/28.20 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.20 U61#(tt,V1) -> ISBINKIND(V1) 27.12/28.20 U62#(tt,V1) -> U63#(isBin(V1)) 27.12/28.20 U62#(tt,V1) -> ISBIN(V1) 27.12/28.20 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.20 U71#(tt,V1,V2) -> ISBINKIND(V1) 27.12/28.20 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.20 U72#(tt,V1,V2) -> ISBINKIND(V2) 27.12/28.20 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.20 U73#(tt,V1,V2) -> ISBINKIND(V2) 27.12/28.20 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.20 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.20 U75#(tt,V2) -> U76#(isBin(V2)) 27.12/28.20 U75#(tt,V2) -> ISBIN(V2) 27.12/28.20 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.20 U81#(tt,V1,V2) -> ISBINKIND(V1) 27.12/28.20 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.20 U82#(tt,V1,V2) -> ISBINKIND(V2) 27.12/28.20 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.20 U83#(tt,V1,V2) -> ISBINKIND(V2) 27.12/28.20 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.20 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.20 U85#(tt,V2) -> U86#(isBin(V2)) 27.12/28.20 U85#(tt,V2) -> ISBIN(V2) 27.12/28.20 U91#(tt,V1) -> U92#(isBagKind(V1),V1) 27.12/28.20 U91#(tt,V1) -> ISBAGKIND(V1) 27.12/28.20 U92#(tt,V1) -> U93#(isBag(V1)) 27.12/28.20 U92#(tt,V1) -> ISBAG(V1) 27.12/28.20 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.20 ISBAG(union(V1,V2)) -> ISBAGKIND(V1) 27.12/28.20 ISBAG(singl(V1)) -> U11#(isBinKind(V1),V1) 27.12/28.20 ISBAG(singl(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBAGKIND(union(V1,V2)) -> U41#(isBagKind(V1),V2) 27.12/28.20 ISBAGKIND(union(V1,V2)) -> ISBAGKIND(V1) 27.12/28.20 ISBAGKIND(singl(V1)) -> U31#(isBinKind(V1)) 27.12/28.20 ISBAGKIND(singl(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.20 ISBIN(0(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.20 ISBIN(mult(V1,V2)) -> ISBINKIND(V1) 27.12/28.20 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.20 ISBIN(plus(V1,V2)) -> ISBINKIND(V1) 27.12/28.20 ISBIN(prod(V1)) -> U91#(isBagKind(V1),V1) 27.12/28.20 ISBIN(prod(V1)) -> ISBAGKIND(V1) 27.12/28.20 ISBIN(sum(V1)) -> U101#(isBagKind(V1),V1) 27.12/28.20 ISBIN(sum(V1)) -> ISBAGKIND(V1) 27.12/28.20 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.20 ISBIN(1(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(0(V1)) -> U111#(isBinKind(V1)) 27.12/28.20 ISBINKIND(0(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(mult(V1,V2)) -> U131#(isBinKind(V1),V2) 27.12/28.20 ISBINKIND(mult(V1,V2)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(plus(V1,V2)) -> U141#(isBinKind(V1),V2) 27.12/28.20 ISBINKIND(plus(V1,V2)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(prod(V1)) -> U151#(isBagKind(V1)) 27.12/28.20 ISBINKIND(prod(V1)) -> ISBAGKIND(V1) 27.12/28.20 ISBINKIND(sum(V1)) -> U161#(isBagKind(V1)) 27.12/28.20 ISBINKIND(sum(V1)) -> ISBAGKIND(V1) 27.12/28.20 ISBINKIND(1(V1)) -> U121#(isBinKind(V1)) 27.12/28.20 ISBINKIND(1(V1)) -> ISBINKIND(V1) 27.12/28.20 MULT(0(X),Y) -> U181#(isBin(X),X,Y) 27.12/28.20 MULT(0(X),Y) -> ISBIN(X) 27.12/28.20 MULT(mult(0(X),Y),x6) -> U181#(isBin(X),X,Y) 27.12/28.20 MULT(mult(0(X),Y),x6) -> ISBIN(X) 27.12/28.20 MULT(mult(0(X),Y),x6) -> MULT(U181(isBin(X),X,Y),x6) 27.12/28.20 MULT(mult(1(X),Y),x6) -> U191#(isBin(X),X,Y) 27.12/28.20 MULT(mult(1(X),Y),x6) -> ISBIN(X) 27.12/28.20 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.12/28.20 MULT(mult(z,X),x6) -> U171#(isBin(X),X) 27.12/28.20 MULT(mult(z,X),x6) -> ISBIN(X) 27.12/28.20 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.12/28.20 MULT(1(X),Y) -> U191#(isBin(X),X,Y) 27.12/28.20 MULT(1(X),Y) -> ISBIN(X) 27.12/28.20 MULT(z,X) -> U171#(isBin(X),X) 27.12/28.20 MULT(z,X) -> ISBIN(X) 27.12/28.20 PLUS(0(X),0(Y)) -> U211#(isBin(X),X,Y) 27.12/28.20 PLUS(0(X),0(Y)) -> ISBIN(X) 27.12/28.20 PLUS(0(X),1(Y)) -> U221#(isBin(X),X,Y) 27.12/28.20 PLUS(0(X),1(Y)) -> ISBIN(X) 27.12/28.20 PLUS(plus(0(X),0(Y)),x6) -> U211#(isBin(X),X,Y) 27.12/28.20 PLUS(plus(0(X),0(Y)),x6) -> ISBIN(X) 27.12/28.20 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.12/28.20 PLUS(plus(0(X),1(Y)),x6) -> U221#(isBin(X),X,Y) 27.12/28.20 PLUS(plus(0(X),1(Y)),x6) -> ISBIN(X) 27.12/28.20 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.12/28.20 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.12/28.20 PLUS(plus(1(X),1(Y)),x6) -> ISBIN(X) 27.12/28.20 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.12/28.20 PLUS(plus(z,X),x6) -> U201#(isBin(X),X) 27.12/28.20 PLUS(plus(z,X),x6) -> ISBIN(X) 27.12/28.20 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.12/28.20 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.12/28.20 PLUS(1(X),1(Y)) -> ISBIN(X) 27.12/28.20 PLUS(z,X) -> U201#(isBin(X),X) 27.12/28.20 PLUS(z,X) -> ISBIN(X) 27.12/28.20 PROD(union(A,B)) -> U251#(isBag(A),A,B) 27.12/28.20 PROD(union(A,B)) -> ISBAG(A) 27.12/28.20 PROD(singl(X)) -> U241#(isBin(X),X) 27.12/28.20 PROD(singl(X)) -> ISBIN(X) 27.12/28.20 SUM(union(A,B)) -> U271#(isBag(A),A,B) 27.12/28.20 SUM(union(A,B)) -> ISBAG(A) 27.12/28.20 SUM(empty) -> 0#(z) 27.12/28.20 SUM(singl(X)) -> U261#(isBin(X),X) 27.12/28.20 SUM(singl(X)) -> ISBIN(X) 27.12/28.20 UNION(union(empty,X),x6) -> UNION(X,x6) 27.12/28.20 UNION(union(X,empty),x6) -> UNION(X,x6) 27.12/28.20 -> EAxioms: 27.12/28.20 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) = mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) = plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) = union(x7,x6) 27.12/28.20 -> Rules: 27.12/28.20 0(z) -> z 27.12/28.20 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.20 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.20 U103(tt) -> tt 27.12/28.20 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.20 U111(tt) -> tt 27.12/28.20 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.20 U121(tt) -> tt 27.12/28.20 U13(tt) -> tt 27.12/28.20 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.20 U132(tt) -> tt 27.12/28.20 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.20 U142(tt) -> tt 27.12/28.20 U151(tt) -> tt 27.12/28.20 U161(tt) -> tt 27.12/28.20 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.20 U172(tt) -> z 27.12/28.20 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.20 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.20 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.20 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.20 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.20 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.20 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.20 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.20 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.20 U202(tt,X) -> X 27.12/28.20 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.20 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.20 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.20 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.20 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.20 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.20 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.20 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.20 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.20 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.20 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.20 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.20 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.20 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.20 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.20 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.20 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.20 U242(tt,X) -> X 27.12/28.20 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.20 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.20 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.20 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.20 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.20 U26(tt) -> tt 27.12/28.20 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.20 U262(tt,X) -> X 27.12/28.20 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.20 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.20 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.20 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.20 U31(tt) -> tt 27.12/28.20 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.20 U42(tt) -> tt 27.12/28.20 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.20 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.20 U53(tt) -> tt 27.12/28.20 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.20 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.20 U63(tt) -> tt 27.12/28.20 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.20 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.20 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.20 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.20 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.20 U76(tt) -> tt 27.12/28.20 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.20 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.20 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.20 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.20 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.20 U86(tt) -> tt 27.12/28.20 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.20 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.20 U93(tt) -> tt 27.12/28.20 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.20 isBag(empty) -> tt 27.12/28.20 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.20 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.20 isBagKind(empty) -> tt 27.12/28.20 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.20 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.20 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.20 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.20 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.20 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.20 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.20 isBin(z) -> tt 27.12/28.20 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.20 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.20 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.20 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.20 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.20 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.20 isBinKind(z) -> tt 27.12/28.20 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.20 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.20 mult(z,X) -> U171(isBin(X),X) 27.12/28.20 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.20 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.20 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.20 plus(z,X) -> U201(isBin(X),X) 27.12/28.20 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.20 prod(empty) -> 1(z) 27.12/28.20 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.20 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.20 sum(empty) -> 0(z) 27.12/28.20 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.20 union(empty,X) -> X 27.12/28.20 union(X,empty) -> X 27.12/28.20 -> SRules: 27.12/28.20 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.12/28.20 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.12/28.20 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.12/28.20 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.12/28.20 UNION(union(x6,x7),x8) -> UNION(x6,x7) 27.12/28.20 UNION(x6,union(x7,x8)) -> UNION(x7,x8) 27.12/28.20 ->Strongly Connected Components: 27.12/28.20 ->->Cycle: 27.12/28.20 ->->-> Pairs: 27.12/28.20 UNION(union(empty,X),x6) -> UNION(X,x6) 27.12/28.20 UNION(union(X,empty),x6) -> UNION(X,x6) 27.12/28.20 -> FAxioms: 27.12/28.20 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) -> mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) -> plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) -> union(x7,x6) 27.12/28.20 UNION(union(x6,x7),x8) -> UNION(x6,union(x7,x8)) 27.12/28.20 UNION(x6,x7) -> UNION(x7,x6) 27.12/28.20 -> EAxioms: 27.12/28.20 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) = mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) = plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) = union(x7,x6) 27.12/28.20 ->->-> Rules: 27.12/28.20 0(z) -> z 27.12/28.20 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.20 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.20 U103(tt) -> tt 27.12/28.20 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.20 U111(tt) -> tt 27.12/28.20 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.20 U121(tt) -> tt 27.12/28.20 U13(tt) -> tt 27.12/28.20 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.20 U132(tt) -> tt 27.12/28.20 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.20 U142(tt) -> tt 27.12/28.20 U151(tt) -> tt 27.12/28.20 U161(tt) -> tt 27.12/28.20 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.20 U172(tt) -> z 27.12/28.20 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.20 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.20 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.20 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.20 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.20 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.20 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.20 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.20 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.20 U202(tt,X) -> X 27.12/28.20 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.20 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.20 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.20 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.20 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.20 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.20 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.20 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.20 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.20 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.20 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.20 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.20 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.20 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.20 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.20 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.20 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.20 U242(tt,X) -> X 27.12/28.20 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.20 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.20 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.20 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.20 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.20 U26(tt) -> tt 27.12/28.20 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.20 U262(tt,X) -> X 27.12/28.20 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.20 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.20 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.20 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.20 U31(tt) -> tt 27.12/28.20 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.20 U42(tt) -> tt 27.12/28.20 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.20 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.20 U53(tt) -> tt 27.12/28.20 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.20 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.20 U63(tt) -> tt 27.12/28.20 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.20 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.20 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.20 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.20 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.20 U76(tt) -> tt 27.12/28.20 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.20 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.20 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.20 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.20 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.20 U86(tt) -> tt 27.12/28.20 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.20 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.20 U93(tt) -> tt 27.12/28.20 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.20 isBag(empty) -> tt 27.12/28.20 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.20 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.20 isBagKind(empty) -> tt 27.12/28.20 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.20 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.20 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.20 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.20 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.20 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.20 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.20 isBin(z) -> tt 27.12/28.20 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.20 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.20 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.20 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.20 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.20 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.20 isBinKind(z) -> tt 27.12/28.20 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.20 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.20 mult(z,X) -> U171(isBin(X),X) 27.12/28.20 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.20 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.20 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.20 plus(z,X) -> U201(isBin(X),X) 27.12/28.20 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.20 prod(empty) -> 1(z) 27.12/28.20 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.20 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.20 sum(empty) -> 0(z) 27.12/28.20 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.20 union(empty,X) -> X 27.12/28.20 union(X,empty) -> X 27.12/28.20 -> SRules: 27.12/28.20 UNION(union(x6,x7),x8) -> UNION(x6,x7) 27.12/28.20 UNION(x6,union(x7,x8)) -> UNION(x7,x8) 27.12/28.20 ->->Cycle: 27.12/28.20 ->->-> Pairs: 27.12/28.20 U131#(tt,V2) -> ISBINKIND(V2) 27.12/28.20 U141#(tt,V2) -> ISBINKIND(V2) 27.12/28.20 U41#(tt,V2) -> ISBAGKIND(V2) 27.12/28.20 ISBAGKIND(union(V1,V2)) -> U41#(isBagKind(V1),V2) 27.12/28.20 ISBAGKIND(union(V1,V2)) -> ISBAGKIND(V1) 27.12/28.20 ISBAGKIND(singl(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(0(V1)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(mult(V1,V2)) -> U131#(isBinKind(V1),V2) 27.12/28.20 ISBINKIND(mult(V1,V2)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(plus(V1,V2)) -> U141#(isBinKind(V1),V2) 27.12/28.20 ISBINKIND(plus(V1,V2)) -> ISBINKIND(V1) 27.12/28.20 ISBINKIND(prod(V1)) -> ISBAGKIND(V1) 27.12/28.20 ISBINKIND(sum(V1)) -> ISBAGKIND(V1) 27.12/28.20 ISBINKIND(1(V1)) -> ISBINKIND(V1) 27.12/28.20 -> FAxioms: 27.12/28.20 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) -> mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) -> plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) -> union(x7,x6) 27.12/28.20 -> EAxioms: 27.12/28.20 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) = mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) = plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) = union(x7,x6) 27.12/28.20 ->->-> Rules: 27.12/28.20 0(z) -> z 27.12/28.20 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.20 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.20 U103(tt) -> tt 27.12/28.20 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.20 U111(tt) -> tt 27.12/28.20 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.20 U121(tt) -> tt 27.12/28.20 U13(tt) -> tt 27.12/28.20 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.20 U132(tt) -> tt 27.12/28.20 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.20 U142(tt) -> tt 27.12/28.20 U151(tt) -> tt 27.12/28.20 U161(tt) -> tt 27.12/28.20 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.20 U172(tt) -> z 27.12/28.20 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.20 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.20 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.20 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.20 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.20 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.20 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.20 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.20 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.20 U202(tt,X) -> X 27.12/28.20 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.20 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.20 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.20 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.20 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.20 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.20 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.20 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.20 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.20 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.20 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.20 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.20 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.20 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.20 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.20 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.20 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.20 U242(tt,X) -> X 27.12/28.20 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.20 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.20 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.20 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.20 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.20 U26(tt) -> tt 27.12/28.20 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.20 U262(tt,X) -> X 27.12/28.20 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.20 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.20 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.20 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.20 U31(tt) -> tt 27.12/28.20 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.20 U42(tt) -> tt 27.12/28.20 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.20 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.20 U53(tt) -> tt 27.12/28.20 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.20 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.20 U63(tt) -> tt 27.12/28.20 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.20 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.20 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.20 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.20 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.20 U76(tt) -> tt 27.12/28.20 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.20 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.20 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.20 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.20 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.20 U86(tt) -> tt 27.12/28.20 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.20 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.20 U93(tt) -> tt 27.12/28.20 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.20 isBag(empty) -> tt 27.12/28.20 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.20 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.20 isBagKind(empty) -> tt 27.12/28.20 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.20 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.20 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.20 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.20 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.20 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.20 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.20 isBin(z) -> tt 27.12/28.20 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.20 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.20 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.20 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.20 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.20 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.20 isBinKind(z) -> tt 27.12/28.20 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.20 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.20 mult(z,X) -> U171(isBin(X),X) 27.12/28.20 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.20 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.20 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.20 plus(z,X) -> U201(isBin(X),X) 27.12/28.20 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.20 prod(empty) -> 1(z) 27.12/28.20 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.20 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.20 sum(empty) -> 0(z) 27.12/28.20 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.20 union(empty,X) -> X 27.12/28.20 union(X,empty) -> X 27.12/28.20 -> SRules: 27.12/28.20 Empty 27.12/28.20 ->->Cycle: 27.12/28.20 ->->-> Pairs: 27.12/28.20 U101#(tt,V1) -> U102#(isBagKind(V1),V1) 27.12/28.20 U102#(tt,V1) -> ISBAG(V1) 27.12/28.20 U11#(tt,V1) -> U12#(isBinKind(V1),V1) 27.12/28.20 U12#(tt,V1) -> ISBIN(V1) 27.12/28.20 U21#(tt,V1,V2) -> U22#(isBagKind(V1),V1,V2) 27.12/28.20 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.20 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.20 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.20 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.20 U25#(tt,V2) -> ISBAG(V2) 27.12/28.20 U51#(tt,V1) -> U52#(isBinKind(V1),V1) 27.12/28.20 U52#(tt,V1) -> ISBIN(V1) 27.12/28.20 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.20 U62#(tt,V1) -> ISBIN(V1) 27.12/28.20 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.20 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.20 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.20 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.20 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.20 U75#(tt,V2) -> ISBIN(V2) 27.12/28.20 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.20 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.20 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.20 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.20 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.20 U85#(tt,V2) -> ISBIN(V2) 27.12/28.20 U91#(tt,V1) -> U92#(isBagKind(V1),V1) 27.12/28.20 U92#(tt,V1) -> ISBAG(V1) 27.12/28.20 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.20 ISBAG(singl(V1)) -> U11#(isBinKind(V1),V1) 27.12/28.20 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.20 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.20 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.20 ISBIN(prod(V1)) -> U91#(isBagKind(V1),V1) 27.12/28.20 ISBIN(sum(V1)) -> U101#(isBagKind(V1),V1) 27.12/28.20 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.20 -> FAxioms: 27.12/28.20 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) -> mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) -> plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) -> union(x7,x6) 27.12/28.20 -> EAxioms: 27.12/28.20 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) = mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) = plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) = union(x7,x6) 27.12/28.20 ->->-> Rules: 27.12/28.20 0(z) -> z 27.12/28.20 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.20 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.20 U103(tt) -> tt 27.12/28.20 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.20 U111(tt) -> tt 27.12/28.20 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.20 U121(tt) -> tt 27.12/28.20 U13(tt) -> tt 27.12/28.20 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.20 U132(tt) -> tt 27.12/28.20 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.20 U142(tt) -> tt 27.12/28.20 U151(tt) -> tt 27.12/28.20 U161(tt) -> tt 27.12/28.20 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.20 U172(tt) -> z 27.12/28.20 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.20 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.20 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.20 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.20 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.20 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.20 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.20 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.20 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.20 U202(tt,X) -> X 27.12/28.20 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.20 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.20 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.20 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.20 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.20 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.20 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.20 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.20 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.20 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.20 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.20 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.20 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.20 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.20 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.20 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.20 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.20 U242(tt,X) -> X 27.12/28.20 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.20 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.20 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.20 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.20 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.20 U26(tt) -> tt 27.12/28.20 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.20 U262(tt,X) -> X 27.12/28.20 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.20 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.20 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.20 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.20 U31(tt) -> tt 27.12/28.20 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.20 U42(tt) -> tt 27.12/28.20 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.20 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.20 U53(tt) -> tt 27.12/28.20 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.20 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.20 U63(tt) -> tt 27.12/28.20 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.20 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.20 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.20 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.20 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.20 U76(tt) -> tt 27.12/28.20 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.20 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.20 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.20 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.20 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.20 U86(tt) -> tt 27.12/28.20 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.20 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.20 U93(tt) -> tt 27.12/28.20 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.20 isBag(empty) -> tt 27.12/28.20 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.20 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.20 isBagKind(empty) -> tt 27.12/28.20 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.20 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.20 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.20 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.20 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.20 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.20 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.20 isBin(z) -> tt 27.12/28.20 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.20 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.20 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.20 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.20 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.20 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.20 isBinKind(z) -> tt 27.12/28.20 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.20 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.20 mult(z,X) -> U171(isBin(X),X) 27.12/28.20 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.20 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.20 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.20 plus(z,X) -> U201(isBin(X),X) 27.12/28.20 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.20 prod(empty) -> 1(z) 27.12/28.20 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.20 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.20 sum(empty) -> 0(z) 27.12/28.20 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.20 union(empty,X) -> X 27.12/28.20 union(X,empty) -> X 27.12/28.20 -> SRules: 27.12/28.20 Empty 27.12/28.20 ->->Cycle: 27.12/28.20 ->->-> Pairs: 27.12/28.20 U211#(tt,X,Y) -> U212#(isBinKind(X),X,Y) 27.12/28.20 U212#(tt,X,Y) -> U213#(isBin(Y),X,Y) 27.12/28.20 U213#(tt,X,Y) -> U214#(isBinKind(Y),X,Y) 27.12/28.20 U214#(tt,X,Y) -> PLUS(X,Y) 27.12/28.20 U221#(tt,X,Y) -> U222#(isBinKind(X),X,Y) 27.12/28.20 U222#(tt,X,Y) -> U223#(isBin(Y),X,Y) 27.12/28.20 U223#(tt,X,Y) -> U224#(isBinKind(Y),X,Y) 27.12/28.20 U224#(tt,X,Y) -> PLUS(X,Y) 27.12/28.20 U231#(tt,X,Y) -> U232#(isBinKind(X),X,Y) 27.12/28.20 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.12/28.20 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.12/28.20 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.12/28.20 U234#(tt,X,Y) -> PLUS(X,Y) 27.12/28.20 PLUS(0(X),0(Y)) -> U211#(isBin(X),X,Y) 27.12/28.20 PLUS(0(X),1(Y)) -> U221#(isBin(X),X,Y) 27.12/28.20 PLUS(plus(0(X),0(Y)),x6) -> U211#(isBin(X),X,Y) 27.12/28.20 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.12/28.20 PLUS(plus(0(X),1(Y)),x6) -> U221#(isBin(X),X,Y) 27.12/28.20 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.12/28.20 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.12/28.20 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.12/28.20 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.12/28.20 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.12/28.20 -> FAxioms: 27.12/28.20 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) -> mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) -> plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) -> union(x7,x6) 27.12/28.20 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8)) 27.12/28.20 PLUS(x6,x7) -> PLUS(x7,x6) 27.12/28.20 -> EAxioms: 27.12/28.20 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) = mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) = plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) = union(x7,x6) 27.12/28.20 ->->-> Rules: 27.12/28.20 0(z) -> z 27.12/28.20 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.20 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.20 U103(tt) -> tt 27.12/28.20 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.20 U111(tt) -> tt 27.12/28.20 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.20 U121(tt) -> tt 27.12/28.20 U13(tt) -> tt 27.12/28.20 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.20 U132(tt) -> tt 27.12/28.20 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.20 U142(tt) -> tt 27.12/28.20 U151(tt) -> tt 27.12/28.20 U161(tt) -> tt 27.12/28.20 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.20 U172(tt) -> z 27.12/28.20 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.20 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.20 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.20 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.20 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.20 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.20 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.20 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.20 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.20 U202(tt,X) -> X 27.12/28.20 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.20 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.20 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.20 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.20 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.20 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.20 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.20 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.20 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.20 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.20 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.20 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.20 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.20 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.20 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.20 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.20 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.20 U242(tt,X) -> X 27.12/28.20 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.20 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.20 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.20 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.20 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.20 U26(tt) -> tt 27.12/28.20 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.20 U262(tt,X) -> X 27.12/28.20 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.20 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.20 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.20 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.20 U31(tt) -> tt 27.12/28.20 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.20 U42(tt) -> tt 27.12/28.20 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.20 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.20 U53(tt) -> tt 27.12/28.20 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.20 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.20 U63(tt) -> tt 27.12/28.20 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.20 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.20 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.20 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.20 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.20 U76(tt) -> tt 27.12/28.20 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.20 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.20 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.20 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.20 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.20 U86(tt) -> tt 27.12/28.20 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.20 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.20 U93(tt) -> tt 27.12/28.20 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.20 isBag(empty) -> tt 27.12/28.20 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.20 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.20 isBagKind(empty) -> tt 27.12/28.20 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.20 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.20 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.20 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.20 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.20 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.20 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.20 isBin(z) -> tt 27.12/28.20 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.20 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.20 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.20 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.20 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.20 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.20 isBinKind(z) -> tt 27.12/28.20 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.20 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.20 mult(z,X) -> U171(isBin(X),X) 27.12/28.20 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.20 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.20 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.20 plus(z,X) -> U201(isBin(X),X) 27.12/28.20 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.20 prod(empty) -> 1(z) 27.12/28.20 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.20 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.20 sum(empty) -> 0(z) 27.12/28.20 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.20 union(empty,X) -> X 27.12/28.20 union(X,empty) -> X 27.12/28.20 -> SRules: 27.12/28.20 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.12/28.20 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.12/28.20 ->->Cycle: 27.12/28.20 ->->-> Pairs: 27.12/28.20 U181#(tt,X,Y) -> U182#(isBinKind(X),X,Y) 27.12/28.20 U182#(tt,X,Y) -> U183#(isBin(Y),X,Y) 27.12/28.20 U183#(tt,X,Y) -> U184#(isBinKind(Y),X,Y) 27.12/28.20 U184#(tt,X,Y) -> MULT(X,Y) 27.12/28.20 U191#(tt,X,Y) -> U192#(isBinKind(X),X,Y) 27.12/28.20 U192#(tt,X,Y) -> U193#(isBin(Y),X,Y) 27.12/28.20 U193#(tt,X,Y) -> U194#(isBinKind(Y),X,Y) 27.12/28.20 U194#(tt,X,Y) -> MULT(X,Y) 27.12/28.20 MULT(0(X),Y) -> U181#(isBin(X),X,Y) 27.12/28.20 MULT(mult(0(X),Y),x6) -> U181#(isBin(X),X,Y) 27.12/28.20 MULT(mult(0(X),Y),x6) -> MULT(U181(isBin(X),X,Y),x6) 27.12/28.20 MULT(mult(1(X),Y),x6) -> U191#(isBin(X),X,Y) 27.12/28.20 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.12/28.20 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.12/28.20 MULT(1(X),Y) -> U191#(isBin(X),X,Y) 27.12/28.20 -> FAxioms: 27.12/28.20 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) -> mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) -> plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) -> union(x7,x6) 27.12/28.20 MULT(mult(x6,x7),x8) -> MULT(x6,mult(x7,x8)) 27.12/28.20 MULT(x6,x7) -> MULT(x7,x6) 27.12/28.20 -> EAxioms: 27.12/28.20 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.20 mult(x6,x7) = mult(x7,x6) 27.12/28.20 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.20 plus(x6,x7) = plus(x7,x6) 27.12/28.20 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.20 union(x6,x7) = union(x7,x6) 27.12/28.20 ->->-> Rules: 27.12/28.20 0(z) -> z 27.12/28.20 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.20 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.20 U103(tt) -> tt 27.12/28.20 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.20 U111(tt) -> tt 27.12/28.20 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.20 U121(tt) -> tt 27.12/28.20 U13(tt) -> tt 27.12/28.20 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.20 U132(tt) -> tt 27.12/28.20 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.20 U142(tt) -> tt 27.12/28.20 U151(tt) -> tt 27.12/28.20 U161(tt) -> tt 27.12/28.20 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.20 U172(tt) -> z 27.12/28.20 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.20 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.20 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.20 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.20 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.20 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.20 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.20 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> SRules: 27.12/28.21 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.12/28.21 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.12/28.21 ->->Cycle: 27.12/28.21 ->->-> Pairs: 27.12/28.21 U251#(tt,A,B) -> U252#(isBagKind(A),A,B) 27.12/28.21 U252#(tt,A,B) -> U253#(isBag(B),A,B) 27.12/28.21 U253#(tt,A,B) -> U254#(isBagKind(B),A,B) 27.12/28.21 U254#(tt,A,B) -> PROD(A) 27.12/28.21 U254#(tt,A,B) -> PROD(B) 27.12/28.21 PROD(union(A,B)) -> U251#(isBag(A),A,B) 27.12/28.21 -> FAxioms: 27.12/28.21 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) -> mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) -> plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) -> union(x7,x6) 27.12/28.21 -> EAxioms: 27.12/28.21 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) = mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) = plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 ->->-> Rules: 27.12/28.21 0(z) -> z 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.21 U172(tt) -> z 27.12/28.21 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.21 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.21 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.21 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.21 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.21 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.21 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.21 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> SRules: 27.12/28.21 Empty 27.12/28.21 ->->Cycle: 27.12/28.21 ->->-> Pairs: 27.12/28.21 U271#(tt,A,B) -> U272#(isBagKind(A),A,B) 27.12/28.21 U272#(tt,A,B) -> U273#(isBag(B),A,B) 27.12/28.21 U273#(tt,A,B) -> U274#(isBagKind(B),A,B) 27.12/28.21 U274#(tt,A,B) -> SUM(A) 27.12/28.21 U274#(tt,A,B) -> SUM(B) 27.12/28.21 SUM(union(A,B)) -> U271#(isBag(A),A,B) 27.12/28.21 -> FAxioms: 27.12/28.21 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) -> mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) -> plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) -> union(x7,x6) 27.12/28.21 -> EAxioms: 27.12/28.21 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) = mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) = plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 ->->-> Rules: 27.12/28.21 0(z) -> z 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.21 U172(tt) -> z 27.12/28.21 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.21 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.21 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.21 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.21 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.21 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.21 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.21 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> SRules: 27.12/28.21 Empty 27.12/28.21 27.12/28.21 27.12/28.21 The problem is decomposed in 7 subproblems. 27.12/28.21 27.12/28.21 Problem 1.1: 27.12/28.21 27.12/28.21 Reduction Pairs Processor: 27.12/28.21 -> FAxioms: 27.12/28.21 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8)) 27.12/28.21 UNION(x6,x7) = UNION(x7,x6) 27.12/28.21 -> Pairs: 27.12/28.21 UNION(union(empty,X),x6) -> UNION(X,x6) 27.12/28.21 UNION(union(X,empty),x6) -> UNION(X,x6) 27.12/28.21 -> EAxioms: 27.12/28.21 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) = mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) = plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 -> Usable Equations: 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 -> Rules: 27.12/28.21 0(z) -> z 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.21 U172(tt) -> z 27.12/28.21 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.21 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.21 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.21 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.21 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.21 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.21 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.21 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> Usable Rules: 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> SRules: 27.12/28.21 UNION(union(x6,x7),x8) -> UNION(x6,x7) 27.12/28.21 UNION(x6,union(x7,x8)) -> UNION(x7,x8) 27.12/28.21 ->Interpretation type: 27.12/28.21 Linear 27.12/28.21 ->Coefficients: 27.12/28.21 Natural Numbers 27.12/28.21 ->Dimension: 27.12/28.21 1 27.12/28.21 ->Bound: 27.12/28.21 2 27.12/28.21 ->Interpretation: 27.12/28.21 27.12/28.21 [0](X) = 0 27.12/28.21 [U101](X1,X2) = 0 27.12/28.21 [U102](X1,X2) = 0 27.12/28.21 [U103](X) = 0 27.12/28.21 [U11](X1,X2) = 0 27.12/28.21 [U111](X) = 0 27.12/28.21 [U12](X1,X2) = 0 27.12/28.21 [U121](X) = 0 27.12/28.21 [U13](X) = 0 27.12/28.21 [U131](X1,X2) = 0 27.12/28.21 [U132](X) = 0 27.12/28.21 [U141](X1,X2) = 0 27.12/28.21 [U142](X) = 0 27.12/28.21 [U151](X) = 0 27.12/28.21 [U161](X) = 0 27.12/28.21 [U171](X1,X2) = 0 27.12/28.21 [U172](X) = 0 27.12/28.21 [U181](X1,X2,X3) = 0 27.12/28.21 [U182](X1,X2,X3) = 0 27.12/28.21 [U183](X1,X2,X3) = 0 27.12/28.21 [U184](X1,X2,X3) = 0 27.12/28.21 [U191](X1,X2,X3) = 0 27.12/28.21 [U192](X1,X2,X3) = 0 27.12/28.21 [U193](X1,X2,X3) = 0 27.12/28.21 [U194](X1,X2,X3) = 0 27.12/28.21 [U201](X1,X2) = 0 27.12/28.21 [U202](X1,X2) = 0 27.12/28.21 [U21](X1,X2,X3) = 0 27.12/28.21 [U211](X1,X2,X3) = 0 27.12/28.21 [U212](X1,X2,X3) = 0 27.12/28.21 [U213](X1,X2,X3) = 0 27.12/28.21 [U214](X1,X2,X3) = 0 27.12/28.21 [U22](X1,X2,X3) = 0 27.12/28.21 [U221](X1,X2,X3) = 0 27.12/28.21 [U222](X1,X2,X3) = 0 27.12/28.21 [U223](X1,X2,X3) = 0 27.12/28.21 [U224](X1,X2,X3) = 0 27.12/28.21 [U23](X1,X2,X3) = 0 27.12/28.21 [U231](X1,X2,X3) = 0 27.12/28.21 [U232](X1,X2,X3) = 0 27.12/28.21 [U233](X1,X2,X3) = 0 27.12/28.21 [U234](X1,X2,X3) = 0 27.12/28.21 [U24](X1,X2,X3) = 0 27.12/28.21 [U241](X1,X2) = 0 27.12/28.21 [U242](X1,X2) = 0 27.12/28.21 [U25](X1,X2) = 0 27.12/28.21 [U251](X1,X2,X3) = 0 27.12/28.21 [U252](X1,X2,X3) = 0 27.12/28.21 [U253](X1,X2,X3) = 0 27.12/28.21 [U254](X1,X2,X3) = 0 27.12/28.21 [U26](X) = 0 27.12/28.21 [U261](X1,X2) = 0 27.12/28.21 [U262](X1,X2) = 0 27.12/28.21 [U271](X1,X2,X3) = 0 27.12/28.21 [U272](X1,X2,X3) = 0 27.12/28.21 [U273](X1,X2,X3) = 0 27.12/28.21 [U274](X1,X2,X3) = 0 27.12/28.21 [U31](X) = 0 27.12/28.21 [U41](X1,X2) = 0 27.12/28.21 [U42](X) = 0 27.12/28.21 [U51](X1,X2) = 0 27.12/28.21 [U52](X1,X2) = 0 27.12/28.21 [U53](X) = 0 27.12/28.21 [U61](X1,X2) = 0 27.12/28.21 [U62](X1,X2) = 0 27.12/28.21 [U63](X) = 0 27.12/28.21 [U71](X1,X2,X3) = 0 27.12/28.21 [U72](X1,X2,X3) = 0 27.12/28.21 [U73](X1,X2,X3) = 0 27.12/28.21 [U74](X1,X2,X3) = 0 27.12/28.21 [U75](X1,X2) = 0 27.12/28.21 [U76](X) = 0 27.12/28.21 [U81](X1,X2,X3) = 0 27.12/28.21 [U82](X1,X2,X3) = 0 27.12/28.21 [U83](X1,X2,X3) = 0 27.12/28.21 [U84](X1,X2,X3) = 0 27.12/28.21 [U85](X1,X2) = 0 27.12/28.21 [U86](X) = 0 27.12/28.21 [U91](X1,X2) = 0 27.12/28.21 [U92](X1,X2) = 0 27.12/28.21 [U93](X) = 0 27.12/28.21 [isBag](X) = 0 27.12/28.21 [isBagKind](X) = 0 27.12/28.21 [isBin](X) = 0 27.12/28.21 [isBinKind](X) = 0 27.12/28.21 [mult](X1,X2) = 0 27.12/28.21 [plus](X1,X2) = 0 27.12/28.21 [prod](X) = 0 27.12/28.21 [sum](X) = 0 27.12/28.21 [union](X1,X2) = X1 + X2 27.12/28.21 [1](X) = 0 27.12/28.21 [empty] = 2 27.12/28.21 [singl](X) = 0 27.12/28.21 [tt] = 0 27.12/28.21 [z] = 0 27.12/28.21 [0#](X) = 0 27.12/28.21 [U101#](X1,X2) = 0 27.12/28.21 [U102#](X1,X2) = 0 27.12/28.21 [U103#](X) = 0 27.12/28.21 [U11#](X1,X2) = 0 27.12/28.21 [U111#](X) = 0 27.12/28.21 [U12#](X1,X2) = 0 27.12/28.21 [U121#](X) = 0 27.12/28.21 [U13#](X) = 0 27.12/28.21 [U131#](X1,X2) = 0 27.12/28.21 [U132#](X) = 0 27.12/28.21 [U141#](X1,X2) = 0 27.12/28.21 [U142#](X) = 0 27.12/28.21 [U151#](X) = 0 27.12/28.21 [U161#](X) = 0 27.12/28.21 [U171#](X1,X2) = 0 27.12/28.21 [U172#](X) = 0 27.12/28.21 [U181#](X1,X2,X3) = 0 27.12/28.21 [U182#](X1,X2,X3) = 0 27.12/28.21 [U183#](X1,X2,X3) = 0 27.12/28.21 [U184#](X1,X2,X3) = 0 27.12/28.21 [U191#](X1,X2,X3) = 0 27.12/28.21 [U192#](X1,X2,X3) = 0 27.12/28.21 [U193#](X1,X2,X3) = 0 27.12/28.21 [U194#](X1,X2,X3) = 0 27.12/28.21 [U201#](X1,X2) = 0 27.12/28.21 [U202#](X1,X2) = 0 27.12/28.21 [U21#](X1,X2,X3) = 0 27.12/28.21 [U211#](X1,X2,X3) = 0 27.12/28.21 [U212#](X1,X2,X3) = 0 27.12/28.21 [U213#](X1,X2,X3) = 0 27.12/28.21 [U214#](X1,X2,X3) = 0 27.12/28.21 [U22#](X1,X2,X3) = 0 27.12/28.21 [U221#](X1,X2,X3) = 0 27.12/28.21 [U222#](X1,X2,X3) = 0 27.12/28.21 [U223#](X1,X2,X3) = 0 27.12/28.21 [U224#](X1,X2,X3) = 0 27.12/28.21 [U23#](X1,X2,X3) = 0 27.12/28.21 [U231#](X1,X2,X3) = 0 27.12/28.21 [U232#](X1,X2,X3) = 0 27.12/28.21 [U233#](X1,X2,X3) = 0 27.12/28.21 [U234#](X1,X2,X3) = 0 27.12/28.21 [U24#](X1,X2,X3) = 0 27.12/28.21 [U241#](X1,X2) = 0 27.12/28.21 [U242#](X1,X2) = 0 27.12/28.21 [U25#](X1,X2) = 0 27.12/28.21 [U251#](X1,X2,X3) = 0 27.12/28.21 [U252#](X1,X2,X3) = 0 27.12/28.21 [U253#](X1,X2,X3) = 0 27.12/28.21 [U254#](X1,X2,X3) = 0 27.12/28.21 [U26#](X) = 0 27.12/28.21 [U261#](X1,X2) = 0 27.12/28.21 [U262#](X1,X2) = 0 27.12/28.21 [U271#](X1,X2,X3) = 0 27.12/28.21 [U272#](X1,X2,X3) = 0 27.12/28.21 [U273#](X1,X2,X3) = 0 27.12/28.21 [U274#](X1,X2,X3) = 0 27.12/28.21 [U31#](X) = 0 27.12/28.21 [U41#](X1,X2) = 0 27.12/28.21 [U42#](X) = 0 27.12/28.21 [U51#](X1,X2) = 0 27.12/28.21 [U52#](X1,X2) = 0 27.12/28.21 [U53#](X) = 0 27.12/28.21 [U61#](X1,X2) = 0 27.12/28.21 [U62#](X1,X2) = 0 27.12/28.21 [U63#](X) = 0 27.12/28.21 [U71#](X1,X2,X3) = 0 27.12/28.21 [U72#](X1,X2,X3) = 0 27.12/28.21 [U73#](X1,X2,X3) = 0 27.12/28.21 [U74#](X1,X2,X3) = 0 27.12/28.21 [U75#](X1,X2) = 0 27.12/28.21 [U76#](X) = 0 27.12/28.21 [U81#](X1,X2,X3) = 0 27.12/28.21 [U82#](X1,X2,X3) = 0 27.12/28.21 [U83#](X1,X2,X3) = 0 27.12/28.21 [U84#](X1,X2,X3) = 0 27.12/28.21 [U85#](X1,X2) = 0 27.12/28.21 [U86#](X) = 0 27.12/28.21 [U91#](X1,X2) = 0 27.12/28.21 [U92#](X1,X2) = 0 27.12/28.21 [U93#](X) = 0 27.12/28.21 [ISBAG](X) = 0 27.12/28.21 [ISBAGKIND](X) = 0 27.12/28.21 [ISBIN](X) = 0 27.12/28.21 [ISBINKIND](X) = 0 27.12/28.21 [MULT](X1,X2) = 0 27.12/28.21 [PLUS](X1,X2) = 0 27.12/28.21 [PROD](X) = 0 27.12/28.21 [SUM](X) = 0 27.12/28.21 [UNION](X1,X2) = 2.X1 + 2.X2 27.12/28.21 27.12/28.21 Problem 1.1: 27.12/28.21 27.12/28.21 SCC Processor: 27.12/28.21 -> FAxioms: 27.12/28.21 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8)) 27.12/28.21 UNION(x6,x7) = UNION(x7,x6) 27.12/28.21 -> Pairs: 27.12/28.21 UNION(union(X,empty),x6) -> UNION(X,x6) 27.12/28.21 -> EAxioms: 27.12/28.21 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) = mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) = plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 -> Rules: 27.12/28.21 0(z) -> z 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.21 U172(tt) -> z 27.12/28.21 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.21 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.21 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.21 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.21 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.21 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.21 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.21 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> SRules: 27.12/28.21 UNION(union(x6,x7),x8) -> UNION(x6,x7) 27.12/28.21 UNION(x6,union(x7,x8)) -> UNION(x7,x8) 27.12/28.21 ->Strongly Connected Components: 27.12/28.21 ->->Cycle: 27.12/28.21 ->->-> Pairs: 27.12/28.21 UNION(union(X,empty),x6) -> UNION(X,x6) 27.12/28.21 -> FAxioms: 27.12/28.21 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) -> mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) -> plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) -> union(x7,x6) 27.12/28.21 UNION(union(x6,x7),x8) -> UNION(x6,union(x7,x8)) 27.12/28.21 UNION(x6,x7) -> UNION(x7,x6) 27.12/28.21 -> EAxioms: 27.12/28.21 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) = mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) = plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 ->->-> Rules: 27.12/28.21 0(z) -> z 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.21 U172(tt) -> z 27.12/28.21 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.21 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.21 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.21 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.21 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.21 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.21 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.21 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> SRules: 27.12/28.21 UNION(union(x6,x7),x8) -> UNION(x6,x7) 27.12/28.21 UNION(x6,union(x7,x8)) -> UNION(x7,x8) 27.12/28.21 27.12/28.21 Problem 1.1: 27.12/28.21 27.12/28.21 Reduction Pairs Processor: 27.12/28.21 -> FAxioms: 27.12/28.21 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8)) 27.12/28.21 UNION(x6,x7) = UNION(x7,x6) 27.12/28.21 -> Pairs: 27.12/28.21 UNION(union(X,empty),x6) -> UNION(X,x6) 27.12/28.21 -> EAxioms: 27.12/28.21 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) = mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) = plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 -> Usable Equations: 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 -> Rules: 27.12/28.21 0(z) -> z 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.21 U172(tt) -> z 27.12/28.21 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.21 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.21 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.21 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.21 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.21 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.21 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.21 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> Usable Rules: 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> SRules: 27.12/28.21 UNION(union(x6,x7),x8) -> UNION(x6,x7) 27.12/28.21 UNION(x6,union(x7,x8)) -> UNION(x7,x8) 27.12/28.21 ->Interpretation type: 27.12/28.21 Linear 27.12/28.21 ->Coefficients: 27.12/28.21 Natural Numbers 27.12/28.21 ->Dimension: 27.12/28.21 1 27.12/28.21 ->Bound: 27.12/28.21 2 27.12/28.21 ->Interpretation: 27.12/28.21 27.12/28.21 [0](X) = 0 27.12/28.21 [U101](X1,X2) = 0 27.12/28.21 [U102](X1,X2) = 0 27.12/28.21 [U103](X) = 0 27.12/28.21 [U11](X1,X2) = 0 27.12/28.21 [U111](X) = 0 27.12/28.21 [U12](X1,X2) = 0 27.12/28.21 [U121](X) = 0 27.12/28.21 [U13](X) = 0 27.12/28.21 [U131](X1,X2) = 0 27.12/28.21 [U132](X) = 0 27.12/28.21 [U141](X1,X2) = 0 27.12/28.21 [U142](X) = 0 27.12/28.21 [U151](X) = 0 27.12/28.21 [U161](X) = 0 27.12/28.21 [U171](X1,X2) = 0 27.12/28.21 [U172](X) = 0 27.12/28.21 [U181](X1,X2,X3) = 0 27.12/28.21 [U182](X1,X2,X3) = 0 27.12/28.21 [U183](X1,X2,X3) = 0 27.12/28.21 [U184](X1,X2,X3) = 0 27.12/28.21 [U191](X1,X2,X3) = 0 27.12/28.21 [U192](X1,X2,X3) = 0 27.12/28.21 [U193](X1,X2,X3) = 0 27.12/28.21 [U194](X1,X2,X3) = 0 27.12/28.21 [U201](X1,X2) = 0 27.12/28.21 [U202](X1,X2) = 0 27.12/28.21 [U21](X1,X2,X3) = 0 27.12/28.21 [U211](X1,X2,X3) = 0 27.12/28.21 [U212](X1,X2,X3) = 0 27.12/28.21 [U213](X1,X2,X3) = 0 27.12/28.21 [U214](X1,X2,X3) = 0 27.12/28.21 [U22](X1,X2,X3) = 0 27.12/28.21 [U221](X1,X2,X3) = 0 27.12/28.21 [U222](X1,X2,X3) = 0 27.12/28.21 [U223](X1,X2,X3) = 0 27.12/28.21 [U224](X1,X2,X3) = 0 27.12/28.21 [U23](X1,X2,X3) = 0 27.12/28.21 [U231](X1,X2,X3) = 0 27.12/28.21 [U232](X1,X2,X3) = 0 27.12/28.21 [U233](X1,X2,X3) = 0 27.12/28.21 [U234](X1,X2,X3) = 0 27.12/28.21 [U24](X1,X2,X3) = 0 27.12/28.21 [U241](X1,X2) = 0 27.12/28.21 [U242](X1,X2) = 0 27.12/28.21 [U25](X1,X2) = 0 27.12/28.21 [U251](X1,X2,X3) = 0 27.12/28.21 [U252](X1,X2,X3) = 0 27.12/28.21 [U253](X1,X2,X3) = 0 27.12/28.21 [U254](X1,X2,X3) = 0 27.12/28.21 [U26](X) = 0 27.12/28.21 [U261](X1,X2) = 0 27.12/28.21 [U262](X1,X2) = 0 27.12/28.21 [U271](X1,X2,X3) = 0 27.12/28.21 [U272](X1,X2,X3) = 0 27.12/28.21 [U273](X1,X2,X3) = 0 27.12/28.21 [U274](X1,X2,X3) = 0 27.12/28.21 [U31](X) = 0 27.12/28.21 [U41](X1,X2) = 0 27.12/28.21 [U42](X) = 0 27.12/28.21 [U51](X1,X2) = 0 27.12/28.21 [U52](X1,X2) = 0 27.12/28.21 [U53](X) = 0 27.12/28.21 [U61](X1,X2) = 0 27.12/28.21 [U62](X1,X2) = 0 27.12/28.21 [U63](X) = 0 27.12/28.21 [U71](X1,X2,X3) = 0 27.12/28.21 [U72](X1,X2,X3) = 0 27.12/28.21 [U73](X1,X2,X3) = 0 27.12/28.21 [U74](X1,X2,X3) = 0 27.12/28.21 [U75](X1,X2) = 0 27.12/28.21 [U76](X) = 0 27.12/28.21 [U81](X1,X2,X3) = 0 27.12/28.21 [U82](X1,X2,X3) = 0 27.12/28.21 [U83](X1,X2,X3) = 0 27.12/28.21 [U84](X1,X2,X3) = 0 27.12/28.21 [U85](X1,X2) = 0 27.12/28.21 [U86](X) = 0 27.12/28.21 [U91](X1,X2) = 0 27.12/28.21 [U92](X1,X2) = 0 27.12/28.21 [U93](X) = 0 27.12/28.21 [isBag](X) = 0 27.12/28.21 [isBagKind](X) = 0 27.12/28.21 [isBin](X) = 0 27.12/28.21 [isBinKind](X) = 0 27.12/28.21 [mult](X1,X2) = 0 27.12/28.21 [plus](X1,X2) = 0 27.12/28.21 [prod](X) = 0 27.12/28.21 [sum](X) = 0 27.12/28.21 [union](X1,X2) = X1 + X2 + 2 27.12/28.21 [1](X) = 0 27.12/28.21 [empty] = 2 27.12/28.21 [singl](X) = 0 27.12/28.21 [tt] = 0 27.12/28.21 [z] = 0 27.12/28.21 [0#](X) = 0 27.12/28.21 [U101#](X1,X2) = 0 27.12/28.21 [U102#](X1,X2) = 0 27.12/28.21 [U103#](X) = 0 27.12/28.21 [U11#](X1,X2) = 0 27.12/28.21 [U111#](X) = 0 27.12/28.21 [U12#](X1,X2) = 0 27.12/28.21 [U121#](X) = 0 27.12/28.21 [U13#](X) = 0 27.12/28.21 [U131#](X1,X2) = 0 27.12/28.21 [U132#](X) = 0 27.12/28.21 [U141#](X1,X2) = 0 27.12/28.21 [U142#](X) = 0 27.12/28.21 [U151#](X) = 0 27.12/28.21 [U161#](X) = 0 27.12/28.21 [U171#](X1,X2) = 0 27.12/28.21 [U172#](X) = 0 27.12/28.21 [U181#](X1,X2,X3) = 0 27.12/28.21 [U182#](X1,X2,X3) = 0 27.12/28.21 [U183#](X1,X2,X3) = 0 27.12/28.21 [U184#](X1,X2,X3) = 0 27.12/28.21 [U191#](X1,X2,X3) = 0 27.12/28.21 [U192#](X1,X2,X3) = 0 27.12/28.21 [U193#](X1,X2,X3) = 0 27.12/28.21 [U194#](X1,X2,X3) = 0 27.12/28.21 [U201#](X1,X2) = 0 27.12/28.21 [U202#](X1,X2) = 0 27.12/28.21 [U21#](X1,X2,X3) = 0 27.12/28.21 [U211#](X1,X2,X3) = 0 27.12/28.21 [U212#](X1,X2,X3) = 0 27.12/28.21 [U213#](X1,X2,X3) = 0 27.12/28.21 [U214#](X1,X2,X3) = 0 27.12/28.21 [U22#](X1,X2,X3) = 0 27.12/28.21 [U221#](X1,X2,X3) = 0 27.12/28.21 [U222#](X1,X2,X3) = 0 27.12/28.21 [U223#](X1,X2,X3) = 0 27.12/28.21 [U224#](X1,X2,X3) = 0 27.12/28.21 [U23#](X1,X2,X3) = 0 27.12/28.21 [U231#](X1,X2,X3) = 0 27.12/28.21 [U232#](X1,X2,X3) = 0 27.12/28.21 [U233#](X1,X2,X3) = 0 27.12/28.21 [U234#](X1,X2,X3) = 0 27.12/28.21 [U24#](X1,X2,X3) = 0 27.12/28.21 [U241#](X1,X2) = 0 27.12/28.21 [U242#](X1,X2) = 0 27.12/28.21 [U25#](X1,X2) = 0 27.12/28.21 [U251#](X1,X2,X3) = 0 27.12/28.21 [U252#](X1,X2,X3) = 0 27.12/28.21 [U253#](X1,X2,X3) = 0 27.12/28.21 [U254#](X1,X2,X3) = 0 27.12/28.21 [U26#](X) = 0 27.12/28.21 [U261#](X1,X2) = 0 27.12/28.21 [U262#](X1,X2) = 0 27.12/28.21 [U271#](X1,X2,X3) = 0 27.12/28.21 [U272#](X1,X2,X3) = 0 27.12/28.21 [U273#](X1,X2,X3) = 0 27.12/28.21 [U274#](X1,X2,X3) = 0 27.12/28.21 [U31#](X) = 0 27.12/28.21 [U41#](X1,X2) = 0 27.12/28.21 [U42#](X) = 0 27.12/28.21 [U51#](X1,X2) = 0 27.12/28.21 [U52#](X1,X2) = 0 27.12/28.21 [U53#](X) = 0 27.12/28.21 [U61#](X1,X2) = 0 27.12/28.21 [U62#](X1,X2) = 0 27.12/28.21 [U63#](X) = 0 27.12/28.21 [U71#](X1,X2,X3) = 0 27.12/28.21 [U72#](X1,X2,X3) = 0 27.12/28.21 [U73#](X1,X2,X3) = 0 27.12/28.21 [U74#](X1,X2,X3) = 0 27.12/28.21 [U75#](X1,X2) = 0 27.12/28.21 [U76#](X) = 0 27.12/28.21 [U81#](X1,X2,X3) = 0 27.12/28.21 [U82#](X1,X2,X3) = 0 27.12/28.21 [U83#](X1,X2,X3) = 0 27.12/28.21 [U84#](X1,X2,X3) = 0 27.12/28.21 [U85#](X1,X2) = 0 27.12/28.21 [U86#](X) = 0 27.12/28.21 [U91#](X1,X2) = 0 27.12/28.21 [U92#](X1,X2) = 0 27.12/28.21 [U93#](X) = 0 27.12/28.21 [ISBAG](X) = 0 27.12/28.21 [ISBAGKIND](X) = 0 27.12/28.21 [ISBIN](X) = 0 27.12/28.21 [ISBINKIND](X) = 0 27.12/28.21 [MULT](X1,X2) = 0 27.12/28.21 [PLUS](X1,X2) = 0 27.12/28.21 [PROD](X) = 0 27.12/28.21 [SUM](X) = 0 27.12/28.21 [UNION](X1,X2) = 2.X1 + 2.X2 27.12/28.21 27.12/28.21 Problem 1.1: 27.12/28.21 27.12/28.21 SCC Processor: 27.12/28.21 -> FAxioms: 27.12/28.21 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8)) 27.12/28.21 UNION(x6,x7) = UNION(x7,x6) 27.12/28.21 -> Pairs: 27.12/28.21 Empty 27.12/28.21 -> EAxioms: 27.12/28.21 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) = mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) = plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 -> Rules: 27.12/28.21 0(z) -> z 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.21 U172(tt) -> z 27.12/28.21 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.21 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.21 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.21 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.21 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.21 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.21 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.21 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> SRules: 27.12/28.21 UNION(union(x6,x7),x8) -> UNION(x6,x7) 27.12/28.21 UNION(x6,union(x7,x8)) -> UNION(x7,x8) 27.12/28.21 ->Strongly Connected Components: 27.12/28.21 There is no strongly connected component 27.12/28.21 27.12/28.21 The problem is finite. 27.12/28.21 27.12/28.21 Problem 1.2: 27.12/28.21 27.12/28.21 Subterm Processor: 27.12/28.21 -> FAxioms: 27.12/28.21 Empty 27.12/28.21 -> Pairs: 27.12/28.21 U131#(tt,V2) -> ISBINKIND(V2) 27.12/28.21 U141#(tt,V2) -> ISBINKIND(V2) 27.12/28.21 U41#(tt,V2) -> ISBAGKIND(V2) 27.12/28.21 ISBAGKIND(union(V1,V2)) -> U41#(isBagKind(V1),V2) 27.12/28.21 ISBAGKIND(union(V1,V2)) -> ISBAGKIND(V1) 27.12/28.21 ISBAGKIND(singl(V1)) -> ISBINKIND(V1) 27.12/28.21 ISBINKIND(0(V1)) -> ISBINKIND(V1) 27.12/28.21 ISBINKIND(mult(V1,V2)) -> U131#(isBinKind(V1),V2) 27.12/28.21 ISBINKIND(mult(V1,V2)) -> ISBINKIND(V1) 27.12/28.21 ISBINKIND(plus(V1,V2)) -> U141#(isBinKind(V1),V2) 27.12/28.21 ISBINKIND(plus(V1,V2)) -> ISBINKIND(V1) 27.12/28.21 ISBINKIND(prod(V1)) -> ISBAGKIND(V1) 27.12/28.21 ISBINKIND(sum(V1)) -> ISBAGKIND(V1) 27.12/28.21 ISBINKIND(1(V1)) -> ISBINKIND(V1) 27.12/28.21 -> EAxioms: 27.12/28.21 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) = mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) = plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 -> Rules: 27.12/28.21 0(z) -> z 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.21 U172(tt) -> z 27.12/28.21 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.21 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.21 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.21 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.21 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.21 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.21 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.21 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> SRules: 27.12/28.21 Empty 27.12/28.21 ->Projection: 27.12/28.21 pi(U131#) = [2] 27.12/28.21 pi(U141#) = [2] 27.12/28.21 pi(U41#) = [2] 27.12/28.21 pi(ISBAGKIND) = [1] 27.12/28.21 pi(ISBINKIND) = [1] 27.12/28.21 27.12/28.21 Problem 1.2: 27.12/28.21 27.12/28.21 SCC Processor: 27.12/28.21 -> FAxioms: 27.12/28.21 Empty 27.12/28.21 -> Pairs: 27.12/28.21 U131#(tt,V2) -> ISBINKIND(V2) 27.12/28.21 U141#(tt,V2) -> ISBINKIND(V2) 27.12/28.21 U41#(tt,V2) -> ISBAGKIND(V2) 27.12/28.21 -> EAxioms: 27.12/28.21 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) = mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) = plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 -> Rules: 27.12/28.21 0(z) -> z 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.21 U172(tt) -> z 27.12/28.21 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.21 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.21 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.21 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.21 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.21 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.21 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.21 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> SRules: 27.12/28.21 Empty 27.12/28.21 ->Strongly Connected Components: 27.12/28.21 There is no strongly connected component 27.12/28.21 27.12/28.21 The problem is finite. 27.12/28.21 27.12/28.21 Problem 1.3: 27.12/28.21 27.12/28.21 Reduction Pairs Processor: 27.12/28.21 -> FAxioms: 27.12/28.21 Empty 27.12/28.21 -> Pairs: 27.12/28.21 U101#(tt,V1) -> U102#(isBagKind(V1),V1) 27.12/28.21 U102#(tt,V1) -> ISBAG(V1) 27.12/28.21 U11#(tt,V1) -> U12#(isBinKind(V1),V1) 27.12/28.21 U12#(tt,V1) -> ISBIN(V1) 27.12/28.21 U21#(tt,V1,V2) -> U22#(isBagKind(V1),V1,V2) 27.12/28.21 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.21 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.21 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.21 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.21 U25#(tt,V2) -> ISBAG(V2) 27.12/28.21 U51#(tt,V1) -> U52#(isBinKind(V1),V1) 27.12/28.21 U52#(tt,V1) -> ISBIN(V1) 27.12/28.21 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.21 U62#(tt,V1) -> ISBIN(V1) 27.12/28.21 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.21 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.21 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.21 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.21 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.21 U75#(tt,V2) -> ISBIN(V2) 27.12/28.21 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.21 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.21 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.21 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.21 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.21 U85#(tt,V2) -> ISBIN(V2) 27.12/28.21 U91#(tt,V1) -> U92#(isBagKind(V1),V1) 27.12/28.21 U92#(tt,V1) -> ISBAG(V1) 27.12/28.21 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.21 ISBAG(singl(V1)) -> U11#(isBinKind(V1),V1) 27.12/28.21 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.21 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.21 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.21 ISBIN(prod(V1)) -> U91#(isBagKind(V1),V1) 27.12/28.21 ISBIN(sum(V1)) -> U101#(isBagKind(V1),V1) 27.12/28.21 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.21 -> EAxioms: 27.12/28.21 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.21 mult(x6,x7) = mult(x7,x6) 27.12/28.21 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.21 plus(x6,x7) = plus(x7,x6) 27.12/28.21 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.21 union(x6,x7) = union(x7,x6) 27.12/28.21 -> Usable Equations: 27.12/28.21 Empty 27.12/28.21 -> Rules: 27.12/28.21 0(z) -> z 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.21 U172(tt) -> z 27.12/28.21 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.21 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.21 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.21 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.21 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.21 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.21 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.21 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.21 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.21 U202(tt,X) -> X 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.21 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.21 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.21 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.21 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.21 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.21 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.21 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.21 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.21 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.21 U242(tt,X) -> X 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.21 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.21 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.21 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.21 U262(tt,X) -> X 27.12/28.21 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.21 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.21 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.21 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.21 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.21 mult(z,X) -> U171(isBin(X),X) 27.12/28.21 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.21 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.21 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.21 plus(z,X) -> U201(isBin(X),X) 27.12/28.21 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.21 prod(empty) -> 1(z) 27.12/28.21 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.21 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.21 sum(empty) -> 0(z) 27.12/28.21 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.21 union(empty,X) -> X 27.12/28.21 union(X,empty) -> X 27.12/28.21 -> Usable Rules: 27.12/28.21 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.21 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.21 U103(tt) -> tt 27.12/28.21 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.21 U111(tt) -> tt 27.12/28.21 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.21 U121(tt) -> tt 27.12/28.21 U13(tt) -> tt 27.12/28.21 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.21 U132(tt) -> tt 27.12/28.21 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.21 U142(tt) -> tt 27.12/28.21 U151(tt) -> tt 27.12/28.21 U161(tt) -> tt 27.12/28.21 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.21 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.21 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.21 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.21 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.21 U26(tt) -> tt 27.12/28.21 U31(tt) -> tt 27.12/28.21 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.21 U42(tt) -> tt 27.12/28.21 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.21 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.21 U53(tt) -> tt 27.12/28.21 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.21 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.21 U63(tt) -> tt 27.12/28.21 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.21 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.21 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.21 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.21 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.21 U76(tt) -> tt 27.12/28.21 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.21 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.21 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.21 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.21 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.21 U86(tt) -> tt 27.12/28.21 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.21 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.21 U93(tt) -> tt 27.12/28.21 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.21 isBag(empty) -> tt 27.12/28.21 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.21 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.21 isBagKind(empty) -> tt 27.12/28.21 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.21 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.21 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.21 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.21 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.21 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.21 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.21 isBin(z) -> tt 27.12/28.21 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.21 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.21 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.21 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.21 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.21 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.21 isBinKind(z) -> tt 27.12/28.21 -> SRules: 27.12/28.21 Empty 27.12/28.21 ->Interpretation type: 27.12/28.21 Linear 27.12/28.21 ->Coefficients: 27.12/28.21 Natural Numbers 27.12/28.21 ->Dimension: 27.12/28.21 1 27.12/28.21 ->Bound: 27.12/28.21 2 27.12/28.21 ->Interpretation: 27.12/28.21 27.12/28.21 [0](X) = 2.X + 2 27.12/28.21 [U101](X1,X2) = 2.X1 + 2 27.12/28.21 [U102](X1,X2) = 2 27.12/28.21 [U103](X) = 2 27.12/28.21 [U11](X1,X2) = X1 + 2.X2 + 2 27.12/28.21 [U111](X) = 2 27.12/28.21 [U12](X1,X2) = X1 + 2 27.12/28.21 [U121](X) = 2.X + 2 27.12/28.21 [U13](X) = 2 27.12/28.21 [U131](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.21 [U132](X) = 2 27.12/28.21 [U141](X1,X2) = X1 + 2.X2 + 2 27.12/28.21 [U142](X) = X + 2 27.12/28.21 [U151](X) = 2.X + 2 27.12/28.21 [U161](X) = 2.X 27.12/28.21 [U171](X1,X2) = 0 27.12/28.21 [U172](X) = 0 27.12/28.21 [U181](X1,X2,X3) = 0 27.12/28.21 [U182](X1,X2,X3) = 0 27.12/28.21 [U183](X1,X2,X3) = 0 27.12/28.21 [U184](X1,X2,X3) = 0 27.12/28.21 [U191](X1,X2,X3) = 0 27.12/28.21 [U192](X1,X2,X3) = 0 27.12/28.21 [U193](X1,X2,X3) = 0 27.12/28.21 [U194](X1,X2,X3) = 0 27.12/28.21 [U201](X1,X2) = 0 27.12/28.21 [U202](X1,X2) = 0 27.12/28.21 [U21](X1,X2,X3) = 2.X1 + 2.X3 + 2 27.12/28.21 [U211](X1,X2,X3) = 0 27.12/28.21 [U212](X1,X2,X3) = 0 27.12/28.21 [U213](X1,X2,X3) = 0 27.12/28.21 [U214](X1,X2,X3) = 0 27.12/28.22 [U22](X1,X2,X3) = 2.X3 + 2 27.12/28.22 [U221](X1,X2,X3) = 0 27.12/28.22 [U222](X1,X2,X3) = 0 27.12/28.22 [U223](X1,X2,X3) = 0 27.12/28.22 [U224](X1,X2,X3) = 0 27.12/28.22 [U23](X1,X2,X3) = X1 + 2 27.12/28.22 [U231](X1,X2,X3) = 0 27.12/28.22 [U232](X1,X2,X3) = 0 27.12/28.22 [U233](X1,X2,X3) = 0 27.12/28.22 [U234](X1,X2,X3) = 0 27.12/28.22 [U24](X1,X2,X3) = 2 27.12/28.22 [U241](X1,X2) = 0 27.12/28.22 [U242](X1,X2) = 0 27.12/28.22 [U25](X1,X2) = 2 27.12/28.22 [U251](X1,X2,X3) = 0 27.12/28.22 [U252](X1,X2,X3) = 0 27.12/28.22 [U253](X1,X2,X3) = 0 27.12/28.22 [U254](X1,X2,X3) = 0 27.12/28.22 [U26](X) = 2 27.12/28.22 [U261](X1,X2) = 0 27.12/28.22 [U262](X1,X2) = 0 27.12/28.22 [U271](X1,X2,X3) = 0 27.12/28.22 [U272](X1,X2,X3) = 0 27.12/28.22 [U273](X1,X2,X3) = 0 27.12/28.22 [U274](X1,X2,X3) = 0 27.12/28.22 [U31](X) = 2.X 27.12/28.22 [U41](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.22 [U42](X) = X + 1 27.12/28.22 [U51](X1,X2) = X1 + 2.X2 + 2 27.12/28.22 [U52](X1,X2) = X1 + 1 27.12/28.22 [U53](X) = 2 27.12/28.22 [U61](X1,X2) = X1 + 2.X2 + 2 27.12/28.22 [U62](X1,X2) = X1 + 2 27.12/28.22 [U63](X) = 2 27.12/28.22 [U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.12/28.22 [U72](X1,X2,X3) = X1 + 2.X3 + 2 27.12/28.22 [U73](X1,X2,X3) = X1 + 2 27.12/28.22 [U74](X1,X2,X3) = 2 27.12/28.22 [U75](X1,X2) = 2 27.12/28.22 [U76](X) = 2 27.12/28.22 [U81](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.12/28.22 [U82](X1,X2,X3) = X1 + 2.X3 + 2 27.12/28.22 [U83](X1,X2,X3) = X1 + 2 27.12/28.22 [U84](X1,X2,X3) = 2 27.12/28.22 [U85](X1,X2) = 2 27.12/28.22 [U86](X) = 2 27.12/28.22 [U91](X1,X2) = 2.X2 + 2 27.12/28.22 [U92](X1,X2) = X1 + 2 27.12/28.22 [U93](X) = 2 27.12/28.22 [isBag](X) = 2.X + 2 27.12/28.22 [isBagKind](X) = 2.X 27.12/28.22 [isBin](X) = 2.X + 2 27.12/28.22 [isBinKind](X) = 2.X + 2 27.12/28.22 [mult](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.22 [plus](X1,X2) = 2.X1 + 2.X2 + 1 27.12/28.22 [prod](X) = 2.X + 1 27.12/28.22 [sum](X) = 2.X + 2 27.12/28.22 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.22 [1](X) = 2.X + 2 27.12/28.22 [empty] = 2 27.12/28.22 [singl](X) = 2.X + 2 27.12/28.22 [tt] = 2 27.12/28.22 [z] = 1 27.12/28.22 [0#](X) = 0 27.12/28.22 [U101#](X1,X2) = X1 + 2.X2 + 2 27.12/28.22 [U102#](X1,X2) = 2.X2 + 1 27.12/28.22 [U103#](X) = 0 27.12/28.22 [U11#](X1,X2) = X1 + 2.X2 + 2 27.12/28.22 [U111#](X) = 0 27.12/28.22 [U12#](X1,X2) = 2.X2 + 2 27.12/28.22 [U121#](X) = 0 27.12/28.22 [U13#](X) = 0 27.12/28.22 [U131#](X1,X2) = 0 27.12/28.22 [U132#](X) = 0 27.12/28.22 [U141#](X1,X2) = 0 27.12/28.22 [U142#](X) = 0 27.12/28.22 [U151#](X) = 0 27.12/28.22 [U161#](X) = 0 27.12/28.22 [U171#](X1,X2) = 0 27.12/28.22 [U172#](X) = 0 27.12/28.22 [U181#](X1,X2,X3) = 0 27.12/28.22 [U182#](X1,X2,X3) = 0 27.12/28.22 [U183#](X1,X2,X3) = 0 27.12/28.22 [U184#](X1,X2,X3) = 0 27.12/28.22 [U191#](X1,X2,X3) = 0 27.12/28.22 [U192#](X1,X2,X3) = 0 27.12/28.22 [U193#](X1,X2,X3) = 0 27.12/28.22 [U194#](X1,X2,X3) = 0 27.12/28.22 [U201#](X1,X2) = 0 27.12/28.22 [U202#](X1,X2) = 0 27.12/28.22 [U21#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.12/28.22 [U211#](X1,X2,X3) = 0 27.12/28.22 [U212#](X1,X2,X3) = 0 27.12/28.22 [U213#](X1,X2,X3) = 0 27.12/28.22 [U214#](X1,X2,X3) = 0 27.12/28.22 [U22#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U221#](X1,X2,X3) = 0 27.12/28.22 [U222#](X1,X2,X3) = 0 27.12/28.22 [U223#](X1,X2,X3) = 0 27.12/28.22 [U224#](X1,X2,X3) = 0 27.12/28.22 [U23#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U231#](X1,X2,X3) = 0 27.12/28.22 [U232#](X1,X2,X3) = 0 27.12/28.22 [U233#](X1,X2,X3) = 0 27.12/28.22 [U234#](X1,X2,X3) = 0 27.12/28.22 [U24#](X1,X2,X3) = 2.X2 + 2.X3 27.12/28.22 [U241#](X1,X2) = 0 27.12/28.22 [U242#](X1,X2) = 0 27.12/28.22 [U25#](X1,X2) = 2.X2 27.12/28.22 [U251#](X1,X2,X3) = 0 27.12/28.22 [U252#](X1,X2,X3) = 0 27.12/28.22 [U253#](X1,X2,X3) = 0 27.12/28.22 [U254#](X1,X2,X3) = 0 27.12/28.22 [U26#](X) = 0 27.12/28.22 [U261#](X1,X2) = 0 27.12/28.22 [U262#](X1,X2) = 0 27.12/28.22 [U271#](X1,X2,X3) = 0 27.12/28.22 [U272#](X1,X2,X3) = 0 27.12/28.22 [U273#](X1,X2,X3) = 0 27.12/28.22 [U274#](X1,X2,X3) = 0 27.12/28.22 [U31#](X) = 0 27.12/28.22 [U41#](X1,X2) = 0 27.12/28.22 [U42#](X) = 0 27.12/28.22 [U51#](X1,X2) = 2.X2 + 2 27.12/28.22 [U52#](X1,X2) = 2.X2 + 2 27.12/28.22 [U53#](X) = 0 27.12/28.22 [U61#](X1,X2) = X1 + 2.X2 + 2 27.12/28.22 [U62#](X1,X2) = 2.X2 + 2 27.12/28.22 [U63#](X) = 0 27.12/28.22 [U71#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.12/28.22 [U72#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U73#](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.12/28.22 [U74#](X1,X2,X3) = 2.X2 + 2.X3 27.12/28.22 [U75#](X1,X2) = 2.X2 27.12/28.22 [U76#](X) = 0 27.12/28.22 [U81#](X1,X2,X3) = X1 + 2.X2 + 2.X3 27.12/28.22 [U82#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U83#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U84#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U85#](X1,X2) = X1 + 2.X2 27.12/28.22 [U86#](X) = 0 27.12/28.22 [U91#](X1,X2) = X1 + 2.X2 + 2 27.12/28.22 [U92#](X1,X2) = 2.X2 + 2 27.12/28.22 [U93#](X) = 0 27.12/28.22 [ISBAG](X) = 2.X 27.12/28.22 [ISBAGKIND](X) = 0 27.12/28.22 [ISBIN](X) = 2.X 27.12/28.22 [ISBINKIND](X) = 0 27.12/28.22 [MULT](X1,X2) = 0 27.12/28.22 [PLUS](X1,X2) = 0 27.12/28.22 [PROD](X) = 0 27.12/28.22 [SUM](X) = 0 27.12/28.22 [UNION](X1,X2) = 0 27.12/28.22 27.12/28.22 Problem 1.3: 27.12/28.22 27.12/28.22 SCC Processor: 27.12/28.22 -> FAxioms: 27.12/28.22 Empty 27.12/28.22 -> Pairs: 27.12/28.22 U102#(tt,V1) -> ISBAG(V1) 27.12/28.22 U11#(tt,V1) -> U12#(isBinKind(V1),V1) 27.12/28.22 U12#(tt,V1) -> ISBIN(V1) 27.12/28.22 U21#(tt,V1,V2) -> U22#(isBagKind(V1),V1,V2) 27.12/28.22 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.22 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.22 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.22 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.22 U25#(tt,V2) -> ISBAG(V2) 27.12/28.22 U51#(tt,V1) -> U52#(isBinKind(V1),V1) 27.12/28.22 U52#(tt,V1) -> ISBIN(V1) 27.12/28.22 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.22 U62#(tt,V1) -> ISBIN(V1) 27.12/28.22 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.22 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.22 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.22 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.22 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U75#(tt,V2) -> ISBIN(V2) 27.12/28.22 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.22 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.22 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.22 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.22 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U85#(tt,V2) -> ISBIN(V2) 27.12/28.22 U91#(tt,V1) -> U92#(isBagKind(V1),V1) 27.12/28.22 U92#(tt,V1) -> ISBAG(V1) 27.12/28.22 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.22 ISBAG(singl(V1)) -> U11#(isBinKind(V1),V1) 27.12/28.22 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.22 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(prod(V1)) -> U91#(isBagKind(V1),V1) 27.12/28.22 ISBIN(sum(V1)) -> U101#(isBagKind(V1),V1) 27.12/28.22 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.22 -> EAxioms: 27.12/28.22 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) = mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) = plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) = union(x7,x6) 27.12/28.22 -> Rules: 27.12/28.22 0(z) -> z 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.22 U172(tt) -> z 27.12/28.22 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.22 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.22 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.22 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.22 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.22 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.22 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.22 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.22 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.22 U202(tt,X) -> X 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.22 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.22 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.22 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.22 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.22 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.22 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.22 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.22 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.22 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.22 U242(tt,X) -> X 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.22 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.22 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.22 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.22 U262(tt,X) -> X 27.12/28.22 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.22 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.22 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.22 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.22 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.22 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.22 isBinKind(z) -> tt 27.12/28.22 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.22 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.22 mult(z,X) -> U171(isBin(X),X) 27.12/28.22 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.22 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.22 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.22 plus(z,X) -> U201(isBin(X),X) 27.12/28.22 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.22 prod(empty) -> 1(z) 27.12/28.22 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.22 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.22 sum(empty) -> 0(z) 27.12/28.22 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.22 union(empty,X) -> X 27.12/28.22 union(X,empty) -> X 27.12/28.22 -> SRules: 27.12/28.22 Empty 27.12/28.22 ->Strongly Connected Components: 27.12/28.22 ->->Cycle: 27.12/28.22 ->->-> Pairs: 27.12/28.22 U11#(tt,V1) -> U12#(isBinKind(V1),V1) 27.12/28.22 U12#(tt,V1) -> ISBIN(V1) 27.12/28.22 U21#(tt,V1,V2) -> U22#(isBagKind(V1),V1,V2) 27.12/28.22 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.22 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.22 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.22 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.22 U25#(tt,V2) -> ISBAG(V2) 27.12/28.22 U51#(tt,V1) -> U52#(isBinKind(V1),V1) 27.12/28.22 U52#(tt,V1) -> ISBIN(V1) 27.12/28.22 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.22 U62#(tt,V1) -> ISBIN(V1) 27.12/28.22 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.22 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.22 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.22 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.22 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U75#(tt,V2) -> ISBIN(V2) 27.12/28.22 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.22 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.22 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.22 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.22 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U85#(tt,V2) -> ISBIN(V2) 27.12/28.22 U91#(tt,V1) -> U92#(isBagKind(V1),V1) 27.12/28.22 U92#(tt,V1) -> ISBAG(V1) 27.12/28.22 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.22 ISBAG(singl(V1)) -> U11#(isBinKind(V1),V1) 27.12/28.22 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.22 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(prod(V1)) -> U91#(isBagKind(V1),V1) 27.12/28.22 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.22 -> FAxioms: 27.12/28.22 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) -> mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) -> plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) -> union(x7,x6) 27.12/28.22 -> EAxioms: 27.12/28.22 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) = mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) = plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) = union(x7,x6) 27.12/28.22 ->->-> Rules: 27.12/28.22 0(z) -> z 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.22 U172(tt) -> z 27.12/28.22 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.22 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.22 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.22 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.22 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.22 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.22 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.22 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.22 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.22 U202(tt,X) -> X 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.22 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.22 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.22 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.22 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.22 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.22 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.22 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.22 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.22 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.22 U242(tt,X) -> X 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.22 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.22 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.22 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.22 U262(tt,X) -> X 27.12/28.22 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.22 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.22 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.22 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.22 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.22 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.22 isBinKind(z) -> tt 27.12/28.22 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.22 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.22 mult(z,X) -> U171(isBin(X),X) 27.12/28.22 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.22 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.22 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.22 plus(z,X) -> U201(isBin(X),X) 27.12/28.22 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.22 prod(empty) -> 1(z) 27.12/28.22 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.22 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.22 sum(empty) -> 0(z) 27.12/28.22 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.22 union(empty,X) -> X 27.12/28.22 union(X,empty) -> X 27.12/28.22 -> SRules: 27.12/28.22 Empty 27.12/28.22 27.12/28.22 Problem 1.3: 27.12/28.22 27.12/28.22 Reduction Pairs Processor: 27.12/28.22 -> FAxioms: 27.12/28.22 Empty 27.12/28.22 -> Pairs: 27.12/28.22 U11#(tt,V1) -> U12#(isBinKind(V1),V1) 27.12/28.22 U12#(tt,V1) -> ISBIN(V1) 27.12/28.22 U21#(tt,V1,V2) -> U22#(isBagKind(V1),V1,V2) 27.12/28.22 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.22 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.22 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.22 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.22 U25#(tt,V2) -> ISBAG(V2) 27.12/28.22 U51#(tt,V1) -> U52#(isBinKind(V1),V1) 27.12/28.22 U52#(tt,V1) -> ISBIN(V1) 27.12/28.22 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.22 U62#(tt,V1) -> ISBIN(V1) 27.12/28.22 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.22 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.22 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.22 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.22 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U75#(tt,V2) -> ISBIN(V2) 27.12/28.22 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.22 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.22 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.22 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.22 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U85#(tt,V2) -> ISBIN(V2) 27.12/28.22 U91#(tt,V1) -> U92#(isBagKind(V1),V1) 27.12/28.22 U92#(tt,V1) -> ISBAG(V1) 27.12/28.22 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.22 ISBAG(singl(V1)) -> U11#(isBinKind(V1),V1) 27.12/28.22 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.22 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(prod(V1)) -> U91#(isBagKind(V1),V1) 27.12/28.22 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.22 -> EAxioms: 27.12/28.22 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) = mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) = plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) = union(x7,x6) 27.12/28.22 -> Usable Equations: 27.12/28.22 Empty 27.12/28.22 -> Rules: 27.12/28.22 0(z) -> z 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.22 U172(tt) -> z 27.12/28.22 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.22 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.22 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.22 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.22 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.22 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.22 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.22 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.22 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.22 U202(tt,X) -> X 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.22 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.22 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.22 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.22 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.22 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.22 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.22 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.22 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.22 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.22 U242(tt,X) -> X 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.22 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.22 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.22 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.22 U262(tt,X) -> X 27.12/28.22 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.22 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.22 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.22 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.22 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.22 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.22 isBinKind(z) -> tt 27.12/28.22 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.22 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.22 mult(z,X) -> U171(isBin(X),X) 27.12/28.22 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.22 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.22 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.22 plus(z,X) -> U201(isBin(X),X) 27.12/28.22 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.22 prod(empty) -> 1(z) 27.12/28.22 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.22 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.22 sum(empty) -> 0(z) 27.12/28.22 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.22 union(empty,X) -> X 27.12/28.22 union(X,empty) -> X 27.12/28.22 -> Usable Rules: 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.22 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.22 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.22 isBinKind(z) -> tt 27.12/28.22 -> SRules: 27.12/28.22 Empty 27.12/28.22 ->Interpretation type: 27.12/28.22 Linear 27.12/28.22 ->Coefficients: 27.12/28.22 Natural Numbers 27.12/28.22 ->Dimension: 27.12/28.22 1 27.12/28.22 ->Bound: 27.12/28.22 2 27.12/28.22 ->Interpretation: 27.12/28.22 27.12/28.22 [0](X) = 2.X + 2 27.12/28.22 [U101](X1,X2) = 2.X1 27.12/28.22 [U102](X1,X2) = X1 + 2 27.12/28.22 [U103](X) = 2.X 27.12/28.22 [U11](X1,X2) = 2 27.12/28.22 [U111](X) = X + 2 27.12/28.22 [U12](X1,X2) = 2 27.12/28.22 [U121](X) = 2.X + 1 27.12/28.22 [U13](X) = 2 27.12/28.22 [U131](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.22 [U132](X) = X + 2 27.12/28.22 [U141](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.22 [U142](X) = X + 2 27.12/28.22 [U151](X) = X + 2 27.12/28.22 [U161](X) = 2.X + 1 27.12/28.22 [U171](X1,X2) = 0 27.12/28.22 [U172](X) = 0 27.12/28.22 [U181](X1,X2,X3) = 0 27.12/28.22 [U182](X1,X2,X3) = 0 27.12/28.22 [U183](X1,X2,X3) = 0 27.12/28.22 [U184](X1,X2,X3) = 0 27.12/28.22 [U191](X1,X2,X3) = 0 27.12/28.22 [U192](X1,X2,X3) = 0 27.12/28.22 [U193](X1,X2,X3) = 0 27.12/28.22 [U194](X1,X2,X3) = 0 27.12/28.22 [U201](X1,X2) = 0 27.12/28.22 [U202](X1,X2) = 0 27.12/28.22 [U21](X1,X2,X3) = X1 27.12/28.22 [U211](X1,X2,X3) = 0 27.12/28.22 [U212](X1,X2,X3) = 0 27.12/28.22 [U213](X1,X2,X3) = 0 27.12/28.22 [U214](X1,X2,X3) = 0 27.12/28.22 [U22](X1,X2,X3) = X1 27.12/28.22 [U221](X1,X2,X3) = 0 27.12/28.22 [U222](X1,X2,X3) = 0 27.12/28.22 [U223](X1,X2,X3) = 0 27.12/28.22 [U224](X1,X2,X3) = 0 27.12/28.22 [U23](X1,X2,X3) = 2 27.12/28.22 [U231](X1,X2,X3) = 0 27.12/28.22 [U232](X1,X2,X3) = 0 27.12/28.22 [U233](X1,X2,X3) = 0 27.12/28.22 [U234](X1,X2,X3) = 0 27.12/28.22 [U24](X1,X2,X3) = X1 27.12/28.22 [U241](X1,X2) = 0 27.12/28.22 [U242](X1,X2) = 0 27.12/28.22 [U25](X1,X2) = 2 27.12/28.22 [U251](X1,X2,X3) = 0 27.12/28.22 [U252](X1,X2,X3) = 0 27.12/28.22 [U253](X1,X2,X3) = 0 27.12/28.22 [U254](X1,X2,X3) = 0 27.12/28.22 [U26](X) = X 27.12/28.22 [U261](X1,X2) = 0 27.12/28.22 [U262](X1,X2) = 0 27.12/28.22 [U271](X1,X2,X3) = 0 27.12/28.22 [U272](X1,X2,X3) = 0 27.12/28.22 [U273](X1,X2,X3) = 0 27.12/28.22 [U274](X1,X2,X3) = 0 27.12/28.22 [U31](X) = 2 27.12/28.22 [U41](X1,X2) = 2 27.12/28.22 [U42](X) = 2 27.12/28.22 [U51](X1,X2) = X1 + 2.X2 + 2 27.12/28.22 [U52](X1,X2) = X1 + 2 27.12/28.22 [U53](X) = 2 27.12/28.22 [U61](X1,X2) = X1 + 2.X2 27.12/28.22 [U62](X1,X2) = 2.X2 27.12/28.22 [U63](X) = X 27.12/28.22 [U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 27.12/28.22 [U72](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U73](X1,X2,X3) = X1 + 2.X2 27.12/28.22 [U74](X1,X2,X3) = 2.X2 + 2 27.12/28.22 [U75](X1,X2) = X1 + 2 27.12/28.22 [U76](X) = 2 27.12/28.22 [U81](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.12/28.22 [U82](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U83](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U84](X1,X2,X3) = X1 + 2.X2 27.12/28.22 [U85](X1,X2) = X1 + 2 27.12/28.22 [U86](X) = 2 27.12/28.22 [U91](X1,X2) = X1 + 2 27.12/28.22 [U92](X1,X2) = X1 + 2 27.12/28.22 [U93](X) = X + 1 27.12/28.22 [isBag](X) = 2 27.12/28.22 [isBagKind](X) = 2 27.12/28.22 [isBin](X) = 2.X 27.12/28.22 [isBinKind](X) = 2.X + 2 27.12/28.22 [mult](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.22 [plus](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.22 [prod](X) = 2.X + 2 27.12/28.22 [sum](X) = 2.X + 2 27.12/28.22 [union](X1,X2) = X1 + X2 + 2 27.12/28.22 [1](X) = 2.X + 2 27.12/28.22 [empty] = 1 27.12/28.22 [singl](X) = 2.X + 2 27.12/28.22 [tt] = 2 27.12/28.22 [z] = 2 27.12/28.22 [0#](X) = 0 27.12/28.22 [U101#](X1,X2) = 0 27.12/28.22 [U102#](X1,X2) = 0 27.12/28.22 [U103#](X) = 0 27.12/28.22 [U11#](X1,X2) = X1 + 2.X2 + 1 27.12/28.22 [U111#](X) = 0 27.12/28.22 [U12#](X1,X2) = 2.X2 + 2 27.12/28.22 [U121#](X) = 0 27.12/28.22 [U13#](X) = 0 27.12/28.22 [U131#](X1,X2) = 0 27.12/28.22 [U132#](X) = 0 27.12/28.22 [U141#](X1,X2) = 0 27.12/28.22 [U142#](X) = 0 27.12/28.22 [U151#](X) = 0 27.12/28.22 [U161#](X) = 0 27.12/28.22 [U171#](X1,X2) = 0 27.12/28.22 [U172#](X) = 0 27.12/28.22 [U181#](X1,X2,X3) = 0 27.12/28.22 [U182#](X1,X2,X3) = 0 27.12/28.22 [U183#](X1,X2,X3) = 0 27.12/28.22 [U184#](X1,X2,X3) = 0 27.12/28.22 [U191#](X1,X2,X3) = 0 27.12/28.22 [U192#](X1,X2,X3) = 0 27.12/28.22 [U193#](X1,X2,X3) = 0 27.12/28.22 [U194#](X1,X2,X3) = 0 27.12/28.22 [U201#](X1,X2) = 0 27.12/28.22 [U202#](X1,X2) = 0 27.12/28.22 [U21#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.12/28.22 [U211#](X1,X2,X3) = 0 27.12/28.22 [U212#](X1,X2,X3) = 0 27.12/28.22 [U213#](X1,X2,X3) = 0 27.12/28.22 [U214#](X1,X2,X3) = 0 27.12/28.22 [U22#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.12/28.22 [U221#](X1,X2,X3) = 0 27.12/28.22 [U222#](X1,X2,X3) = 0 27.12/28.22 [U223#](X1,X2,X3) = 0 27.12/28.22 [U224#](X1,X2,X3) = 0 27.12/28.22 [U23#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.12/28.22 [U231#](X1,X2,X3) = 0 27.12/28.22 [U232#](X1,X2,X3) = 0 27.12/28.22 [U233#](X1,X2,X3) = 0 27.12/28.22 [U234#](X1,X2,X3) = 0 27.12/28.22 [U24#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 27.12/28.22 [U241#](X1,X2) = 0 27.12/28.22 [U242#](X1,X2) = 0 27.12/28.22 [U25#](X1,X2) = X1 + 2.X2 + 2 27.12/28.22 [U251#](X1,X2,X3) = 0 27.12/28.22 [U252#](X1,X2,X3) = 0 27.12/28.22 [U253#](X1,X2,X3) = 0 27.12/28.22 [U254#](X1,X2,X3) = 0 27.12/28.22 [U26#](X) = 0 27.12/28.22 [U261#](X1,X2) = 0 27.12/28.22 [U262#](X1,X2) = 0 27.12/28.22 [U271#](X1,X2,X3) = 0 27.12/28.22 [U272#](X1,X2,X3) = 0 27.12/28.22 [U273#](X1,X2,X3) = 0 27.12/28.22 [U274#](X1,X2,X3) = 0 27.12/28.22 [U31#](X) = 0 27.12/28.22 [U41#](X1,X2) = 0 27.12/28.22 [U42#](X) = 0 27.12/28.22 [U51#](X1,X2) = 2.X2 + 2 27.12/28.22 [U52#](X1,X2) = 2.X2 + 2 27.12/28.22 [U53#](X) = 0 27.12/28.22 [U61#](X1,X2) = X1 + 2.X2 + 1 27.12/28.22 [U62#](X1,X2) = 2.X2 + 2 27.12/28.22 [U63#](X) = 0 27.12/28.22 [U71#](X1,X2,X3) = X1 + 2.X2 + 2.X3 27.12/28.22 [U72#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U73#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U74#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U75#](X1,X2) = X1 + 2.X2 + 2 27.12/28.22 [U76#](X) = 0 27.12/28.22 [U81#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.12/28.22 [U82#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U83#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U84#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U85#](X1,X2) = X1 + 2.X2 27.12/28.22 [U86#](X) = 0 27.12/28.22 [U91#](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.22 [U92#](X1,X2) = 2.X1 + 2.X2 + 1 27.12/28.22 [U93#](X) = 0 27.12/28.22 [ISBAG](X) = 2.X + 2 27.12/28.22 [ISBAGKIND](X) = 0 27.12/28.22 [ISBIN](X) = 2.X + 2 27.12/28.22 [ISBINKIND](X) = 0 27.12/28.22 [MULT](X1,X2) = 0 27.12/28.22 [PLUS](X1,X2) = 0 27.12/28.22 [PROD](X) = 0 27.12/28.22 [SUM](X) = 0 27.12/28.22 [UNION](X1,X2) = 0 27.12/28.22 27.12/28.22 Problem 1.3: 27.12/28.22 27.12/28.22 SCC Processor: 27.12/28.22 -> FAxioms: 27.12/28.22 Empty 27.12/28.22 -> Pairs: 27.12/28.22 U12#(tt,V1) -> ISBIN(V1) 27.12/28.22 U21#(tt,V1,V2) -> U22#(isBagKind(V1),V1,V2) 27.12/28.22 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.22 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.22 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.22 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.22 U25#(tt,V2) -> ISBAG(V2) 27.12/28.22 U51#(tt,V1) -> U52#(isBinKind(V1),V1) 27.12/28.22 U52#(tt,V1) -> ISBIN(V1) 27.12/28.22 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.22 U62#(tt,V1) -> ISBIN(V1) 27.12/28.22 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.22 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.22 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.22 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.22 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U75#(tt,V2) -> ISBIN(V2) 27.12/28.22 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.22 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.22 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.22 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.22 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U85#(tt,V2) -> ISBIN(V2) 27.12/28.22 U91#(tt,V1) -> U92#(isBagKind(V1),V1) 27.12/28.22 U92#(tt,V1) -> ISBAG(V1) 27.12/28.22 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.22 ISBAG(singl(V1)) -> U11#(isBinKind(V1),V1) 27.12/28.22 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.22 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(prod(V1)) -> U91#(isBagKind(V1),V1) 27.12/28.22 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.22 -> EAxioms: 27.12/28.22 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) = mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) = plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) = union(x7,x6) 27.12/28.22 -> Rules: 27.12/28.22 0(z) -> z 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.22 U172(tt) -> z 27.12/28.22 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.22 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.22 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.22 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.22 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.22 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.22 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.22 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.22 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.22 U202(tt,X) -> X 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.22 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.22 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.22 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.22 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.22 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.22 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.22 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.22 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.22 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.22 U242(tt,X) -> X 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.22 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.22 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.22 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.22 U262(tt,X) -> X 27.12/28.22 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.22 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.22 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.22 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.22 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.22 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.22 isBinKind(z) -> tt 27.12/28.22 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.22 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.22 mult(z,X) -> U171(isBin(X),X) 27.12/28.22 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.22 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.22 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.22 plus(z,X) -> U201(isBin(X),X) 27.12/28.22 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.22 prod(empty) -> 1(z) 27.12/28.22 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.22 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.22 sum(empty) -> 0(z) 27.12/28.22 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.22 union(empty,X) -> X 27.12/28.22 union(X,empty) -> X 27.12/28.22 -> SRules: 27.12/28.22 Empty 27.12/28.22 ->Strongly Connected Components: 27.12/28.22 ->->Cycle: 27.12/28.22 ->->-> Pairs: 27.12/28.22 U21#(tt,V1,V2) -> U22#(isBagKind(V1),V1,V2) 27.12/28.22 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.22 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.22 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.22 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.22 U25#(tt,V2) -> ISBAG(V2) 27.12/28.22 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.22 -> FAxioms: 27.12/28.22 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) -> mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) -> plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) -> union(x7,x6) 27.12/28.22 -> EAxioms: 27.12/28.22 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) = mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) = plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) = union(x7,x6) 27.12/28.22 ->->-> Rules: 27.12/28.22 0(z) -> z 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.22 U172(tt) -> z 27.12/28.22 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.22 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.22 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.22 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.22 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.22 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.22 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.22 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.22 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.22 U202(tt,X) -> X 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.22 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.22 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.22 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.22 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.22 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.22 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.22 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.22 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.22 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.22 U242(tt,X) -> X 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.22 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.22 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.22 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.22 U262(tt,X) -> X 27.12/28.22 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.22 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.22 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.22 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.22 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.22 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.22 isBinKind(z) -> tt 27.12/28.22 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.22 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.22 mult(z,X) -> U171(isBin(X),X) 27.12/28.22 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.22 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.22 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.22 plus(z,X) -> U201(isBin(X),X) 27.12/28.22 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.22 prod(empty) -> 1(z) 27.12/28.22 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.22 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.22 sum(empty) -> 0(z) 27.12/28.22 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.22 union(empty,X) -> X 27.12/28.22 union(X,empty) -> X 27.12/28.22 -> SRules: 27.12/28.22 Empty 27.12/28.22 ->->Cycle: 27.12/28.22 ->->-> Pairs: 27.12/28.22 U51#(tt,V1) -> U52#(isBinKind(V1),V1) 27.12/28.22 U52#(tt,V1) -> ISBIN(V1) 27.12/28.22 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.22 U62#(tt,V1) -> ISBIN(V1) 27.12/28.22 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.22 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.22 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.22 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.22 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U75#(tt,V2) -> ISBIN(V2) 27.12/28.22 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.22 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.22 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.22 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.22 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U85#(tt,V2) -> ISBIN(V2) 27.12/28.22 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.22 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.22 -> FAxioms: 27.12/28.22 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) -> mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) -> plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) -> union(x7,x6) 27.12/28.22 -> EAxioms: 27.12/28.22 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) = mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) = plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) = union(x7,x6) 27.12/28.22 ->->-> Rules: 27.12/28.22 0(z) -> z 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.22 U172(tt) -> z 27.12/28.22 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.22 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.22 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.22 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.22 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.22 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.22 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.22 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.22 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.22 U202(tt,X) -> X 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.22 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.22 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.22 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.22 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.22 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.22 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.22 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.22 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.22 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.22 U242(tt,X) -> X 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.22 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.22 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.22 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.22 U262(tt,X) -> X 27.12/28.22 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.22 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.22 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.22 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.22 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.22 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.22 isBinKind(z) -> tt 27.12/28.22 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.22 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.22 mult(z,X) -> U171(isBin(X),X) 27.12/28.22 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.22 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.22 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.22 plus(z,X) -> U201(isBin(X),X) 27.12/28.22 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.22 prod(empty) -> 1(z) 27.12/28.22 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.22 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.22 sum(empty) -> 0(z) 27.12/28.22 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.22 union(empty,X) -> X 27.12/28.22 union(X,empty) -> X 27.12/28.22 -> SRules: 27.12/28.22 Empty 27.12/28.22 27.12/28.22 27.12/28.22 The problem is decomposed in 2 subproblems. 27.12/28.22 27.12/28.22 Problem 1.3.1: 27.12/28.22 27.12/28.22 Reduction Pairs Processor: 27.12/28.22 -> FAxioms: 27.12/28.22 Empty 27.12/28.22 -> Pairs: 27.12/28.22 U21#(tt,V1,V2) -> U22#(isBagKind(V1),V1,V2) 27.12/28.22 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.22 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.22 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.22 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.22 U25#(tt,V2) -> ISBAG(V2) 27.12/28.22 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.22 -> EAxioms: 27.12/28.22 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) = mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) = plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) = union(x7,x6) 27.12/28.22 -> Usable Equations: 27.12/28.22 Empty 27.12/28.22 -> Rules: 27.12/28.22 0(z) -> z 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.22 U172(tt) -> z 27.12/28.22 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.22 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.22 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.22 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.22 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.22 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.22 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.22 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.22 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.22 U202(tt,X) -> X 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.22 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.22 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.22 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.22 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.22 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.22 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.22 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.22 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.22 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.22 U242(tt,X) -> X 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.22 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.22 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.22 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.22 U262(tt,X) -> X 27.12/28.22 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.22 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.22 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.22 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.22 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.22 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.22 isBinKind(z) -> tt 27.12/28.22 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.22 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.22 mult(z,X) -> U171(isBin(X),X) 27.12/28.22 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.22 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.22 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.22 plus(z,X) -> U201(isBin(X),X) 27.12/28.22 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.22 prod(empty) -> 1(z) 27.12/28.22 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.22 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.22 sum(empty) -> 0(z) 27.12/28.22 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.22 union(empty,X) -> X 27.12/28.22 union(X,empty) -> X 27.12/28.22 -> Usable Rules: 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.22 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.22 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.22 isBinKind(z) -> tt 27.12/28.22 -> SRules: 27.12/28.22 Empty 27.12/28.22 ->Interpretation type: 27.12/28.22 Linear 27.12/28.22 ->Coefficients: 27.12/28.22 Natural Numbers 27.12/28.22 ->Dimension: 27.12/28.22 1 27.12/28.22 ->Bound: 27.12/28.22 2 27.12/28.22 ->Interpretation: 27.12/28.22 27.12/28.22 [0](X) = 2.X + 2 27.12/28.22 [U101](X1,X2) = 2 27.12/28.22 [U102](X1,X2) = 2 27.12/28.22 [U103](X) = 2 27.12/28.22 [U11](X1,X2) = 2.X1 + 2 27.12/28.22 [U111](X) = 2.X + 1 27.12/28.22 [U12](X1,X2) = 2 27.12/28.22 [U121](X) = 2.X + 1 27.12/28.22 [U13](X) = X 27.12/28.22 [U131](X1,X2) = 2.X1 + 1 27.12/28.22 [U132](X) = 2 27.12/28.22 [U141](X1,X2) = 2.X2 + 1 27.12/28.22 [U142](X) = X + 1 27.12/28.22 [U151](X) = X + 1 27.12/28.22 [U161](X) = 2 27.12/28.22 [U171](X1,X2) = 0 27.12/28.22 [U172](X) = 0 27.12/28.22 [U181](X1,X2,X3) = 0 27.12/28.22 [U182](X1,X2,X3) = 0 27.12/28.22 [U183](X1,X2,X3) = 0 27.12/28.22 [U184](X1,X2,X3) = 0 27.12/28.22 [U191](X1,X2,X3) = 0 27.12/28.22 [U192](X1,X2,X3) = 0 27.12/28.22 [U193](X1,X2,X3) = 0 27.12/28.22 [U194](X1,X2,X3) = 0 27.12/28.22 [U201](X1,X2) = 0 27.12/28.22 [U202](X1,X2) = 0 27.12/28.22 [U21](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 27.12/28.22 [U211](X1,X2,X3) = 0 27.12/28.22 [U212](X1,X2,X3) = 0 27.12/28.22 [U213](X1,X2,X3) = 0 27.12/28.22 [U214](X1,X2,X3) = 0 27.12/28.22 [U22](X1,X2,X3) = X1 + 2.X3 + 2 27.12/28.22 [U221](X1,X2,X3) = 0 27.12/28.22 [U222](X1,X2,X3) = 0 27.12/28.22 [U223](X1,X2,X3) = 0 27.12/28.22 [U224](X1,X2,X3) = 0 27.12/28.22 [U23](X1,X2,X3) = X1 + X3 + 2 27.12/28.22 [U231](X1,X2,X3) = 0 27.12/28.22 [U232](X1,X2,X3) = 0 27.12/28.22 [U233](X1,X2,X3) = 0 27.12/28.22 [U234](X1,X2,X3) = 0 27.12/28.22 [U24](X1,X2,X3) = X1 + 2 27.12/28.22 [U241](X1,X2) = 0 27.12/28.22 [U242](X1,X2) = 0 27.12/28.22 [U25](X1,X2) = 2 27.12/28.22 [U251](X1,X2,X3) = 0 27.12/28.22 [U252](X1,X2,X3) = 0 27.12/28.22 [U253](X1,X2,X3) = 0 27.12/28.22 [U254](X1,X2,X3) = 0 27.12/28.22 [U26](X) = 2 27.12/28.22 [U261](X1,X2) = 0 27.12/28.22 [U262](X1,X2) = 0 27.12/28.22 [U271](X1,X2,X3) = 0 27.12/28.22 [U272](X1,X2,X3) = 0 27.12/28.22 [U273](X1,X2,X3) = 0 27.12/28.22 [U274](X1,X2,X3) = 0 27.12/28.22 [U31](X) = X + 2 27.12/28.22 [U41](X1,X2) = X1 + 2.X2 + 2 27.12/28.22 [U42](X) = 2.X 27.12/28.22 [U51](X1,X2) = 2 27.12/28.22 [U52](X1,X2) = 2 27.12/28.22 [U53](X) = 2 27.12/28.22 [U61](X1,X2) = 2 27.12/28.22 [U62](X1,X2) = 2 27.12/28.22 [U63](X) = 2 27.12/28.22 [U71](X1,X2,X3) = 2 27.12/28.22 [U72](X1,X2,X3) = 2 27.12/28.22 [U73](X1,X2,X3) = 2 27.12/28.22 [U74](X1,X2,X3) = 2 27.12/28.22 [U75](X1,X2) = 2 27.12/28.22 [U76](X) = 2 27.12/28.22 [U81](X1,X2,X3) = 2 27.12/28.22 [U82](X1,X2,X3) = 2 27.12/28.22 [U83](X1,X2,X3) = 2 27.12/28.22 [U84](X1,X2,X3) = 2 27.12/28.22 [U85](X1,X2) = 2 27.12/28.22 [U86](X) = X 27.12/28.22 [U91](X1,X2) = 2 27.12/28.22 [U92](X1,X2) = 2 27.12/28.22 [U93](X) = 2 27.12/28.22 [isBag](X) = 2.X + 2 27.12/28.22 [isBagKind](X) = X + 2 27.12/28.22 [isBin](X) = 2 27.12/28.22 [isBinKind](X) = 2.X 27.12/28.22 [mult](X1,X2) = 2.X1 + 2 27.12/28.22 [plus](X1,X2) = 2.X1 + 2.X2 + 1 27.12/28.22 [prod](X) = 2.X + 2 27.12/28.22 [sum](X) = 2 27.12/28.22 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.22 [1](X) = 2.X + 2 27.12/28.22 [empty] = 0 27.12/28.22 [singl](X) = 2.X + 1 27.12/28.22 [tt] = 2 27.12/28.22 [z] = 2 27.12/28.22 [0#](X) = 0 27.12/28.22 [U101#](X1,X2) = 0 27.12/28.22 [U102#](X1,X2) = 0 27.12/28.22 [U103#](X) = 0 27.12/28.22 [U11#](X1,X2) = 0 27.12/28.22 [U111#](X) = 0 27.12/28.22 [U12#](X1,X2) = 0 27.12/28.22 [U121#](X) = 0 27.12/28.22 [U13#](X) = 0 27.12/28.22 [U131#](X1,X2) = 0 27.12/28.22 [U132#](X) = 0 27.12/28.22 [U141#](X1,X2) = 0 27.12/28.22 [U142#](X) = 0 27.12/28.22 [U151#](X) = 0 27.12/28.22 [U161#](X) = 0 27.12/28.22 [U171#](X1,X2) = 0 27.12/28.22 [U172#](X) = 0 27.12/28.22 [U181#](X1,X2,X3) = 0 27.12/28.22 [U182#](X1,X2,X3) = 0 27.12/28.22 [U183#](X1,X2,X3) = 0 27.12/28.22 [U184#](X1,X2,X3) = 0 27.12/28.22 [U191#](X1,X2,X3) = 0 27.12/28.22 [U192#](X1,X2,X3) = 0 27.12/28.22 [U193#](X1,X2,X3) = 0 27.12/28.22 [U194#](X1,X2,X3) = 0 27.12/28.22 [U201#](X1,X2) = 0 27.12/28.22 [U202#](X1,X2) = 0 27.12/28.22 [U21#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.22 [U211#](X1,X2,X3) = 0 27.12/28.22 [U212#](X1,X2,X3) = 0 27.12/28.22 [U213#](X1,X2,X3) = 0 27.12/28.22 [U214#](X1,X2,X3) = 0 27.12/28.22 [U22#](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.12/28.22 [U221#](X1,X2,X3) = 0 27.12/28.22 [U222#](X1,X2,X3) = 0 27.12/28.22 [U223#](X1,X2,X3) = 0 27.12/28.22 [U224#](X1,X2,X3) = 0 27.12/28.22 [U23#](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.12/28.22 [U231#](X1,X2,X3) = 0 27.12/28.22 [U232#](X1,X2,X3) = 0 27.12/28.22 [U233#](X1,X2,X3) = 0 27.12/28.22 [U234#](X1,X2,X3) = 0 27.12/28.22 [U24#](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.12/28.22 [U241#](X1,X2) = 0 27.12/28.22 [U242#](X1,X2) = 0 27.12/28.22 [U25#](X1,X2) = 2.X2 + 1 27.12/28.22 [U251#](X1,X2,X3) = 0 27.12/28.22 [U252#](X1,X2,X3) = 0 27.12/28.22 [U253#](X1,X2,X3) = 0 27.12/28.22 [U254#](X1,X2,X3) = 0 27.12/28.22 [U26#](X) = 0 27.12/28.22 [U261#](X1,X2) = 0 27.12/28.22 [U262#](X1,X2) = 0 27.12/28.22 [U271#](X1,X2,X3) = 0 27.12/28.22 [U272#](X1,X2,X3) = 0 27.12/28.22 [U273#](X1,X2,X3) = 0 27.12/28.22 [U274#](X1,X2,X3) = 0 27.12/28.22 [U31#](X) = 0 27.12/28.22 [U41#](X1,X2) = 0 27.12/28.22 [U42#](X) = 0 27.12/28.22 [U51#](X1,X2) = 0 27.12/28.22 [U52#](X1,X2) = 0 27.12/28.22 [U53#](X) = 0 27.12/28.22 [U61#](X1,X2) = 0 27.12/28.22 [U62#](X1,X2) = 0 27.12/28.22 [U63#](X) = 0 27.12/28.22 [U71#](X1,X2,X3) = 0 27.12/28.22 [U72#](X1,X2,X3) = 0 27.12/28.22 [U73#](X1,X2,X3) = 0 27.12/28.22 [U74#](X1,X2,X3) = 0 27.12/28.22 [U75#](X1,X2) = 0 27.12/28.22 [U76#](X) = 0 27.12/28.22 [U81#](X1,X2,X3) = 0 27.12/28.22 [U82#](X1,X2,X3) = 0 27.12/28.22 [U83#](X1,X2,X3) = 0 27.12/28.22 [U84#](X1,X2,X3) = 0 27.12/28.22 [U85#](X1,X2) = 0 27.12/28.22 [U86#](X) = 0 27.12/28.22 [U91#](X1,X2) = 0 27.12/28.22 [U92#](X1,X2) = 0 27.12/28.22 [U93#](X) = 0 27.12/28.22 [ISBAG](X) = 2.X + 1 27.12/28.22 [ISBAGKIND](X) = 0 27.12/28.22 [ISBIN](X) = 0 27.12/28.22 [ISBINKIND](X) = 0 27.12/28.22 [MULT](X1,X2) = 0 27.12/28.22 [PLUS](X1,X2) = 0 27.12/28.22 [PROD](X) = 0 27.12/28.22 [SUM](X) = 0 27.12/28.22 [UNION](X1,X2) = 0 27.12/28.22 27.12/28.22 Problem 1.3.1: 27.12/28.22 27.12/28.22 SCC Processor: 27.12/28.22 -> FAxioms: 27.12/28.22 Empty 27.12/28.22 -> Pairs: 27.12/28.22 U22#(tt,V1,V2) -> U23#(isBagKind(V2),V1,V2) 27.12/28.22 U23#(tt,V1,V2) -> U24#(isBagKind(V2),V1,V2) 27.12/28.22 U24#(tt,V1,V2) -> U25#(isBag(V1),V2) 27.12/28.22 U24#(tt,V1,V2) -> ISBAG(V1) 27.12/28.22 U25#(tt,V2) -> ISBAG(V2) 27.12/28.22 ISBAG(union(V1,V2)) -> U21#(isBagKind(V1),V1,V2) 27.12/28.22 -> EAxioms: 27.12/28.22 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) = mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) = plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) = union(x7,x6) 27.12/28.22 -> Rules: 27.12/28.22 0(z) -> z 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.22 U172(tt) -> z 27.12/28.22 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.22 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.22 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.22 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.22 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.22 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.22 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.22 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.22 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.22 U202(tt,X) -> X 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.22 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.22 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.22 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.22 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.22 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.22 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.22 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.22 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.22 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.22 U242(tt,X) -> X 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.22 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.22 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.22 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.22 U262(tt,X) -> X 27.12/28.22 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.22 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.22 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.22 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.22 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.22 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.22 isBinKind(z) -> tt 27.12/28.22 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.22 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.22 mult(z,X) -> U171(isBin(X),X) 27.12/28.22 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.22 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.22 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.22 plus(z,X) -> U201(isBin(X),X) 27.12/28.22 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.22 prod(empty) -> 1(z) 27.12/28.22 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.22 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.22 sum(empty) -> 0(z) 27.12/28.22 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.22 union(empty,X) -> X 27.12/28.22 union(X,empty) -> X 27.12/28.22 -> SRules: 27.12/28.22 Empty 27.12/28.22 ->Strongly Connected Components: 27.12/28.22 There is no strongly connected component 27.12/28.22 27.12/28.22 The problem is finite. 27.12/28.22 27.12/28.22 Problem 1.3.2: 27.12/28.22 27.12/28.22 Reduction Pairs Processor: 27.12/28.22 -> FAxioms: 27.12/28.22 Empty 27.12/28.22 -> Pairs: 27.12/28.22 U51#(tt,V1) -> U52#(isBinKind(V1),V1) 27.12/28.22 U52#(tt,V1) -> ISBIN(V1) 27.12/28.22 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.22 U62#(tt,V1) -> ISBIN(V1) 27.12/28.22 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.22 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.22 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.22 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.22 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U75#(tt,V2) -> ISBIN(V2) 27.12/28.22 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.22 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.22 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.22 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.22 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.22 U85#(tt,V2) -> ISBIN(V2) 27.12/28.22 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.22 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.22 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.22 -> EAxioms: 27.12/28.22 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.22 mult(x6,x7) = mult(x7,x6) 27.12/28.22 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.22 plus(x6,x7) = plus(x7,x6) 27.12/28.22 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.22 union(x6,x7) = union(x7,x6) 27.12/28.22 -> Usable Equations: 27.12/28.22 Empty 27.12/28.22 -> Rules: 27.12/28.22 0(z) -> z 27.12/28.22 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.22 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.22 U103(tt) -> tt 27.12/28.22 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.22 U111(tt) -> tt 27.12/28.22 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.22 U121(tt) -> tt 27.12/28.22 U13(tt) -> tt 27.12/28.22 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.22 U132(tt) -> tt 27.12/28.22 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.22 U142(tt) -> tt 27.12/28.22 U151(tt) -> tt 27.12/28.22 U161(tt) -> tt 27.12/28.22 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.22 U172(tt) -> z 27.12/28.22 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.22 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.22 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.22 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.22 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.22 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.22 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.22 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.22 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.22 U202(tt,X) -> X 27.12/28.22 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.22 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.22 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.22 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.22 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.22 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.22 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.22 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.22 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.22 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.22 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.22 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.22 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.22 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.22 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.22 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.22 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.22 U242(tt,X) -> X 27.12/28.22 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.22 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.22 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.22 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.22 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.22 U26(tt) -> tt 27.12/28.22 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.22 U262(tt,X) -> X 27.12/28.22 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.22 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.22 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.22 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.22 U31(tt) -> tt 27.12/28.22 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.22 U42(tt) -> tt 27.12/28.22 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.22 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.22 U53(tt) -> tt 27.12/28.22 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.22 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.22 U63(tt) -> tt 27.12/28.22 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.22 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.22 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.22 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.22 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.22 U76(tt) -> tt 27.12/28.22 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.22 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.22 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.22 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.22 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.22 U86(tt) -> tt 27.12/28.22 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.22 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.22 U93(tt) -> tt 27.12/28.22 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.22 isBag(empty) -> tt 27.12/28.22 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.22 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.22 isBagKind(empty) -> tt 27.12/28.22 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.22 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.22 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.22 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.22 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.22 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.22 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.22 isBin(z) -> tt 27.12/28.22 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.22 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.22 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.22 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.23 isBinKind(z) -> tt 27.12/28.23 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.23 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.23 mult(z,X) -> U171(isBin(X),X) 27.12/28.23 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.23 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.23 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.23 plus(z,X) -> U201(isBin(X),X) 27.12/28.23 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.23 prod(empty) -> 1(z) 27.12/28.23 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.23 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.23 sum(empty) -> 0(z) 27.12/28.23 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.23 union(empty,X) -> X 27.12/28.23 union(X,empty) -> X 27.12/28.23 -> Usable Rules: 27.12/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.23 U103(tt) -> tt 27.12/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.23 U111(tt) -> tt 27.12/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.23 U121(tt) -> tt 27.12/28.23 U13(tt) -> tt 27.12/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.23 U132(tt) -> tt 27.12/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.23 U142(tt) -> tt 27.12/28.23 U151(tt) -> tt 27.12/28.23 U161(tt) -> tt 27.12/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.23 U26(tt) -> tt 27.12/28.23 U31(tt) -> tt 27.12/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.23 U42(tt) -> tt 27.12/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.23 U53(tt) -> tt 27.12/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.23 U63(tt) -> tt 27.12/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.23 U76(tt) -> tt 27.12/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.23 U86(tt) -> tt 27.12/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.23 U93(tt) -> tt 27.12/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.23 isBag(empty) -> tt 27.12/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.23 isBagKind(empty) -> tt 27.12/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.23 isBin(z) -> tt 27.12/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.23 isBinKind(z) -> tt 27.12/28.23 -> SRules: 27.12/28.23 Empty 27.12/28.23 ->Interpretation type: 27.12/28.23 Linear 27.12/28.23 ->Coefficients: 27.12/28.23 Natural Numbers 27.12/28.23 ->Dimension: 27.12/28.23 1 27.12/28.23 ->Bound: 27.12/28.23 2 27.12/28.23 ->Interpretation: 27.12/28.23 27.12/28.23 [0](X) = 2.X + 2 27.12/28.23 [U101](X1,X2) = 2.X2 + 2 27.12/28.23 [U102](X1,X2) = 2.X2 + 2 27.12/28.23 [U103](X) = X 27.12/28.23 [U11](X1,X2) = 2 27.12/28.23 [U111](X) = 2.X + 2 27.12/28.23 [U12](X1,X2) = 2 27.12/28.23 [U121](X) = 2.X + 2 27.12/28.23 [U13](X) = 2 27.12/28.23 [U131](X1,X2) = 2.X1 + 2.X2 27.12/28.23 [U132](X) = X + 1 27.12/28.23 [U141](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [U142](X) = X + 2 27.12/28.23 [U151](X) = 2.X + 2 27.12/28.23 [U161](X) = 2 27.12/28.23 [U171](X1,X2) = 0 27.12/28.23 [U172](X) = 0 27.12/28.23 [U181](X1,X2,X3) = 0 27.12/28.23 [U182](X1,X2,X3) = 0 27.12/28.23 [U183](X1,X2,X3) = 0 27.12/28.23 [U184](X1,X2,X3) = 0 27.12/28.23 [U191](X1,X2,X3) = 0 27.12/28.23 [U192](X1,X2,X3) = 0 27.12/28.23 [U193](X1,X2,X3) = 0 27.12/28.23 [U194](X1,X2,X3) = 0 27.12/28.23 [U201](X1,X2) = 0 27.12/28.23 [U202](X1,X2) = 0 27.12/28.23 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.12/28.23 [U211](X1,X2,X3) = 0 27.12/28.23 [U212](X1,X2,X3) = 0 27.12/28.23 [U213](X1,X2,X3) = 0 27.12/28.23 [U214](X1,X2,X3) = 0 27.12/28.23 [U22](X1,X2,X3) = X1 + X2 + 2.X3 27.12/28.23 [U221](X1,X2,X3) = 0 27.12/28.23 [U222](X1,X2,X3) = 0 27.12/28.23 [U223](X1,X2,X3) = 0 27.12/28.23 [U224](X1,X2,X3) = 0 27.12/28.23 [U23](X1,X2,X3) = X1 + X2 + X3 + 2 27.12/28.23 [U231](X1,X2,X3) = 0 27.12/28.23 [U232](X1,X2,X3) = 0 27.12/28.23 [U233](X1,X2,X3) = 0 27.12/28.23 [U234](X1,X2,X3) = 0 27.12/28.23 [U24](X1,X2,X3) = X1 + X2 + 2 27.12/28.23 [U241](X1,X2) = 0 27.12/28.23 [U242](X1,X2) = 0 27.12/28.23 [U25](X1,X2) = 2 27.12/28.23 [U251](X1,X2,X3) = 0 27.12/28.23 [U252](X1,X2,X3) = 0 27.12/28.23 [U253](X1,X2,X3) = 0 27.12/28.23 [U254](X1,X2,X3) = 0 27.12/28.23 [U26](X) = 2 27.12/28.23 [U261](X1,X2) = 0 27.12/28.23 [U262](X1,X2) = 0 27.12/28.23 [U271](X1,X2,X3) = 0 27.12/28.23 [U272](X1,X2,X3) = 0 27.12/28.23 [U273](X1,X2,X3) = 0 27.12/28.23 [U274](X1,X2,X3) = 0 27.12/28.23 [U31](X) = 2 27.12/28.23 [U41](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [U42](X) = 2.X + 2 27.12/28.23 [U51](X1,X2) = 2.X2 + 2 27.12/28.23 [U52](X1,X2) = X1 27.12/28.23 [U53](X) = 2 27.12/28.23 [U61](X1,X2) = 2.X2 + 2 27.12/28.23 [U62](X1,X2) = 2.X2 + 2 27.12/28.23 [U63](X) = 2.X 27.12/28.23 [U71](X1,X2,X3) = X1 + 2.X3 + 1 27.12/28.23 [U72](X1,X2,X3) = 2.X3 + 2 27.12/28.23 [U73](X1,X2,X3) = 2.X3 + 2 27.12/28.23 [U74](X1,X2,X3) = X1 27.12/28.23 [U75](X1,X2) = 2 27.12/28.23 [U76](X) = 2 27.12/28.23 [U81](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U82](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U83](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U84](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U85](X1,X2) = 2.X1 + 2.X2 27.12/28.23 [U86](X) = 2.X + 2 27.12/28.23 [U91](X1,X2) = 2.X2 + 2 27.12/28.23 [U92](X1,X2) = 2.X1 + 2 27.12/28.23 [U93](X) = 2 27.12/28.23 [isBag](X) = 2.X + 2 27.12/28.23 [isBagKind](X) = X 27.12/28.23 [isBin](X) = X + 1 27.12/28.23 [isBinKind](X) = 2.X + 2 27.12/28.23 [mult](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [plus](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [prod](X) = 2.X + 2 27.12/28.23 [sum](X) = 2.X + 2 27.12/28.23 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [1](X) = 2.X + 2 27.12/28.23 [empty] = 2 27.12/28.23 [singl](X) = 2 27.12/28.23 [tt] = 2 27.12/28.23 [z] = 2 27.12/28.23 [0#](X) = 0 27.12/28.23 [U101#](X1,X2) = 0 27.12/28.23 [U102#](X1,X2) = 0 27.12/28.23 [U103#](X) = 0 27.12/28.23 [U11#](X1,X2) = 0 27.12/28.23 [U111#](X) = 0 27.12/28.23 [U12#](X1,X2) = 0 27.12/28.23 [U121#](X) = 0 27.12/28.23 [U13#](X) = 0 27.12/28.23 [U131#](X1,X2) = 0 27.12/28.23 [U132#](X) = 0 27.12/28.23 [U141#](X1,X2) = 0 27.12/28.23 [U142#](X) = 0 27.12/28.23 [U151#](X) = 0 27.12/28.23 [U161#](X) = 0 27.12/28.23 [U171#](X1,X2) = 0 27.12/28.23 [U172#](X) = 0 27.12/28.23 [U181#](X1,X2,X3) = 0 27.12/28.23 [U182#](X1,X2,X3) = 0 27.12/28.23 [U183#](X1,X2,X3) = 0 27.12/28.23 [U184#](X1,X2,X3) = 0 27.12/28.23 [U191#](X1,X2,X3) = 0 27.12/28.23 [U192#](X1,X2,X3) = 0 27.12/28.23 [U193#](X1,X2,X3) = 0 27.12/28.23 [U194#](X1,X2,X3) = 0 27.12/28.23 [U201#](X1,X2) = 0 27.12/28.23 [U202#](X1,X2) = 0 27.12/28.23 [U21#](X1,X2,X3) = 0 27.12/28.23 [U211#](X1,X2,X3) = 0 27.12/28.23 [U212#](X1,X2,X3) = 0 27.12/28.23 [U213#](X1,X2,X3) = 0 27.12/28.23 [U214#](X1,X2,X3) = 0 27.12/28.23 [U22#](X1,X2,X3) = 0 27.12/28.23 [U221#](X1,X2,X3) = 0 27.12/28.23 [U222#](X1,X2,X3) = 0 27.12/28.23 [U223#](X1,X2,X3) = 0 27.12/28.23 [U224#](X1,X2,X3) = 0 27.12/28.23 [U23#](X1,X2,X3) = 0 27.12/28.23 [U231#](X1,X2,X3) = 0 27.12/28.23 [U232#](X1,X2,X3) = 0 27.12/28.23 [U233#](X1,X2,X3) = 0 27.12/28.23 [U234#](X1,X2,X3) = 0 27.12/28.23 [U24#](X1,X2,X3) = 0 27.12/28.23 [U241#](X1,X2) = 0 27.12/28.23 [U242#](X1,X2) = 0 27.12/28.23 [U25#](X1,X2) = 0 27.12/28.23 [U251#](X1,X2,X3) = 0 27.12/28.23 [U252#](X1,X2,X3) = 0 27.12/28.23 [U253#](X1,X2,X3) = 0 27.12/28.23 [U254#](X1,X2,X3) = 0 27.12/28.23 [U26#](X) = 0 27.12/28.23 [U261#](X1,X2) = 0 27.12/28.23 [U262#](X1,X2) = 0 27.12/28.23 [U271#](X1,X2,X3) = 0 27.12/28.23 [U272#](X1,X2,X3) = 0 27.12/28.23 [U273#](X1,X2,X3) = 0 27.12/28.23 [U274#](X1,X2,X3) = 0 27.12/28.23 [U31#](X) = 0 27.12/28.23 [U41#](X1,X2) = 0 27.12/28.23 [U42#](X) = 0 27.12/28.23 [U51#](X1,X2) = X1 + 2.X2 + 2 27.12/28.23 [U52#](X1,X2) = 2.X2 + 2 27.12/28.23 [U53#](X) = 0 27.12/28.23 [U61#](X1,X2) = X1 + 2.X2 + 2 27.12/28.23 [U62#](X1,X2) = 2.X2 + 2 27.12/28.23 [U63#](X) = 0 27.12/28.23 [U71#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.12/28.23 [U72#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U73#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U74#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U75#](X1,X2) = 2.X2 + 2 27.12/28.23 [U76#](X) = 0 27.12/28.23 [U81#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 27.12/28.23 [U82#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U83#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U84#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U85#](X1,X2) = X1 + 2.X2 + 1 27.12/28.23 [U86#](X) = 0 27.12/28.23 [U91#](X1,X2) = 0 27.12/28.23 [U92#](X1,X2) = 0 27.12/28.23 [U93#](X) = 0 27.12/28.23 [ISBAG](X) = 0 27.12/28.23 [ISBAGKIND](X) = 0 27.12/28.23 [ISBIN](X) = 2.X + 2 27.12/28.23 [ISBINKIND](X) = 0 27.12/28.23 [MULT](X1,X2) = 0 27.12/28.23 [PLUS](X1,X2) = 0 27.12/28.23 [PROD](X) = 0 27.12/28.23 [SUM](X) = 0 27.12/28.23 [UNION](X1,X2) = 0 27.12/28.23 27.12/28.23 Problem 1.3.2: 27.12/28.23 27.12/28.23 SCC Processor: 27.12/28.23 -> FAxioms: 27.12/28.23 Empty 27.12/28.23 -> Pairs: 27.12/28.23 U52#(tt,V1) -> ISBIN(V1) 27.12/28.23 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.23 U62#(tt,V1) -> ISBIN(V1) 27.12/28.23 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.23 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.23 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.23 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.23 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.23 U75#(tt,V2) -> ISBIN(V2) 27.12/28.23 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.23 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.23 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.23 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.23 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.23 U85#(tt,V2) -> ISBIN(V2) 27.12/28.23 ISBIN(0(V1)) -> U51#(isBinKind(V1),V1) 27.12/28.23 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.23 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.23 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.23 -> EAxioms: 27.12/28.23 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.23 mult(x6,x7) = mult(x7,x6) 27.12/28.23 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.23 plus(x6,x7) = plus(x7,x6) 27.12/28.23 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.23 union(x6,x7) = union(x7,x6) 27.12/28.23 -> Rules: 27.12/28.23 0(z) -> z 27.12/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.23 U103(tt) -> tt 27.12/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.23 U111(tt) -> tt 27.12/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.23 U121(tt) -> tt 27.12/28.23 U13(tt) -> tt 27.12/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.23 U132(tt) -> tt 27.12/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.23 U142(tt) -> tt 27.12/28.23 U151(tt) -> tt 27.12/28.23 U161(tt) -> tt 27.12/28.23 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.23 U172(tt) -> z 27.12/28.23 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.23 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.23 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.23 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.23 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.23 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.23 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.23 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.23 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.23 U202(tt,X) -> X 27.12/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.23 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.23 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.23 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.23 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.23 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.23 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.23 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.23 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.23 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.23 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.23 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.23 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.23 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.23 U242(tt,X) -> X 27.12/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.23 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.23 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.23 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.23 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.23 U26(tt) -> tt 27.12/28.23 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.23 U262(tt,X) -> X 27.12/28.23 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.23 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.23 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.23 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.23 U31(tt) -> tt 27.12/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.23 U42(tt) -> tt 27.12/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.23 U53(tt) -> tt 27.12/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.23 U63(tt) -> tt 27.12/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.23 U76(tt) -> tt 27.12/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.23 U86(tt) -> tt 27.12/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.23 U93(tt) -> tt 27.12/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.23 isBag(empty) -> tt 27.12/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.23 isBagKind(empty) -> tt 27.12/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.23 isBin(z) -> tt 27.12/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.23 isBinKind(z) -> tt 27.12/28.23 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.23 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.23 mult(z,X) -> U171(isBin(X),X) 27.12/28.23 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.23 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.23 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.23 plus(z,X) -> U201(isBin(X),X) 27.12/28.23 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.23 prod(empty) -> 1(z) 27.12/28.23 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.23 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.23 sum(empty) -> 0(z) 27.12/28.23 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.23 union(empty,X) -> X 27.12/28.23 union(X,empty) -> X 27.12/28.23 -> SRules: 27.12/28.23 Empty 27.12/28.23 ->Strongly Connected Components: 27.12/28.23 ->->Cycle: 27.12/28.23 ->->-> Pairs: 27.12/28.23 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.23 U62#(tt,V1) -> ISBIN(V1) 27.12/28.23 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.23 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.23 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.23 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.23 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.23 U75#(tt,V2) -> ISBIN(V2) 27.12/28.23 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.23 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.23 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.23 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.23 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.23 U85#(tt,V2) -> ISBIN(V2) 27.12/28.23 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.23 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.23 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.23 -> FAxioms: 27.12/28.23 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.12/28.23 mult(x6,x7) -> mult(x7,x6) 27.12/28.23 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.12/28.23 plus(x6,x7) -> plus(x7,x6) 27.12/28.23 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.12/28.23 union(x6,x7) -> union(x7,x6) 27.12/28.23 -> EAxioms: 27.12/28.23 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.23 mult(x6,x7) = mult(x7,x6) 27.12/28.23 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.23 plus(x6,x7) = plus(x7,x6) 27.12/28.23 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.23 union(x6,x7) = union(x7,x6) 27.12/28.23 ->->-> Rules: 27.12/28.23 0(z) -> z 27.12/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.23 U103(tt) -> tt 27.12/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.23 U111(tt) -> tt 27.12/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.23 U121(tt) -> tt 27.12/28.23 U13(tt) -> tt 27.12/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.23 U132(tt) -> tt 27.12/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.23 U142(tt) -> tt 27.12/28.23 U151(tt) -> tt 27.12/28.23 U161(tt) -> tt 27.12/28.23 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.23 U172(tt) -> z 27.12/28.23 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.23 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.23 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.23 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.23 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.23 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.23 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.23 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.23 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.23 U202(tt,X) -> X 27.12/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.23 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.23 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.23 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.23 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.23 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.23 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.23 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.23 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.23 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.23 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.23 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.23 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.23 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.23 U242(tt,X) -> X 27.12/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.23 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.23 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.23 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.23 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.23 U26(tt) -> tt 27.12/28.23 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.23 U262(tt,X) -> X 27.12/28.23 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.23 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.23 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.23 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.23 U31(tt) -> tt 27.12/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.23 U42(tt) -> tt 27.12/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.23 U53(tt) -> tt 27.12/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.23 U63(tt) -> tt 27.12/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.23 U76(tt) -> tt 27.12/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.23 U86(tt) -> tt 27.12/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.23 U93(tt) -> tt 27.12/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.23 isBag(empty) -> tt 27.12/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.23 isBagKind(empty) -> tt 27.12/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.23 isBin(z) -> tt 27.12/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.23 isBinKind(z) -> tt 27.12/28.23 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.23 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.23 mult(z,X) -> U171(isBin(X),X) 27.12/28.23 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.23 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.23 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.23 plus(z,X) -> U201(isBin(X),X) 27.12/28.23 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.23 prod(empty) -> 1(z) 27.12/28.23 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.23 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.23 sum(empty) -> 0(z) 27.12/28.23 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.23 union(empty,X) -> X 27.12/28.23 union(X,empty) -> X 27.12/28.23 -> SRules: 27.12/28.23 Empty 27.12/28.23 27.12/28.23 Problem 1.3.2: 27.12/28.23 27.12/28.23 Reduction Pairs Processor: 27.12/28.23 -> FAxioms: 27.12/28.23 Empty 27.12/28.23 -> Pairs: 27.12/28.23 U61#(tt,V1) -> U62#(isBinKind(V1),V1) 27.12/28.23 U62#(tt,V1) -> ISBIN(V1) 27.12/28.23 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.23 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.23 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.23 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.23 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.23 U75#(tt,V2) -> ISBIN(V2) 27.12/28.23 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.23 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.23 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.23 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.23 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.23 U85#(tt,V2) -> ISBIN(V2) 27.12/28.23 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.23 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.23 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.23 -> EAxioms: 27.12/28.23 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.23 mult(x6,x7) = mult(x7,x6) 27.12/28.23 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.23 plus(x6,x7) = plus(x7,x6) 27.12/28.23 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.23 union(x6,x7) = union(x7,x6) 27.12/28.23 -> Usable Equations: 27.12/28.23 Empty 27.12/28.23 -> Rules: 27.12/28.23 0(z) -> z 27.12/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.23 U103(tt) -> tt 27.12/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.23 U111(tt) -> tt 27.12/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.23 U121(tt) -> tt 27.12/28.23 U13(tt) -> tt 27.12/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.23 U132(tt) -> tt 27.12/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.23 U142(tt) -> tt 27.12/28.23 U151(tt) -> tt 27.12/28.23 U161(tt) -> tt 27.12/28.23 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.23 U172(tt) -> z 27.12/28.23 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.23 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.23 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.23 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.23 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.23 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.23 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.23 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.23 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.23 U202(tt,X) -> X 27.12/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.23 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.23 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.23 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.23 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.23 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.23 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.23 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.23 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.23 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.23 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.23 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.23 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.23 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.23 U242(tt,X) -> X 27.12/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.23 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.23 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.23 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.23 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.23 U26(tt) -> tt 27.12/28.23 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.23 U262(tt,X) -> X 27.12/28.23 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.23 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.23 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.23 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.23 U31(tt) -> tt 27.12/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.23 U42(tt) -> tt 27.12/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.23 U53(tt) -> tt 27.12/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.23 U63(tt) -> tt 27.12/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.23 U76(tt) -> tt 27.12/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.23 U86(tt) -> tt 27.12/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.23 U93(tt) -> tt 27.12/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.23 isBag(empty) -> tt 27.12/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.23 isBagKind(empty) -> tt 27.12/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.23 isBin(z) -> tt 27.12/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.23 isBinKind(z) -> tt 27.12/28.23 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.12/28.23 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.12/28.23 mult(z,X) -> U171(isBin(X),X) 27.12/28.23 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.12/28.23 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.12/28.23 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.12/28.23 plus(z,X) -> U201(isBin(X),X) 27.12/28.23 prod(union(A,B)) -> U251(isBag(A),A,B) 27.12/28.23 prod(empty) -> 1(z) 27.12/28.23 prod(singl(X)) -> U241(isBin(X),X) 27.12/28.23 sum(union(A,B)) -> U271(isBag(A),A,B) 27.12/28.23 sum(empty) -> 0(z) 27.12/28.23 sum(singl(X)) -> U261(isBin(X),X) 27.12/28.23 union(empty,X) -> X 27.12/28.23 union(X,empty) -> X 27.12/28.23 -> Usable Rules: 27.12/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.23 U103(tt) -> tt 27.12/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.23 U111(tt) -> tt 27.12/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.23 U121(tt) -> tt 27.12/28.23 U13(tt) -> tt 27.12/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.23 U132(tt) -> tt 27.12/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.23 U142(tt) -> tt 27.12/28.23 U151(tt) -> tt 27.12/28.23 U161(tt) -> tt 27.12/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.23 U26(tt) -> tt 27.12/28.23 U31(tt) -> tt 27.12/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.23 U42(tt) -> tt 27.12/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.23 U53(tt) -> tt 27.12/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.23 U63(tt) -> tt 27.12/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.23 U76(tt) -> tt 27.12/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.23 U86(tt) -> tt 27.12/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.23 U93(tt) -> tt 27.12/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.23 isBag(empty) -> tt 27.12/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.23 isBagKind(empty) -> tt 27.12/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.23 isBin(z) -> tt 27.12/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.12/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.12/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.12/28.23 isBinKind(z) -> tt 27.12/28.23 -> SRules: 27.12/28.23 Empty 27.12/28.23 ->Interpretation type: 27.12/28.23 Linear 27.12/28.23 ->Coefficients: 27.12/28.23 Natural Numbers 27.12/28.23 ->Dimension: 27.12/28.23 1 27.12/28.23 ->Bound: 27.12/28.23 2 27.12/28.23 ->Interpretation: 27.12/28.23 27.12/28.23 [0](X) = 2.X 27.12/28.23 [U101](X1,X2) = 2.X2 + 2 27.12/28.23 [U102](X1,X2) = X1 + 2 27.12/28.23 [U103](X) = 2 27.12/28.23 [U11](X1,X2) = X1 + 2.X2 + 2 27.12/28.23 [U111](X) = 2 27.12/28.23 [U12](X1,X2) = X1 + 2 27.12/28.23 [U121](X) = 2.X + 2 27.12/28.23 [U13](X) = 2 27.12/28.23 [U131](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [U132](X) = X + 2 27.12/28.23 [U141](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [U142](X) = X + 1 27.12/28.23 [U151](X) = 2.X + 2 27.12/28.23 [U161](X) = 2 27.12/28.23 [U171](X1,X2) = 0 27.12/28.23 [U172](X) = 0 27.12/28.23 [U181](X1,X2,X3) = 0 27.12/28.23 [U182](X1,X2,X3) = 0 27.12/28.23 [U183](X1,X2,X3) = 0 27.12/28.23 [U184](X1,X2,X3) = 0 27.12/28.23 [U191](X1,X2,X3) = 0 27.12/28.23 [U192](X1,X2,X3) = 0 27.12/28.23 [U193](X1,X2,X3) = 0 27.12/28.23 [U194](X1,X2,X3) = 0 27.12/28.23 [U201](X1,X2) = 0 27.12/28.23 [U202](X1,X2) = 0 27.12/28.23 [U21](X1,X2,X3) = 2.X1 + 2.X3 + 2 27.12/28.23 [U211](X1,X2,X3) = 0 27.12/28.23 [U212](X1,X2,X3) = 0 27.12/28.23 [U213](X1,X2,X3) = 0 27.12/28.23 [U214](X1,X2,X3) = 0 27.12/28.23 [U22](X1,X2,X3) = 2.X3 + 2 27.12/28.23 [U221](X1,X2,X3) = 0 27.12/28.23 [U222](X1,X2,X3) = 0 27.12/28.23 [U223](X1,X2,X3) = 0 27.12/28.23 [U224](X1,X2,X3) = 0 27.12/28.23 [U23](X1,X2,X3) = X1 + 2 27.12/28.23 [U231](X1,X2,X3) = 0 27.12/28.23 [U232](X1,X2,X3) = 0 27.12/28.23 [U233](X1,X2,X3) = 0 27.12/28.23 [U234](X1,X2,X3) = 0 27.12/28.23 [U24](X1,X2,X3) = 2 27.12/28.23 [U241](X1,X2) = 0 27.12/28.23 [U242](X1,X2) = 0 27.12/28.23 [U25](X1,X2) = 2 27.12/28.23 [U251](X1,X2,X3) = 0 27.12/28.23 [U252](X1,X2,X3) = 0 27.12/28.23 [U253](X1,X2,X3) = 0 27.12/28.23 [U254](X1,X2,X3) = 0 27.12/28.23 [U26](X) = 2 27.12/28.23 [U261](X1,X2) = 0 27.12/28.23 [U262](X1,X2) = 0 27.12/28.23 [U271](X1,X2,X3) = 0 27.12/28.23 [U272](X1,X2,X3) = 0 27.12/28.23 [U273](X1,X2,X3) = 0 27.12/28.23 [U274](X1,X2,X3) = 0 27.12/28.23 [U31](X) = X + 1 27.12/28.23 [U41](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [U42](X) = X + 1 27.12/28.23 [U51](X1,X2) = X1 27.12/28.23 [U52](X1,X2) = 2 27.12/28.23 [U53](X) = 2 27.12/28.23 [U61](X1,X2) = X1 + 2.X2 + 1 27.12/28.23 [U62](X1,X2) = X2 + 2 27.12/28.23 [U63](X) = 2 27.12/28.23 [U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.12/28.23 [U72](X1,X2,X3) = X1 + 2.X3 + 2 27.12/28.23 [U73](X1,X2,X3) = 2.X3 + 2 27.12/28.23 [U74](X1,X2,X3) = X1 27.12/28.23 [U75](X1,X2) = 2 27.12/28.23 [U76](X) = 2 27.12/28.23 [U81](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.12/28.23 [U82](X1,X2,X3) = X1 + 2.X3 + 1 27.12/28.23 [U83](X1,X2,X3) = 2.X3 + 2 27.12/28.23 [U84](X1,X2,X3) = X1 27.12/28.23 [U85](X1,X2) = 2 27.12/28.23 [U86](X) = 2 27.12/28.23 [U91](X1,X2) = X1 + 2.X2 27.12/28.23 [U92](X1,X2) = X1 + 1 27.12/28.23 [U93](X) = 2 27.12/28.23 [isBag](X) = 2.X + 2 27.12/28.23 [isBagKind](X) = 2.X 27.12/28.23 [isBin](X) = 2.X + 2 27.12/28.23 [isBinKind](X) = 2.X + 2 27.12/28.23 [mult](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [plus](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [prod](X) = 2.X + 2 27.12/28.23 [sum](X) = X + 2 27.12/28.23 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.12/28.23 [1](X) = 2.X + 2 27.12/28.23 [empty] = 1 27.12/28.23 [singl](X) = 2.X + 2 27.12/28.23 [tt] = 2 27.12/28.23 [z] = 0 27.12/28.23 [0#](X) = 0 27.12/28.23 [U101#](X1,X2) = 0 27.12/28.23 [U102#](X1,X2) = 0 27.12/28.23 [U103#](X) = 0 27.12/28.23 [U11#](X1,X2) = 0 27.12/28.23 [U111#](X) = 0 27.12/28.23 [U12#](X1,X2) = 0 27.12/28.23 [U121#](X) = 0 27.12/28.23 [U13#](X) = 0 27.12/28.23 [U131#](X1,X2) = 0 27.12/28.23 [U132#](X) = 0 27.12/28.23 [U141#](X1,X2) = 0 27.12/28.23 [U142#](X) = 0 27.12/28.23 [U151#](X) = 0 27.12/28.23 [U161#](X) = 0 27.12/28.23 [U171#](X1,X2) = 0 27.12/28.23 [U172#](X) = 0 27.12/28.23 [U181#](X1,X2,X3) = 0 27.12/28.23 [U182#](X1,X2,X3) = 0 27.12/28.23 [U183#](X1,X2,X3) = 0 27.12/28.23 [U184#](X1,X2,X3) = 0 27.12/28.23 [U191#](X1,X2,X3) = 0 27.12/28.23 [U192#](X1,X2,X3) = 0 27.12/28.23 [U193#](X1,X2,X3) = 0 27.12/28.23 [U194#](X1,X2,X3) = 0 27.12/28.23 [U201#](X1,X2) = 0 27.12/28.23 [U202#](X1,X2) = 0 27.12/28.23 [U21#](X1,X2,X3) = 0 27.12/28.23 [U211#](X1,X2,X3) = 0 27.12/28.23 [U212#](X1,X2,X3) = 0 27.12/28.23 [U213#](X1,X2,X3) = 0 27.12/28.23 [U214#](X1,X2,X3) = 0 27.12/28.23 [U22#](X1,X2,X3) = 0 27.12/28.23 [U221#](X1,X2,X3) = 0 27.12/28.23 [U222#](X1,X2,X3) = 0 27.12/28.23 [U223#](X1,X2,X3) = 0 27.12/28.23 [U224#](X1,X2,X3) = 0 27.12/28.23 [U23#](X1,X2,X3) = 0 27.12/28.23 [U231#](X1,X2,X3) = 0 27.12/28.23 [U232#](X1,X2,X3) = 0 27.12/28.23 [U233#](X1,X2,X3) = 0 27.12/28.23 [U234#](X1,X2,X3) = 0 27.12/28.23 [U24#](X1,X2,X3) = 0 27.12/28.23 [U241#](X1,X2) = 0 27.12/28.23 [U242#](X1,X2) = 0 27.12/28.23 [U25#](X1,X2) = 0 27.12/28.23 [U251#](X1,X2,X3) = 0 27.12/28.23 [U252#](X1,X2,X3) = 0 27.12/28.23 [U253#](X1,X2,X3) = 0 27.12/28.23 [U254#](X1,X2,X3) = 0 27.12/28.23 [U26#](X) = 0 27.12/28.23 [U261#](X1,X2) = 0 27.12/28.23 [U262#](X1,X2) = 0 27.12/28.23 [U271#](X1,X2,X3) = 0 27.12/28.23 [U272#](X1,X2,X3) = 0 27.12/28.23 [U273#](X1,X2,X3) = 0 27.12/28.23 [U274#](X1,X2,X3) = 0 27.12/28.23 [U31#](X) = 0 27.12/28.23 [U41#](X1,X2) = 0 27.12/28.23 [U42#](X) = 0 27.12/28.23 [U51#](X1,X2) = 0 27.12/28.23 [U52#](X1,X2) = 0 27.12/28.23 [U53#](X) = 0 27.12/28.23 [U61#](X1,X2) = X1 + 2.X2 + 1 27.12/28.23 [U62#](X1,X2) = 2.X2 + 2 27.12/28.23 [U63#](X) = 0 27.12/28.23 [U71#](X1,X2,X3) = X1 + 2.X2 + 2.X3 27.12/28.23 [U72#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U73#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U74#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U75#](X1,X2) = 2.X2 + 1 27.12/28.23 [U76#](X) = 0 27.12/28.23 [U81#](X1,X2,X3) = X1 + 2.X2 + 2.X3 27.12/28.23 [U82#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U83#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U84#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.12/28.23 [U85#](X1,X2) = 2.X2 + 2 27.12/28.23 [U86#](X) = 0 27.12/28.23 [U91#](X1,X2) = 0 27.12/28.23 [U92#](X1,X2) = 0 27.12/28.23 [U93#](X) = 0 27.12/28.23 [ISBAG](X) = 0 27.12/28.23 [ISBAGKIND](X) = 0 27.12/28.23 [ISBIN](X) = 2.X + 1 27.12/28.23 [ISBINKIND](X) = 0 27.12/28.23 [MULT](X1,X2) = 0 27.12/28.23 [PLUS](X1,X2) = 0 27.12/28.23 [PROD](X) = 0 27.12/28.23 [SUM](X) = 0 27.12/28.23 [UNION](X1,X2) = 0 27.12/28.23 27.12/28.23 Problem 1.3.2: 27.12/28.23 27.12/28.23 SCC Processor: 27.12/28.23 -> FAxioms: 27.12/28.23 Empty 27.12/28.23 -> Pairs: 27.12/28.23 U62#(tt,V1) -> ISBIN(V1) 27.12/28.23 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.12/28.23 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.12/28.23 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.12/28.23 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.12/28.23 U74#(tt,V1,V2) -> ISBIN(V1) 27.12/28.23 U75#(tt,V2) -> ISBIN(V2) 27.12/28.23 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.12/28.23 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.12/28.23 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.12/28.23 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.12/28.23 U84#(tt,V1,V2) -> ISBIN(V1) 27.12/28.23 U85#(tt,V2) -> ISBIN(V2) 27.12/28.23 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.12/28.23 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.12/28.23 ISBIN(1(V1)) -> U61#(isBinKind(V1),V1) 27.12/28.23 -> EAxioms: 27.12/28.23 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.12/28.23 mult(x6,x7) = mult(x7,x6) 27.12/28.23 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.12/28.23 plus(x6,x7) = plus(x7,x6) 27.12/28.23 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.12/28.23 union(x6,x7) = union(x7,x6) 27.12/28.23 -> Rules: 27.12/28.23 0(z) -> z 27.12/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.12/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.12/28.23 U103(tt) -> tt 27.12/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.12/28.23 U111(tt) -> tt 27.12/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.12/28.23 U121(tt) -> tt 27.12/28.23 U13(tt) -> tt 27.12/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.12/28.23 U132(tt) -> tt 27.12/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.12/28.23 U142(tt) -> tt 27.12/28.23 U151(tt) -> tt 27.12/28.23 U161(tt) -> tt 27.12/28.23 U171(tt,X) -> U172(isBinKind(X)) 27.12/28.23 U172(tt) -> z 27.12/28.23 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.12/28.23 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.12/28.23 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.12/28.23 U184(tt,X,Y) -> 0(mult(X,Y)) 27.12/28.23 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.12/28.23 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.12/28.23 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.12/28.23 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.12/28.23 U201(tt,X) -> U202(isBinKind(X),X) 27.12/28.23 U202(tt,X) -> X 27.12/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.12/28.23 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.12/28.23 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.12/28.23 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.12/28.23 U214(tt,X,Y) -> 0(plus(X,Y)) 27.12/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.12/28.23 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.12/28.23 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.12/28.23 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.12/28.23 U224(tt,X,Y) -> 1(plus(X,Y)) 27.12/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.12/28.23 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.12/28.23 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.12/28.23 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.12/28.23 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.12/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.12/28.23 U241(tt,X) -> U242(isBinKind(X),X) 27.12/28.23 U242(tt,X) -> X 27.12/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.12/28.23 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.12/28.23 U252(tt,A,B) -> U253(isBag(B),A,B) 27.12/28.23 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.12/28.23 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.12/28.23 U26(tt) -> tt 27.12/28.23 U261(tt,X) -> U262(isBinKind(X),X) 27.12/28.23 U262(tt,X) -> X 27.12/28.23 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.12/28.23 U272(tt,A,B) -> U273(isBag(B),A,B) 27.12/28.23 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.12/28.23 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.12/28.23 U31(tt) -> tt 27.12/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.12/28.23 U42(tt) -> tt 27.12/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.12/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.12/28.23 U53(tt) -> tt 27.12/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.12/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.12/28.23 U63(tt) -> tt 27.12/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.12/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.12/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.12/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.12/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.12/28.23 U76(tt) -> tt 27.12/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.12/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.12/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.12/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.12/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.12/28.23 U86(tt) -> tt 27.12/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.12/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.12/28.23 U93(tt) -> tt 27.12/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.12/28.23 isBag(empty) -> tt 27.12/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.12/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.12/28.23 isBagKind(empty) -> tt 27.12/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.12/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.12/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.12/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.12/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.12/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.12/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.12/28.23 isBin(z) -> tt 27.12/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.12/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.12/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.12/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.23 isBinKind(z) -> tt 27.20/28.23 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.23 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.23 mult(z,X) -> U171(isBin(X),X) 27.20/28.23 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.23 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.23 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.23 plus(z,X) -> U201(isBin(X),X) 27.20/28.23 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.23 prod(empty) -> 1(z) 27.20/28.23 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.23 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.23 sum(empty) -> 0(z) 27.20/28.23 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.23 union(empty,X) -> X 27.20/28.23 union(X,empty) -> X 27.20/28.23 -> SRules: 27.20/28.23 Empty 27.20/28.23 ->Strongly Connected Components: 27.20/28.23 ->->Cycle: 27.20/28.23 ->->-> Pairs: 27.20/28.23 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.20/28.23 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.20/28.23 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.20/28.23 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.20/28.23 U74#(tt,V1,V2) -> ISBIN(V1) 27.20/28.23 U75#(tt,V2) -> ISBIN(V2) 27.20/28.23 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.20/28.23 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.20/28.23 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.20/28.23 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.20/28.23 U84#(tt,V1,V2) -> ISBIN(V1) 27.20/28.23 U85#(tt,V2) -> ISBIN(V2) 27.20/28.23 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.20/28.23 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.20/28.23 -> FAxioms: 27.20/28.23 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.23 mult(x6,x7) -> mult(x7,x6) 27.20/28.23 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.23 plus(x6,x7) -> plus(x7,x6) 27.20/28.23 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.23 union(x6,x7) -> union(x7,x6) 27.20/28.23 -> EAxioms: 27.20/28.23 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.23 mult(x6,x7) = mult(x7,x6) 27.20/28.23 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.23 plus(x6,x7) = plus(x7,x6) 27.20/28.23 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.23 union(x6,x7) = union(x7,x6) 27.20/28.23 ->->-> Rules: 27.20/28.23 0(z) -> z 27.20/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.23 U103(tt) -> tt 27.20/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.23 U111(tt) -> tt 27.20/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.23 U121(tt) -> tt 27.20/28.23 U13(tt) -> tt 27.20/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.23 U132(tt) -> tt 27.20/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.23 U142(tt) -> tt 27.20/28.23 U151(tt) -> tt 27.20/28.23 U161(tt) -> tt 27.20/28.23 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.23 U172(tt) -> z 27.20/28.23 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.23 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.23 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.23 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.23 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.23 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.23 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.23 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.23 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.23 U202(tt,X) -> X 27.20/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.23 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.23 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.23 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.23 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.23 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.23 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.23 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.23 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.23 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.23 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.23 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.23 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.23 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.23 U242(tt,X) -> X 27.20/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.23 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.23 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.23 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.23 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.23 U26(tt) -> tt 27.20/28.23 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.23 U262(tt,X) -> X 27.20/28.23 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.23 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.23 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.23 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.23 U31(tt) -> tt 27.20/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.23 U42(tt) -> tt 27.20/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.23 U53(tt) -> tt 27.20/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.23 U63(tt) -> tt 27.20/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.23 U76(tt) -> tt 27.20/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.23 U86(tt) -> tt 27.20/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.23 U93(tt) -> tt 27.20/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.23 isBag(empty) -> tt 27.20/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.23 isBagKind(empty) -> tt 27.20/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.23 isBin(z) -> tt 27.20/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.23 isBinKind(z) -> tt 27.20/28.23 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.23 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.23 mult(z,X) -> U171(isBin(X),X) 27.20/28.23 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.23 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.23 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.23 plus(z,X) -> U201(isBin(X),X) 27.20/28.23 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.23 prod(empty) -> 1(z) 27.20/28.23 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.23 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.23 sum(empty) -> 0(z) 27.20/28.23 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.23 union(empty,X) -> X 27.20/28.23 union(X,empty) -> X 27.20/28.23 -> SRules: 27.20/28.23 Empty 27.20/28.23 27.20/28.23 Problem 1.3.2: 27.20/28.23 27.20/28.23 Reduction Pairs Processor: 27.20/28.23 -> FAxioms: 27.20/28.23 Empty 27.20/28.23 -> Pairs: 27.20/28.23 U71#(tt,V1,V2) -> U72#(isBinKind(V1),V1,V2) 27.20/28.23 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.20/28.23 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.20/28.23 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.20/28.23 U74#(tt,V1,V2) -> ISBIN(V1) 27.20/28.23 U75#(tt,V2) -> ISBIN(V2) 27.20/28.23 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.20/28.23 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.20/28.23 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.20/28.23 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.20/28.23 U84#(tt,V1,V2) -> ISBIN(V1) 27.20/28.23 U85#(tt,V2) -> ISBIN(V2) 27.20/28.23 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.20/28.23 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.20/28.23 -> EAxioms: 27.20/28.23 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.23 mult(x6,x7) = mult(x7,x6) 27.20/28.23 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.23 plus(x6,x7) = plus(x7,x6) 27.20/28.23 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.23 union(x6,x7) = union(x7,x6) 27.20/28.23 -> Usable Equations: 27.20/28.23 Empty 27.20/28.23 -> Rules: 27.20/28.23 0(z) -> z 27.20/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.23 U103(tt) -> tt 27.20/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.23 U111(tt) -> tt 27.20/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.23 U121(tt) -> tt 27.20/28.23 U13(tt) -> tt 27.20/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.23 U132(tt) -> tt 27.20/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.23 U142(tt) -> tt 27.20/28.23 U151(tt) -> tt 27.20/28.23 U161(tt) -> tt 27.20/28.23 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.23 U172(tt) -> z 27.20/28.23 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.23 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.23 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.23 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.23 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.23 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.23 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.23 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.23 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.23 U202(tt,X) -> X 27.20/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.23 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.23 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.23 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.23 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.23 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.23 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.23 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.23 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.23 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.23 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.23 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.23 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.23 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.23 U242(tt,X) -> X 27.20/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.23 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.23 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.23 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.23 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.23 U26(tt) -> tt 27.20/28.23 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.23 U262(tt,X) -> X 27.20/28.23 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.23 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.23 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.23 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.23 U31(tt) -> tt 27.20/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.23 U42(tt) -> tt 27.20/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.23 U53(tt) -> tt 27.20/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.23 U63(tt) -> tt 27.20/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.23 U76(tt) -> tt 27.20/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.23 U86(tt) -> tt 27.20/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.23 U93(tt) -> tt 27.20/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.23 isBag(empty) -> tt 27.20/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.23 isBagKind(empty) -> tt 27.20/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.23 isBin(z) -> tt 27.20/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.23 isBinKind(z) -> tt 27.20/28.23 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.23 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.23 mult(z,X) -> U171(isBin(X),X) 27.20/28.23 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.23 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.23 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.23 plus(z,X) -> U201(isBin(X),X) 27.20/28.23 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.23 prod(empty) -> 1(z) 27.20/28.23 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.23 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.23 sum(empty) -> 0(z) 27.20/28.23 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.23 union(empty,X) -> X 27.20/28.23 union(X,empty) -> X 27.20/28.23 -> Usable Rules: 27.20/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.23 U103(tt) -> tt 27.20/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.23 U111(tt) -> tt 27.20/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.23 U121(tt) -> tt 27.20/28.23 U13(tt) -> tt 27.20/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.23 U132(tt) -> tt 27.20/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.23 U142(tt) -> tt 27.20/28.23 U151(tt) -> tt 27.20/28.23 U161(tt) -> tt 27.20/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.23 U26(tt) -> tt 27.20/28.23 U31(tt) -> tt 27.20/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.23 U42(tt) -> tt 27.20/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.23 U53(tt) -> tt 27.20/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.23 U63(tt) -> tt 27.20/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.23 U76(tt) -> tt 27.20/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.23 U86(tt) -> tt 27.20/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.23 U93(tt) -> tt 27.20/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.23 isBag(empty) -> tt 27.20/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.23 isBagKind(empty) -> tt 27.20/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.23 isBin(z) -> tt 27.20/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.23 isBinKind(z) -> tt 27.20/28.23 -> SRules: 27.20/28.23 Empty 27.20/28.23 ->Interpretation type: 27.20/28.23 Linear 27.20/28.23 ->Coefficients: 27.20/28.23 Natural Numbers 27.20/28.23 ->Dimension: 27.20/28.23 1 27.20/28.23 ->Bound: 27.20/28.23 2 27.20/28.23 ->Interpretation: 27.20/28.23 27.20/28.23 [0](X) = X + 2 27.20/28.23 [U101](X1,X2) = 2.X1 + 2.X2 27.20/28.23 [U102](X1,X2) = 2.X2 + 2 27.20/28.23 [U103](X) = X 27.20/28.23 [U11](X1,X2) = 2.X1 + 2 27.20/28.23 [U111](X) = X + 1 27.20/28.23 [U12](X1,X2) = 2 27.20/28.23 [U121](X) = X + 1 27.20/28.23 [U13](X) = 2 27.20/28.23 [U131](X1,X2) = 2.X1 + 2.X2 + 1 27.20/28.23 [U132](X) = 2.X + 2 27.20/28.23 [U141](X1,X2) = 2.X1 + 2.X2 + 1 27.20/28.23 [U142](X) = 2.X + 2 27.20/28.23 [U151](X) = 2.X + 2 27.20/28.23 [U161](X) = 2 27.20/28.23 [U171](X1,X2) = 0 27.20/28.23 [U172](X) = 0 27.20/28.23 [U181](X1,X2,X3) = 0 27.20/28.23 [U182](X1,X2,X3) = 0 27.20/28.23 [U183](X1,X2,X3) = 0 27.20/28.23 [U184](X1,X2,X3) = 0 27.20/28.23 [U191](X1,X2,X3) = 0 27.20/28.23 [U192](X1,X2,X3) = 0 27.20/28.23 [U193](X1,X2,X3) = 0 27.20/28.23 [U194](X1,X2,X3) = 0 27.20/28.23 [U201](X1,X2) = 0 27.20/28.23 [U202](X1,X2) = 0 27.20/28.23 [U21](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 27.20/28.23 [U211](X1,X2,X3) = 0 27.20/28.23 [U212](X1,X2,X3) = 0 27.20/28.23 [U213](X1,X2,X3) = 0 27.20/28.23 [U214](X1,X2,X3) = 0 27.20/28.23 [U22](X1,X2,X3) = 2.X1 + X3 + 2 27.20/28.23 [U221](X1,X2,X3) = 0 27.20/28.23 [U222](X1,X2,X3) = 0 27.20/28.23 [U223](X1,X2,X3) = 0 27.20/28.23 [U224](X1,X2,X3) = 0 27.20/28.23 [U23](X1,X2,X3) = X3 + 2 27.20/28.23 [U231](X1,X2,X3) = 0 27.20/28.23 [U232](X1,X2,X3) = 0 27.20/28.23 [U233](X1,X2,X3) = 0 27.20/28.23 [U234](X1,X2,X3) = 0 27.20/28.23 [U24](X1,X2,X3) = X3 + 2 27.20/28.23 [U241](X1,X2) = 0 27.20/28.23 [U242](X1,X2) = 0 27.20/28.23 [U25](X1,X2) = X2 + 2 27.20/28.23 [U251](X1,X2,X3) = 0 27.20/28.23 [U252](X1,X2,X3) = 0 27.20/28.23 [U253](X1,X2,X3) = 0 27.20/28.23 [U254](X1,X2,X3) = 0 27.20/28.23 [U26](X) = 2 27.20/28.23 [U261](X1,X2) = 0 27.20/28.23 [U262](X1,X2) = 0 27.20/28.23 [U271](X1,X2,X3) = 0 27.20/28.23 [U272](X1,X2,X3) = 0 27.20/28.23 [U273](X1,X2,X3) = 0 27.20/28.23 [U274](X1,X2,X3) = 0 27.20/28.23 [U31](X) = X + 1 27.20/28.23 [U41](X1,X2) = 2.X2 + 2 27.20/28.23 [U42](X) = 2.X + 2 27.20/28.23 [U51](X1,X2) = X1 + X2 + 2 27.20/28.23 [U52](X1,X2) = X1 + 2 27.20/28.23 [U53](X) = 2 27.20/28.23 [U61](X1,X2) = 2.X1 + 2.X2 27.20/28.23 [U62](X1,X2) = X1 + X2 + 2 27.20/28.23 [U63](X) = 2 27.20/28.23 [U71](X1,X2,X3) = 2.X1 + 2.X2 + 2 27.20/28.23 [U72](X1,X2,X3) = 2.X1 + 2 27.20/28.23 [U73](X1,X2,X3) = 2 27.20/28.23 [U74](X1,X2,X3) = 2 27.20/28.23 [U75](X1,X2) = 2 27.20/28.23 [U76](X) = 2 27.20/28.23 [U81](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 27.20/28.23 [U82](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.23 [U83](X1,X2,X3) = X1 + 2.X2 + X3 27.20/28.23 [U84](X1,X2,X3) = X1 + 2.X2 27.20/28.23 [U85](X1,X2) = X1 + 2 27.20/28.23 [U86](X) = 2 27.20/28.23 [U91](X1,X2) = 2.X1 + X2 27.20/28.23 [U92](X1,X2) = X1 + 2 27.20/28.23 [U93](X) = 2 27.20/28.23 [isBag](X) = 2.X + 2 27.20/28.23 [isBagKind](X) = X 27.20/28.23 [isBin](X) = 2.X 27.20/28.23 [isBinKind](X) = X + 1 27.20/28.23 [mult](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.23 [plus](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.23 [prod](X) = 2.X + 2 27.20/28.23 [sum](X) = 2.X + 1 27.20/28.23 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.23 [1](X) = 2.X + 1 27.20/28.23 [empty] = 2 27.20/28.23 [singl](X) = X + 2 27.20/28.23 [tt] = 2 27.20/28.23 [z] = 2 27.20/28.23 [0#](X) = 0 27.20/28.23 [U101#](X1,X2) = 0 27.20/28.23 [U102#](X1,X2) = 0 27.20/28.23 [U103#](X) = 0 27.20/28.23 [U11#](X1,X2) = 0 27.20/28.23 [U111#](X) = 0 27.20/28.23 [U12#](X1,X2) = 0 27.20/28.23 [U121#](X) = 0 27.20/28.23 [U13#](X) = 0 27.20/28.23 [U131#](X1,X2) = 0 27.20/28.23 [U132#](X) = 0 27.20/28.23 [U141#](X1,X2) = 0 27.20/28.23 [U142#](X) = 0 27.20/28.23 [U151#](X) = 0 27.20/28.23 [U161#](X) = 0 27.20/28.23 [U171#](X1,X2) = 0 27.20/28.23 [U172#](X) = 0 27.20/28.23 [U181#](X1,X2,X3) = 0 27.20/28.23 [U182#](X1,X2,X3) = 0 27.20/28.23 [U183#](X1,X2,X3) = 0 27.20/28.23 [U184#](X1,X2,X3) = 0 27.20/28.23 [U191#](X1,X2,X3) = 0 27.20/28.23 [U192#](X1,X2,X3) = 0 27.20/28.23 [U193#](X1,X2,X3) = 0 27.20/28.23 [U194#](X1,X2,X3) = 0 27.20/28.23 [U201#](X1,X2) = 0 27.20/28.23 [U202#](X1,X2) = 0 27.20/28.23 [U21#](X1,X2,X3) = 0 27.20/28.23 [U211#](X1,X2,X3) = 0 27.20/28.23 [U212#](X1,X2,X3) = 0 27.20/28.23 [U213#](X1,X2,X3) = 0 27.20/28.23 [U214#](X1,X2,X3) = 0 27.20/28.23 [U22#](X1,X2,X3) = 0 27.20/28.23 [U221#](X1,X2,X3) = 0 27.20/28.23 [U222#](X1,X2,X3) = 0 27.20/28.23 [U223#](X1,X2,X3) = 0 27.20/28.23 [U224#](X1,X2,X3) = 0 27.20/28.23 [U23#](X1,X2,X3) = 0 27.20/28.23 [U231#](X1,X2,X3) = 0 27.20/28.23 [U232#](X1,X2,X3) = 0 27.20/28.23 [U233#](X1,X2,X3) = 0 27.20/28.23 [U234#](X1,X2,X3) = 0 27.20/28.23 [U24#](X1,X2,X3) = 0 27.20/28.23 [U241#](X1,X2) = 0 27.20/28.23 [U242#](X1,X2) = 0 27.20/28.23 [U25#](X1,X2) = 0 27.20/28.23 [U251#](X1,X2,X3) = 0 27.20/28.23 [U252#](X1,X2,X3) = 0 27.20/28.23 [U253#](X1,X2,X3) = 0 27.20/28.23 [U254#](X1,X2,X3) = 0 27.20/28.23 [U26#](X) = 0 27.20/28.23 [U261#](X1,X2) = 0 27.20/28.23 [U262#](X1,X2) = 0 27.20/28.23 [U271#](X1,X2,X3) = 0 27.20/28.23 [U272#](X1,X2,X3) = 0 27.20/28.23 [U273#](X1,X2,X3) = 0 27.20/28.23 [U274#](X1,X2,X3) = 0 27.20/28.23 [U31#](X) = 0 27.20/28.23 [U41#](X1,X2) = 0 27.20/28.23 [U42#](X) = 0 27.20/28.23 [U51#](X1,X2) = 0 27.20/28.23 [U52#](X1,X2) = 0 27.20/28.23 [U53#](X) = 0 27.20/28.23 [U61#](X1,X2) = 0 27.20/28.23 [U62#](X1,X2) = 0 27.20/28.23 [U63#](X) = 0 27.20/28.23 [U71#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.23 [U72#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.23 [U73#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.23 [U74#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.23 [U75#](X1,X2) = X1 + 2.X2 + 2 27.20/28.23 [U76#](X) = 0 27.20/28.23 [U81#](X1,X2,X3) = X1 + 2.X2 + 2.X3 27.20/28.23 [U82#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.23 [U83#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.23 [U84#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.23 [U85#](X1,X2) = X1 + 2.X2 + 2 27.20/28.23 [U86#](X) = 0 27.20/28.23 [U91#](X1,X2) = 0 27.20/28.23 [U92#](X1,X2) = 0 27.20/28.23 [U93#](X) = 0 27.20/28.23 [ISBAG](X) = 0 27.20/28.23 [ISBAGKIND](X) = 0 27.20/28.23 [ISBIN](X) = 2.X + 2 27.20/28.23 [ISBINKIND](X) = 0 27.20/28.23 [MULT](X1,X2) = 0 27.20/28.23 [PLUS](X1,X2) = 0 27.20/28.23 [PROD](X) = 0 27.20/28.23 [SUM](X) = 0 27.20/28.23 [UNION](X1,X2) = 0 27.20/28.23 27.20/28.23 Problem 1.3.2: 27.20/28.23 27.20/28.23 SCC Processor: 27.20/28.23 -> FAxioms: 27.20/28.23 Empty 27.20/28.23 -> Pairs: 27.20/28.23 U72#(tt,V1,V2) -> U73#(isBinKind(V2),V1,V2) 27.20/28.23 U73#(tt,V1,V2) -> U74#(isBinKind(V2),V1,V2) 27.20/28.23 U74#(tt,V1,V2) -> U75#(isBin(V1),V2) 27.20/28.23 U74#(tt,V1,V2) -> ISBIN(V1) 27.20/28.23 U75#(tt,V2) -> ISBIN(V2) 27.20/28.23 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.20/28.23 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.20/28.23 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.20/28.23 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.20/28.23 U84#(tt,V1,V2) -> ISBIN(V1) 27.20/28.23 U85#(tt,V2) -> ISBIN(V2) 27.20/28.23 ISBIN(mult(V1,V2)) -> U71#(isBinKind(V1),V1,V2) 27.20/28.23 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.20/28.23 -> EAxioms: 27.20/28.23 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.23 mult(x6,x7) = mult(x7,x6) 27.20/28.23 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.23 plus(x6,x7) = plus(x7,x6) 27.20/28.23 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.23 union(x6,x7) = union(x7,x6) 27.20/28.23 -> Rules: 27.20/28.23 0(z) -> z 27.20/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.23 U103(tt) -> tt 27.20/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.23 U111(tt) -> tt 27.20/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.23 U121(tt) -> tt 27.20/28.23 U13(tt) -> tt 27.20/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.23 U132(tt) -> tt 27.20/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.23 U142(tt) -> tt 27.20/28.23 U151(tt) -> tt 27.20/28.23 U161(tt) -> tt 27.20/28.23 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.23 U172(tt) -> z 27.20/28.23 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.23 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.23 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.23 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.23 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.23 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.23 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.23 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.23 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.23 U202(tt,X) -> X 27.20/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.23 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.23 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.23 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.23 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.23 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.23 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.23 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.23 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.23 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.23 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.23 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.23 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.23 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.23 U242(tt,X) -> X 27.20/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.23 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.23 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.23 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.23 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.23 U26(tt) -> tt 27.20/28.23 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.23 U262(tt,X) -> X 27.20/28.23 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.23 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.23 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.23 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.23 U31(tt) -> tt 27.20/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.23 U42(tt) -> tt 27.20/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.23 U53(tt) -> tt 27.20/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.23 U63(tt) -> tt 27.20/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.23 U76(tt) -> tt 27.20/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.23 U86(tt) -> tt 27.20/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.23 U93(tt) -> tt 27.20/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.23 isBag(empty) -> tt 27.20/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.23 isBagKind(empty) -> tt 27.20/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.23 isBin(z) -> tt 27.20/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.23 isBinKind(z) -> tt 27.20/28.23 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.23 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.23 mult(z,X) -> U171(isBin(X),X) 27.20/28.23 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.23 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.23 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.23 plus(z,X) -> U201(isBin(X),X) 27.20/28.23 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.23 prod(empty) -> 1(z) 27.20/28.23 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.23 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.23 sum(empty) -> 0(z) 27.20/28.23 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.23 union(empty,X) -> X 27.20/28.23 union(X,empty) -> X 27.20/28.23 -> SRules: 27.20/28.23 Empty 27.20/28.23 ->Strongly Connected Components: 27.20/28.23 ->->Cycle: 27.20/28.23 ->->-> Pairs: 27.20/28.23 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.20/28.23 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.20/28.23 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.20/28.23 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.20/28.23 U84#(tt,V1,V2) -> ISBIN(V1) 27.20/28.23 U85#(tt,V2) -> ISBIN(V2) 27.20/28.23 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.20/28.23 -> FAxioms: 27.20/28.23 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.23 mult(x6,x7) -> mult(x7,x6) 27.20/28.23 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.23 plus(x6,x7) -> plus(x7,x6) 27.20/28.23 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.23 union(x6,x7) -> union(x7,x6) 27.20/28.23 -> EAxioms: 27.20/28.23 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.23 mult(x6,x7) = mult(x7,x6) 27.20/28.23 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.23 plus(x6,x7) = plus(x7,x6) 27.20/28.23 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.23 union(x6,x7) = union(x7,x6) 27.20/28.23 ->->-> Rules: 27.20/28.23 0(z) -> z 27.20/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.23 U103(tt) -> tt 27.20/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.23 U111(tt) -> tt 27.20/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.23 U121(tt) -> tt 27.20/28.23 U13(tt) -> tt 27.20/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.23 U132(tt) -> tt 27.20/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.23 U142(tt) -> tt 27.20/28.23 U151(tt) -> tt 27.20/28.23 U161(tt) -> tt 27.20/28.23 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.23 U172(tt) -> z 27.20/28.23 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.23 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.23 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.23 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.23 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.23 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.23 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.23 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.23 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.23 U202(tt,X) -> X 27.20/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.23 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.23 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.23 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.23 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.23 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.23 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.23 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.23 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.23 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.23 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.23 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.23 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.23 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.23 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.23 U242(tt,X) -> X 27.20/28.23 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.23 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.23 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.23 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.23 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.23 U26(tt) -> tt 27.20/28.23 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.23 U262(tt,X) -> X 27.20/28.23 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.23 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.23 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.23 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.23 U31(tt) -> tt 27.20/28.23 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.23 U42(tt) -> tt 27.20/28.23 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.23 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.23 U53(tt) -> tt 27.20/28.23 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.23 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.23 U63(tt) -> tt 27.20/28.23 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.23 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.23 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.23 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.23 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.23 U76(tt) -> tt 27.20/28.23 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.23 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.23 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.23 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.23 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.23 U86(tt) -> tt 27.20/28.23 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.23 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.23 U93(tt) -> tt 27.20/28.23 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.23 isBag(empty) -> tt 27.20/28.23 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.23 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.23 isBagKind(empty) -> tt 27.20/28.23 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.23 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.23 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.23 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.23 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.23 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.23 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.23 isBin(z) -> tt 27.20/28.23 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.23 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.23 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.23 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.23 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.23 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.23 isBinKind(z) -> tt 27.20/28.23 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.23 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.23 mult(z,X) -> U171(isBin(X),X) 27.20/28.23 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.23 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.23 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.23 plus(z,X) -> U201(isBin(X),X) 27.20/28.23 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.23 prod(empty) -> 1(z) 27.20/28.23 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.23 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.23 sum(empty) -> 0(z) 27.20/28.23 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.23 union(empty,X) -> X 27.20/28.23 union(X,empty) -> X 27.20/28.23 -> SRules: 27.20/28.23 Empty 27.20/28.23 27.20/28.23 Problem 1.3.2: 27.20/28.23 27.20/28.23 Reduction Pairs Processor: 27.20/28.23 -> FAxioms: 27.20/28.23 Empty 27.20/28.23 -> Pairs: 27.20/28.23 U81#(tt,V1,V2) -> U82#(isBinKind(V1),V1,V2) 27.20/28.23 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.20/28.23 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.20/28.23 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.20/28.23 U84#(tt,V1,V2) -> ISBIN(V1) 27.20/28.23 U85#(tt,V2) -> ISBIN(V2) 27.20/28.23 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.20/28.23 -> EAxioms: 27.20/28.23 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.23 mult(x6,x7) = mult(x7,x6) 27.20/28.23 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.23 plus(x6,x7) = plus(x7,x6) 27.20/28.23 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.23 union(x6,x7) = union(x7,x6) 27.20/28.23 -> Usable Equations: 27.20/28.23 Empty 27.20/28.23 -> Rules: 27.20/28.23 0(z) -> z 27.20/28.23 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.23 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.23 U103(tt) -> tt 27.20/28.23 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.23 U111(tt) -> tt 27.20/28.23 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.23 U121(tt) -> tt 27.20/28.23 U13(tt) -> tt 27.20/28.23 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.23 U132(tt) -> tt 27.20/28.23 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.23 U142(tt) -> tt 27.20/28.23 U151(tt) -> tt 27.20/28.23 U161(tt) -> tt 27.20/28.23 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.23 U172(tt) -> z 27.20/28.23 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.23 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.23 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.23 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.23 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.23 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.23 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.23 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.23 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.23 U202(tt,X) -> X 27.20/28.23 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.23 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.23 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.23 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.23 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.23 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.23 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.23 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.23 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.23 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.23 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.23 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.23 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.23 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.23 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.24 U242(tt,X) -> X 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.24 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.24 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.24 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.24 U262(tt,X) -> X 27.20/28.24 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.24 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.24 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.24 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.24 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.24 mult(z,X) -> U171(isBin(X),X) 27.20/28.24 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.24 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.24 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.24 plus(z,X) -> U201(isBin(X),X) 27.20/28.24 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.24 prod(empty) -> 1(z) 27.20/28.24 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.24 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.24 sum(empty) -> 0(z) 27.20/28.24 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.24 union(empty,X) -> X 27.20/28.24 union(X,empty) -> X 27.20/28.24 -> Usable Rules: 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 -> SRules: 27.20/28.24 Empty 27.20/28.24 ->Interpretation type: 27.20/28.24 Linear 27.20/28.24 ->Coefficients: 27.20/28.24 Natural Numbers 27.20/28.24 ->Dimension: 27.20/28.24 1 27.20/28.24 ->Bound: 27.20/28.24 2 27.20/28.24 ->Interpretation: 27.20/28.24 27.20/28.24 [0](X) = 0 27.20/28.24 [U101](X1,X2) = 2 27.20/28.24 [U102](X1,X2) = 2 27.20/28.24 [U103](X) = 2 27.20/28.24 [U11](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.24 [U111](X) = 2 27.20/28.24 [U12](X1,X2) = 2.X2 + 2 27.20/28.24 [U121](X) = X 27.20/28.24 [U13](X) = X 27.20/28.24 [U131](X1,X2) = 2 27.20/28.24 [U132](X) = 2 27.20/28.24 [U141](X1,X2) = 2 27.20/28.24 [U142](X) = 2 27.20/28.24 [U151](X) = 2 27.20/28.24 [U161](X) = 2 27.20/28.24 [U171](X1,X2) = 0 27.20/28.24 [U172](X) = 0 27.20/28.24 [U181](X1,X2,X3) = 0 27.20/28.24 [U182](X1,X2,X3) = 0 27.20/28.24 [U183](X1,X2,X3) = 0 27.20/28.24 [U184](X1,X2,X3) = 0 27.20/28.24 [U191](X1,X2,X3) = 0 27.20/28.24 [U192](X1,X2,X3) = 0 27.20/28.24 [U193](X1,X2,X3) = 0 27.20/28.24 [U194](X1,X2,X3) = 0 27.20/28.24 [U201](X1,X2) = 0 27.20/28.24 [U202](X1,X2) = 0 27.20/28.24 [U21](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 27.20/28.24 [U211](X1,X2,X3) = 0 27.20/28.24 [U212](X1,X2,X3) = 0 27.20/28.24 [U213](X1,X2,X3) = 0 27.20/28.24 [U214](X1,X2,X3) = 0 27.20/28.24 [U22](X1,X2,X3) = X1 + 2.X3 + 2 27.20/28.24 [U221](X1,X2,X3) = 0 27.20/28.24 [U222](X1,X2,X3) = 0 27.20/28.24 [U223](X1,X2,X3) = 0 27.20/28.24 [U224](X1,X2,X3) = 0 27.20/28.24 [U23](X1,X2,X3) = 2.X1 + 2 27.20/28.24 [U231](X1,X2,X3) = 0 27.20/28.24 [U232](X1,X2,X3) = 0 27.20/28.24 [U233](X1,X2,X3) = 0 27.20/28.24 [U234](X1,X2,X3) = 0 27.20/28.24 [U24](X1,X2,X3) = 2 27.20/28.24 [U241](X1,X2) = 0 27.20/28.24 [U242](X1,X2) = 0 27.20/28.24 [U25](X1,X2) = 2 27.20/28.24 [U251](X1,X2,X3) = 0 27.20/28.24 [U252](X1,X2,X3) = 0 27.20/28.24 [U253](X1,X2,X3) = 0 27.20/28.24 [U254](X1,X2,X3) = 0 27.20/28.24 [U26](X) = 2 27.20/28.24 [U261](X1,X2) = 0 27.20/28.24 [U262](X1,X2) = 0 27.20/28.24 [U271](X1,X2,X3) = 0 27.20/28.24 [U272](X1,X2,X3) = 0 27.20/28.24 [U273](X1,X2,X3) = 0 27.20/28.24 [U274](X1,X2,X3) = 0 27.20/28.24 [U31](X) = 2 27.20/28.24 [U41](X1,X2) = X1 + 2.X2 + 2 27.20/28.24 [U42](X) = 2.X + 2 27.20/28.24 [U51](X1,X2) = 2 27.20/28.24 [U52](X1,X2) = 2 27.20/28.24 [U53](X) = 2 27.20/28.24 [U61](X1,X2) = X1 + 2.X2 + 2 27.20/28.24 [U62](X1,X2) = X1 + 2.X2 + 2 27.20/28.24 [U63](X) = 2 27.20/28.24 [U71](X1,X2,X3) = 2.X1 + 2.X2 27.20/28.24 [U72](X1,X2,X3) = X1 + 2.X2 + 2 27.20/28.24 [U73](X1,X2,X3) = X1 + 2.X2 + 2 27.20/28.24 [U74](X1,X2,X3) = X1 + 2.X2 + 2 27.20/28.24 [U75](X1,X2) = 2.X1 27.20/28.24 [U76](X) = 2 27.20/28.24 [U81](X1,X2,X3) = 2.X1 + X2 27.20/28.24 [U82](X1,X2,X3) = X1 + X2 + 2 27.20/28.24 [U83](X1,X2,X3) = 2.X1 + X2 27.20/28.24 [U84](X1,X2,X3) = X1 + X2 + 2 27.20/28.24 [U85](X1,X2) = X1 27.20/28.24 [U86](X) = 2 27.20/28.24 [U91](X1,X2) = X1 + X2 + 2 27.20/28.24 [U92](X1,X2) = X1 + 1 27.20/28.24 [U93](X) = 2 27.20/28.24 [isBag](X) = 2.X + 2 27.20/28.24 [isBagKind](X) = X 27.20/28.24 [isBin](X) = X + 2 27.20/28.24 [isBinKind](X) = 2 27.20/28.24 [mult](X1,X2) = 2.X1 + 2 27.20/28.24 [plus](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.24 [prod](X) = 2.X 27.20/28.24 [sum](X) = 0 27.20/28.24 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.24 [1](X) = 2.X + 2 27.20/28.24 [empty] = 2 27.20/28.24 [singl](X) = 2.X + 2 27.20/28.24 [tt] = 2 27.20/28.24 [z] = 2 27.20/28.24 [0#](X) = 0 27.20/28.24 [U101#](X1,X2) = 0 27.20/28.24 [U102#](X1,X2) = 0 27.20/28.24 [U103#](X) = 0 27.20/28.24 [U11#](X1,X2) = 0 27.20/28.24 [U111#](X) = 0 27.20/28.24 [U12#](X1,X2) = 0 27.20/28.24 [U121#](X) = 0 27.20/28.24 [U13#](X) = 0 27.20/28.24 [U131#](X1,X2) = 0 27.20/28.24 [U132#](X) = 0 27.20/28.24 [U141#](X1,X2) = 0 27.20/28.24 [U142#](X) = 0 27.20/28.24 [U151#](X) = 0 27.20/28.24 [U161#](X) = 0 27.20/28.24 [U171#](X1,X2) = 0 27.20/28.24 [U172#](X) = 0 27.20/28.24 [U181#](X1,X2,X3) = 0 27.20/28.24 [U182#](X1,X2,X3) = 0 27.20/28.24 [U183#](X1,X2,X3) = 0 27.20/28.24 [U184#](X1,X2,X3) = 0 27.20/28.24 [U191#](X1,X2,X3) = 0 27.20/28.24 [U192#](X1,X2,X3) = 0 27.20/28.24 [U193#](X1,X2,X3) = 0 27.20/28.24 [U194#](X1,X2,X3) = 0 27.20/28.24 [U201#](X1,X2) = 0 27.20/28.24 [U202#](X1,X2) = 0 27.20/28.24 [U21#](X1,X2,X3) = 0 27.20/28.24 [U211#](X1,X2,X3) = 0 27.20/28.24 [U212#](X1,X2,X3) = 0 27.20/28.24 [U213#](X1,X2,X3) = 0 27.20/28.24 [U214#](X1,X2,X3) = 0 27.20/28.24 [U22#](X1,X2,X3) = 0 27.20/28.24 [U221#](X1,X2,X3) = 0 27.20/28.24 [U222#](X1,X2,X3) = 0 27.20/28.24 [U223#](X1,X2,X3) = 0 27.20/28.24 [U224#](X1,X2,X3) = 0 27.20/28.24 [U23#](X1,X2,X3) = 0 27.20/28.24 [U231#](X1,X2,X3) = 0 27.20/28.24 [U232#](X1,X2,X3) = 0 27.20/28.24 [U233#](X1,X2,X3) = 0 27.20/28.24 [U234#](X1,X2,X3) = 0 27.20/28.24 [U24#](X1,X2,X3) = 0 27.20/28.24 [U241#](X1,X2) = 0 27.20/28.24 [U242#](X1,X2) = 0 27.20/28.24 [U25#](X1,X2) = 0 27.20/28.24 [U251#](X1,X2,X3) = 0 27.20/28.24 [U252#](X1,X2,X3) = 0 27.20/28.24 [U253#](X1,X2,X3) = 0 27.20/28.24 [U254#](X1,X2,X3) = 0 27.20/28.24 [U26#](X) = 0 27.20/28.24 [U261#](X1,X2) = 0 27.20/28.24 [U262#](X1,X2) = 0 27.20/28.24 [U271#](X1,X2,X3) = 0 27.20/28.24 [U272#](X1,X2,X3) = 0 27.20/28.24 [U273#](X1,X2,X3) = 0 27.20/28.24 [U274#](X1,X2,X3) = 0 27.20/28.24 [U31#](X) = 0 27.20/28.24 [U41#](X1,X2) = 0 27.20/28.24 [U42#](X) = 0 27.20/28.24 [U51#](X1,X2) = 0 27.20/28.24 [U52#](X1,X2) = 0 27.20/28.24 [U53#](X) = 0 27.20/28.24 [U61#](X1,X2) = 0 27.20/28.24 [U62#](X1,X2) = 0 27.20/28.24 [U63#](X) = 0 27.20/28.24 [U71#](X1,X2,X3) = 0 27.20/28.24 [U72#](X1,X2,X3) = 0 27.20/28.24 [U73#](X1,X2,X3) = 0 27.20/28.24 [U74#](X1,X2,X3) = 0 27.20/28.24 [U75#](X1,X2) = 0 27.20/28.24 [U76#](X) = 0 27.20/28.24 [U81#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.24 [U82#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 27.20/28.24 [U83#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 27.20/28.24 [U84#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 27.20/28.24 [U85#](X1,X2) = 2.X1 + 2.X2 + 1 27.20/28.24 [U86#](X) = 0 27.20/28.24 [U91#](X1,X2) = 0 27.20/28.24 [U92#](X1,X2) = 0 27.20/28.24 [U93#](X) = 0 27.20/28.24 [ISBAG](X) = 0 27.20/28.24 [ISBAGKIND](X) = 0 27.20/28.24 [ISBIN](X) = 2.X + 2 27.20/28.24 [ISBINKIND](X) = 0 27.20/28.24 [MULT](X1,X2) = 0 27.20/28.24 [PLUS](X1,X2) = 0 27.20/28.24 [PROD](X) = 0 27.20/28.24 [SUM](X) = 0 27.20/28.24 [UNION](X1,X2) = 0 27.20/28.24 27.20/28.24 Problem 1.3.2: 27.20/28.24 27.20/28.24 SCC Processor: 27.20/28.24 -> FAxioms: 27.20/28.24 Empty 27.20/28.24 -> Pairs: 27.20/28.24 U82#(tt,V1,V2) -> U83#(isBinKind(V2),V1,V2) 27.20/28.24 U83#(tt,V1,V2) -> U84#(isBinKind(V2),V1,V2) 27.20/28.24 U84#(tt,V1,V2) -> U85#(isBin(V1),V2) 27.20/28.24 U84#(tt,V1,V2) -> ISBIN(V1) 27.20/28.24 U85#(tt,V2) -> ISBIN(V2) 27.20/28.24 ISBIN(plus(V1,V2)) -> U81#(isBinKind(V1),V1,V2) 27.20/28.24 -> EAxioms: 27.20/28.24 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.24 mult(x6,x7) = mult(x7,x6) 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.24 union(x6,x7) = union(x7,x6) 27.20/28.24 -> Rules: 27.20/28.24 0(z) -> z 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.24 U172(tt) -> z 27.20/28.24 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.24 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.24 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.24 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.24 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.24 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.24 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.24 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.24 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.24 U202(tt,X) -> X 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.24 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.24 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.24 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.24 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.24 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.24 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.24 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.24 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.24 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.24 U242(tt,X) -> X 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.24 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.24 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.24 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.24 U262(tt,X) -> X 27.20/28.24 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.24 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.24 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.24 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.24 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.24 mult(z,X) -> U171(isBin(X),X) 27.20/28.24 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.24 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.24 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.24 plus(z,X) -> U201(isBin(X),X) 27.20/28.24 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.24 prod(empty) -> 1(z) 27.20/28.24 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.24 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.24 sum(empty) -> 0(z) 27.20/28.24 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.24 union(empty,X) -> X 27.20/28.24 union(X,empty) -> X 27.20/28.24 -> SRules: 27.20/28.24 Empty 27.20/28.24 ->Strongly Connected Components: 27.20/28.24 There is no strongly connected component 27.20/28.24 27.20/28.24 The problem is finite. 27.20/28.24 27.20/28.24 Problem 1.4: 27.20/28.24 27.20/28.24 Reduction Pairs Processor: 27.20/28.24 -> FAxioms: 27.20/28.24 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.24 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.24 -> Pairs: 27.20/28.24 U211#(tt,X,Y) -> U212#(isBinKind(X),X,Y) 27.20/28.24 U212#(tt,X,Y) -> U213#(isBin(Y),X,Y) 27.20/28.24 U213#(tt,X,Y) -> U214#(isBinKind(Y),X,Y) 27.20/28.24 U214#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 U221#(tt,X,Y) -> U222#(isBinKind(X),X,Y) 27.20/28.24 U222#(tt,X,Y) -> U223#(isBin(Y),X,Y) 27.20/28.24 U223#(tt,X,Y) -> U224#(isBinKind(Y),X,Y) 27.20/28.24 U224#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 U231#(tt,X,Y) -> U232#(isBinKind(X),X,Y) 27.20/28.24 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.20/28.24 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.20/28.24 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.20/28.24 U234#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 PLUS(0(X),0(Y)) -> U211#(isBin(X),X,Y) 27.20/28.24 PLUS(0(X),1(Y)) -> U221#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),0(Y)),x6) -> U211#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> U221#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.24 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.20/28.24 -> EAxioms: 27.20/28.24 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.24 mult(x6,x7) = mult(x7,x6) 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.24 union(x6,x7) = union(x7,x6) 27.20/28.24 -> Usable Equations: 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 -> Rules: 27.20/28.24 0(z) -> z 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.24 U172(tt) -> z 27.20/28.24 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.24 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.24 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.24 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.24 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.24 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.24 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.24 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.24 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.24 U202(tt,X) -> X 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.24 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.24 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.24 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.24 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.24 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.24 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.24 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.24 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.24 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.24 U242(tt,X) -> X 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.24 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.24 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.24 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.24 U262(tt,X) -> X 27.20/28.24 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.24 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.24 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.24 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.24 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.24 mult(z,X) -> U171(isBin(X),X) 27.20/28.24 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.24 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.24 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.24 plus(z,X) -> U201(isBin(X),X) 27.20/28.24 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.24 prod(empty) -> 1(z) 27.20/28.24 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.24 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.24 sum(empty) -> 0(z) 27.20/28.24 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.24 union(empty,X) -> X 27.20/28.24 union(X,empty) -> X 27.20/28.24 -> Usable Rules: 27.20/28.24 0(z) -> z 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.24 U202(tt,X) -> X 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.24 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.24 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.24 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.24 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.24 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.24 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.24 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.24 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.24 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.24 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.24 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.24 plus(z,X) -> U201(isBin(X),X) 27.20/28.24 -> SRules: 27.20/28.24 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.24 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.24 ->Interpretation type: 27.20/28.24 Linear 27.20/28.24 ->Coefficients: 27.20/28.24 Natural Numbers 27.20/28.24 ->Dimension: 27.20/28.24 1 27.20/28.24 ->Bound: 27.20/28.24 2 27.20/28.24 ->Interpretation: 27.20/28.24 27.20/28.24 [0](X) = X + 2 27.20/28.24 [U101](X1,X2) = X1 27.20/28.24 [U102](X1,X2) = 2 27.20/28.24 [U103](X) = 2 27.20/28.24 [U11](X1,X2) = 2.X1 + 2 27.20/28.24 [U111](X) = 2 27.20/28.24 [U12](X1,X2) = 2.X1 + 2 27.20/28.24 [U121](X) = 2 27.20/28.24 [U13](X) = 2.X + 2 27.20/28.24 [U131](X1,X2) = 2 27.20/28.24 [U132](X) = 2 27.20/28.24 [U141](X1,X2) = 2 27.20/28.24 [U142](X) = 2 27.20/28.24 [U151](X) = 2 27.20/28.24 [U161](X) = 2 27.20/28.24 [U171](X1,X2) = 0 27.20/28.24 [U172](X) = 0 27.20/28.24 [U181](X1,X2,X3) = 0 27.20/28.24 [U182](X1,X2,X3) = 0 27.20/28.24 [U183](X1,X2,X3) = 0 27.20/28.24 [U184](X1,X2,X3) = 0 27.20/28.24 [U191](X1,X2,X3) = 0 27.20/28.24 [U192](X1,X2,X3) = 0 27.20/28.24 [U193](X1,X2,X3) = 0 27.20/28.24 [U194](X1,X2,X3) = 0 27.20/28.24 [U201](X1,X2) = X2 27.20/28.24 [U202](X1,X2) = X2 27.20/28.24 [U21](X1,X2,X3) = 2.X1 + 2.X3 + 2 27.20/28.24 [U211](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.24 [U212](X1,X2,X3) = 2.X1 + X2 + X3 27.20/28.24 [U213](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.24 [U214](X1,X2,X3) = 2.X1 + X2 + X3 27.20/28.24 [U22](X1,X2,X3) = 2.X1 + 2.X3 + 2 27.20/28.24 [U221](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.24 [U222](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.24 [U223](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.24 [U224](X1,X2,X3) = X1 + X2 + X3 + 1 27.20/28.24 [U23](X1,X2,X3) = X1 + 2.X3 + 2 27.20/28.24 [U231](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.24 [U232](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.24 [U233](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.24 [U234](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.24 [U24](X1,X2,X3) = X1 + X3 + 2 27.20/28.24 [U241](X1,X2) = 0 27.20/28.24 [U242](X1,X2) = 0 27.20/28.24 [U25](X1,X2) = 2 27.20/28.24 [U251](X1,X2,X3) = 0 27.20/28.24 [U252](X1,X2,X3) = 0 27.20/28.24 [U253](X1,X2,X3) = 0 27.20/28.24 [U254](X1,X2,X3) = 0 27.20/28.24 [U26](X) = 2 27.20/28.24 [U261](X1,X2) = 0 27.20/28.24 [U262](X1,X2) = 0 27.20/28.24 [U271](X1,X2,X3) = 0 27.20/28.24 [U272](X1,X2,X3) = 0 27.20/28.24 [U273](X1,X2,X3) = 0 27.20/28.24 [U274](X1,X2,X3) = 0 27.20/28.24 [U31](X) = 2 27.20/28.24 [U41](X1,X2) = 2 27.20/28.24 [U42](X) = X 27.20/28.24 [U51](X1,X2) = X1 27.20/28.24 [U52](X1,X2) = 2 27.20/28.24 [U53](X) = 2 27.20/28.24 [U61](X1,X2) = 2 27.20/28.24 [U62](X1,X2) = 2 27.20/28.24 [U63](X) = 2 27.20/28.24 [U71](X1,X2,X3) = X1 27.20/28.24 [U72](X1,X2,X3) = 2 27.20/28.24 [U73](X1,X2,X3) = 2 27.20/28.24 [U74](X1,X2,X3) = 2 27.20/28.24 [U75](X1,X2) = 2 27.20/28.24 [U76](X) = X 27.20/28.24 [U81](X1,X2,X3) = X1 27.20/28.24 [U82](X1,X2,X3) = 2 27.20/28.24 [U83](X1,X2,X3) = X1 27.20/28.24 [U84](X1,X2,X3) = X1 27.20/28.24 [U85](X1,X2) = X1 27.20/28.24 [U86](X) = 2 27.20/28.24 [U91](X1,X2) = 2 27.20/28.24 [U92](X1,X2) = 2 27.20/28.24 [U93](X) = 2 27.20/28.24 [isBag](X) = 2.X + 2 27.20/28.24 [isBagKind](X) = 2 27.20/28.24 [isBin](X) = 2 27.20/28.24 [isBinKind](X) = 2 27.20/28.24 [mult](X1,X2) = 2.X2 + 2 27.20/28.24 [plus](X1,X2) = X1 + X2 27.20/28.24 [prod](X) = 2.X 27.20/28.24 [sum](X) = 2.X + 2 27.20/28.24 [union](X1,X2) = 2.X1 + X2 + 2 27.20/28.24 [1](X) = X + 2 27.20/28.24 [empty] = 0 27.20/28.24 [singl](X) = 2 27.20/28.24 [tt] = 2 27.20/28.24 [z] = 0 27.20/28.24 [0#](X) = 0 27.20/28.24 [U101#](X1,X2) = 0 27.20/28.24 [U102#](X1,X2) = 0 27.20/28.24 [U103#](X) = 0 27.20/28.24 [U11#](X1,X2) = 0 27.20/28.24 [U111#](X) = 0 27.20/28.24 [U12#](X1,X2) = 0 27.20/28.24 [U121#](X) = 0 27.20/28.24 [U13#](X) = 0 27.20/28.24 [U131#](X1,X2) = 0 27.20/28.24 [U132#](X) = 0 27.20/28.24 [U141#](X1,X2) = 0 27.20/28.24 [U142#](X) = 0 27.20/28.24 [U151#](X) = 0 27.20/28.24 [U161#](X) = 0 27.20/28.24 [U171#](X1,X2) = 0 27.20/28.24 [U172#](X) = 0 27.20/28.24 [U181#](X1,X2,X3) = 0 27.20/28.24 [U182#](X1,X2,X3) = 0 27.20/28.24 [U183#](X1,X2,X3) = 0 27.20/28.24 [U184#](X1,X2,X3) = 0 27.20/28.24 [U191#](X1,X2,X3) = 0 27.20/28.24 [U192#](X1,X2,X3) = 0 27.20/28.24 [U193#](X1,X2,X3) = 0 27.20/28.24 [U194#](X1,X2,X3) = 0 27.20/28.24 [U201#](X1,X2) = 0 27.20/28.24 [U202#](X1,X2) = 0 27.20/28.24 [U21#](X1,X2,X3) = 0 27.20/28.24 [U211#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.24 [U212#](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.20/28.24 [U213#](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.20/28.24 [U214#](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.20/28.24 [U22#](X1,X2,X3) = 0 27.20/28.24 [U221#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.24 [U222#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 27.20/28.24 [U223#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 27.20/28.24 [U224#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.24 [U23#](X1,X2,X3) = 0 27.20/28.24 [U231#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.24 [U232#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.24 [U233#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.24 [U234#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.24 [U24#](X1,X2,X3) = 0 27.20/28.24 [U241#](X1,X2) = 0 27.20/28.24 [U242#](X1,X2) = 0 27.20/28.24 [U25#](X1,X2) = 0 27.20/28.24 [U251#](X1,X2,X3) = 0 27.20/28.24 [U252#](X1,X2,X3) = 0 27.20/28.24 [U253#](X1,X2,X3) = 0 27.20/28.24 [U254#](X1,X2,X3) = 0 27.20/28.24 [U26#](X) = 0 27.20/28.24 [U261#](X1,X2) = 0 27.20/28.24 [U262#](X1,X2) = 0 27.20/28.24 [U271#](X1,X2,X3) = 0 27.20/28.24 [U272#](X1,X2,X3) = 0 27.20/28.24 [U273#](X1,X2,X3) = 0 27.20/28.24 [U274#](X1,X2,X3) = 0 27.20/28.24 [U31#](X) = 0 27.20/28.24 [U41#](X1,X2) = 0 27.20/28.24 [U42#](X) = 0 27.20/28.24 [U51#](X1,X2) = 0 27.20/28.24 [U52#](X1,X2) = 0 27.20/28.24 [U53#](X) = 0 27.20/28.24 [U61#](X1,X2) = 0 27.20/28.24 [U62#](X1,X2) = 0 27.20/28.24 [U63#](X) = 0 27.20/28.24 [U71#](X1,X2,X3) = 0 27.20/28.24 [U72#](X1,X2,X3) = 0 27.20/28.24 [U73#](X1,X2,X3) = 0 27.20/28.24 [U74#](X1,X2,X3) = 0 27.20/28.24 [U75#](X1,X2) = 0 27.20/28.24 [U76#](X) = 0 27.20/28.24 [U81#](X1,X2,X3) = 0 27.20/28.24 [U82#](X1,X2,X3) = 0 27.20/28.24 [U83#](X1,X2,X3) = 0 27.20/28.24 [U84#](X1,X2,X3) = 0 27.20/28.24 [U85#](X1,X2) = 0 27.20/28.24 [U86#](X) = 0 27.20/28.24 [U91#](X1,X2) = 0 27.20/28.24 [U92#](X1,X2) = 0 27.20/28.24 [U93#](X) = 0 27.20/28.24 [ISBAG](X) = 0 27.20/28.24 [ISBAGKIND](X) = 0 27.20/28.24 [ISBIN](X) = 0 27.20/28.24 [ISBINKIND](X) = 0 27.20/28.24 [MULT](X1,X2) = 0 27.20/28.24 [PLUS](X1,X2) = 2.X1 + 2.X2 + 1 27.20/28.24 [PROD](X) = 0 27.20/28.24 [SUM](X) = 0 27.20/28.24 [UNION](X1,X2) = 0 27.20/28.24 27.20/28.24 Problem 1.4: 27.20/28.24 27.20/28.24 SCC Processor: 27.20/28.24 -> FAxioms: 27.20/28.24 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.24 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.24 -> Pairs: 27.20/28.24 U212#(tt,X,Y) -> U213#(isBin(Y),X,Y) 27.20/28.24 U213#(tt,X,Y) -> U214#(isBinKind(Y),X,Y) 27.20/28.24 U214#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 U221#(tt,X,Y) -> U222#(isBinKind(X),X,Y) 27.20/28.24 U222#(tt,X,Y) -> U223#(isBin(Y),X,Y) 27.20/28.24 U223#(tt,X,Y) -> U224#(isBinKind(Y),X,Y) 27.20/28.24 U224#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 U231#(tt,X,Y) -> U232#(isBinKind(X),X,Y) 27.20/28.24 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.20/28.24 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.20/28.24 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.20/28.24 U234#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 PLUS(0(X),0(Y)) -> U211#(isBin(X),X,Y) 27.20/28.24 PLUS(0(X),1(Y)) -> U221#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),0(Y)),x6) -> U211#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> U221#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.24 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.20/28.24 -> EAxioms: 27.20/28.24 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.24 mult(x6,x7) = mult(x7,x6) 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.24 union(x6,x7) = union(x7,x6) 27.20/28.24 -> Rules: 27.20/28.24 0(z) -> z 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.24 U172(tt) -> z 27.20/28.24 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.24 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.24 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.24 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.24 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.24 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.24 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.24 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.24 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.24 U202(tt,X) -> X 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.24 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.24 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.24 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.24 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.24 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.24 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.24 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.24 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.24 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.24 U242(tt,X) -> X 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.24 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.24 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.24 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.24 U262(tt,X) -> X 27.20/28.24 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.24 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.24 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.24 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.24 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.24 mult(z,X) -> U171(isBin(X),X) 27.20/28.24 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.24 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.24 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.24 plus(z,X) -> U201(isBin(X),X) 27.20/28.24 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.24 prod(empty) -> 1(z) 27.20/28.24 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.24 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.24 sum(empty) -> 0(z) 27.20/28.24 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.24 union(empty,X) -> X 27.20/28.24 union(X,empty) -> X 27.20/28.24 -> SRules: 27.20/28.24 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.24 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.24 ->Strongly Connected Components: 27.20/28.24 ->->Cycle: 27.20/28.24 ->->-> Pairs: 27.20/28.24 U221#(tt,X,Y) -> U222#(isBinKind(X),X,Y) 27.20/28.24 U222#(tt,X,Y) -> U223#(isBin(Y),X,Y) 27.20/28.24 U223#(tt,X,Y) -> U224#(isBinKind(Y),X,Y) 27.20/28.24 U224#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 U231#(tt,X,Y) -> U232#(isBinKind(X),X,Y) 27.20/28.24 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.20/28.24 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.20/28.24 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.20/28.24 U234#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 PLUS(0(X),1(Y)) -> U221#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> U221#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.24 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.20/28.24 -> FAxioms: 27.20/28.24 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.24 mult(x6,x7) -> mult(x7,x6) 27.20/28.24 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) -> plus(x7,x6) 27.20/28.24 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.24 union(x6,x7) -> union(x7,x6) 27.20/28.24 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8)) 27.20/28.24 PLUS(x6,x7) -> PLUS(x7,x6) 27.20/28.24 -> EAxioms: 27.20/28.24 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.24 mult(x6,x7) = mult(x7,x6) 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.24 union(x6,x7) = union(x7,x6) 27.20/28.24 ->->-> Rules: 27.20/28.24 0(z) -> z 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.24 U172(tt) -> z 27.20/28.24 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.24 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.24 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.24 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.24 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.24 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.24 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.24 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.24 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.24 U202(tt,X) -> X 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.24 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.24 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.24 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.24 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.24 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.24 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.24 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.24 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.24 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.24 U242(tt,X) -> X 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.24 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.24 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.24 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.24 U262(tt,X) -> X 27.20/28.24 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.24 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.24 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.24 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.24 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.24 mult(z,X) -> U171(isBin(X),X) 27.20/28.24 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.24 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.24 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.24 plus(z,X) -> U201(isBin(X),X) 27.20/28.24 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.24 prod(empty) -> 1(z) 27.20/28.24 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.24 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.24 sum(empty) -> 0(z) 27.20/28.24 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.24 union(empty,X) -> X 27.20/28.24 union(X,empty) -> X 27.20/28.24 -> SRules: 27.20/28.24 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.24 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.24 27.20/28.24 Problem 1.4: 27.20/28.24 27.20/28.24 Reduction Pairs Processor: 27.20/28.24 -> FAxioms: 27.20/28.24 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.24 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.24 -> Pairs: 27.20/28.24 U221#(tt,X,Y) -> U222#(isBinKind(X),X,Y) 27.20/28.24 U222#(tt,X,Y) -> U223#(isBin(Y),X,Y) 27.20/28.24 U223#(tt,X,Y) -> U224#(isBinKind(Y),X,Y) 27.20/28.24 U224#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 U231#(tt,X,Y) -> U232#(isBinKind(X),X,Y) 27.20/28.24 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.20/28.24 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.20/28.24 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.20/28.24 U234#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 PLUS(0(X),1(Y)) -> U221#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> U221#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.24 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.20/28.24 -> EAxioms: 27.20/28.24 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.24 mult(x6,x7) = mult(x7,x6) 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.24 union(x6,x7) = union(x7,x6) 27.20/28.24 -> Usable Equations: 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 -> Rules: 27.20/28.24 0(z) -> z 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.24 U172(tt) -> z 27.20/28.24 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.24 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.24 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.24 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.24 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.24 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.24 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.24 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.24 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.24 U202(tt,X) -> X 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.24 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.24 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.24 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.24 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.24 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.24 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.24 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.24 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.24 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.24 U242(tt,X) -> X 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.24 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.24 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.24 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.24 U262(tt,X) -> X 27.20/28.24 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.24 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.24 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.24 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.24 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.24 mult(z,X) -> U171(isBin(X),X) 27.20/28.24 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.24 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.24 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.24 plus(z,X) -> U201(isBin(X),X) 27.20/28.24 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.24 prod(empty) -> 1(z) 27.20/28.24 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.24 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.24 sum(empty) -> 0(z) 27.20/28.24 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.24 union(empty,X) -> X 27.20/28.24 union(X,empty) -> X 27.20/28.24 -> Usable Rules: 27.20/28.24 0(z) -> z 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.24 U202(tt,X) -> X 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.24 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.24 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.24 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.24 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.24 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.24 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.24 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.24 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.24 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.24 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.24 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.24 plus(z,X) -> U201(isBin(X),X) 27.20/28.24 -> SRules: 27.20/28.24 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.24 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.24 ->Interpretation type: 27.20/28.24 Linear 27.20/28.24 ->Coefficients: 27.20/28.24 Natural Numbers 27.20/28.24 ->Dimension: 27.20/28.24 1 27.20/28.24 ->Bound: 27.20/28.24 2 27.20/28.24 ->Interpretation: 27.20/28.24 27.20/28.24 [0](X) = 2.X 27.20/28.24 [U101](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.24 [U102](X1,X2) = 2.X1 + 2.X2 + 1 27.20/28.24 [U103](X) = X + 2 27.20/28.24 [U11](X1,X2) = X1 + 2.X2 + 2 27.20/28.24 [U111](X) = 2 27.20/28.24 [U12](X1,X2) = X1 + 2 27.20/28.24 [U121](X) = X + 1 27.20/28.24 [U13](X) = 2 27.20/28.24 [U131](X1,X2) = 2.X1 + X2 + 2 27.20/28.24 [U132](X) = 2 27.20/28.24 [U141](X1,X2) = X1 + 2.X2 27.20/28.24 [U142](X) = 2 27.20/28.24 [U151](X) = 2.X + 2 27.20/28.24 [U161](X) = 2.X + 1 27.20/28.24 [U171](X1,X2) = 0 27.20/28.24 [U172](X) = 0 27.20/28.24 [U181](X1,X2,X3) = 0 27.20/28.24 [U182](X1,X2,X3) = 0 27.20/28.24 [U183](X1,X2,X3) = 0 27.20/28.24 [U184](X1,X2,X3) = 0 27.20/28.24 [U191](X1,X2,X3) = 0 27.20/28.24 [U192](X1,X2,X3) = 0 27.20/28.24 [U193](X1,X2,X3) = 0 27.20/28.24 [U194](X1,X2,X3) = 0 27.20/28.24 [U201](X1,X2) = X2 27.20/28.24 [U202](X1,X2) = X2 27.20/28.24 [U21](X1,X2,X3) = X1 + 2.X2 + 2 27.20/28.24 [U211](X1,X2,X3) = 2.X2 + 2.X3 27.20/28.24 [U212](X1,X2,X3) = 2.X2 + 2.X3 27.20/28.24 [U213](X1,X2,X3) = 2.X2 + 2.X3 27.20/28.24 [U214](X1,X2,X3) = 2.X2 + 2.X3 27.20/28.24 [U22](X1,X2,X3) = X1 + 2.X2 + 2 27.20/28.24 [U221](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.20/28.24 [U222](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.20/28.24 [U223](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.20/28.24 [U224](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.20/28.24 [U23](X1,X2,X3) = 2.X2 + 2 27.20/28.24 [U231](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.24 [U232](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.24 [U233](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.24 [U234](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.24 [U24](X1,X2,X3) = 2.X2 + 2 27.20/28.24 [U241](X1,X2) = 0 27.20/28.24 [U242](X1,X2) = 0 27.20/28.24 [U25](X1,X2) = X1 27.20/28.24 [U251](X1,X2,X3) = 0 27.20/28.24 [U252](X1,X2,X3) = 0 27.20/28.24 [U253](X1,X2,X3) = 0 27.20/28.24 [U254](X1,X2,X3) = 0 27.20/28.24 [U26](X) = 2 27.20/28.24 [U261](X1,X2) = 0 27.20/28.24 [U262](X1,X2) = 0 27.20/28.24 [U271](X1,X2,X3) = 0 27.20/28.24 [U272](X1,X2,X3) = 0 27.20/28.24 [U273](X1,X2,X3) = 0 27.20/28.24 [U274](X1,X2,X3) = 0 27.20/28.24 [U31](X) = 2 27.20/28.24 [U41](X1,X2) = X1 27.20/28.24 [U42](X) = 2 27.20/28.24 [U51](X1,X2) = 2.X2 + 2 27.20/28.24 [U52](X1,X2) = X1 27.20/28.24 [U53](X) = 2 27.20/28.24 [U61](X1,X2) = X1 + 2.X2 + 2 27.20/28.24 [U62](X1,X2) = X1 + 2 27.20/28.24 [U63](X) = 2 27.20/28.24 [U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 27.20/28.24 [U72](X1,X2,X3) = X1 + 2.X3 27.20/28.24 [U73](X1,X2,X3) = 2.X3 + 2 27.20/28.24 [U74](X1,X2,X3) = X1 27.20/28.24 [U75](X1,X2) = 2 27.20/28.24 [U76](X) = 2 27.20/28.24 [U81](X1,X2,X3) = X1 + 2.X3 27.20/28.24 [U82](X1,X2,X3) = 2.X3 + 2 27.20/28.24 [U83](X1,X2,X3) = 2.X3 + 2 27.20/28.24 [U84](X1,X2,X3) = 2.X3 + 2 27.20/28.24 [U85](X1,X2) = 2.X2 + 2 27.20/28.24 [U86](X) = 2 27.20/28.24 [U91](X1,X2) = 2.X1 + 2.X2 + 1 27.20/28.24 [U92](X1,X2) = 2.X1 + 2.X2 27.20/28.24 [U93](X) = 2 27.20/28.24 [isBag](X) = 2.X + 2 27.20/28.24 [isBagKind](X) = 2 27.20/28.24 [isBin](X) = 2.X + 2 27.20/28.24 [isBinKind](X) = 2.X + 2 27.20/28.24 [mult](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.24 [plus](X1,X2) = X1 + X2 27.20/28.24 [prod](X) = 2.X + 2 27.20/28.24 [sum](X) = X + 2 27.20/28.24 [union](X1,X2) = 2.X1 + 2 27.20/28.24 [1](X) = 2.X + 1 27.20/28.24 [empty] = 2 27.20/28.24 [singl](X) = 2.X + 2 27.20/28.24 [tt] = 2 27.20/28.24 [z] = 0 27.20/28.24 [0#](X) = 0 27.20/28.24 [U101#](X1,X2) = 0 27.20/28.24 [U102#](X1,X2) = 0 27.20/28.24 [U103#](X) = 0 27.20/28.24 [U11#](X1,X2) = 0 27.20/28.24 [U111#](X) = 0 27.20/28.24 [U12#](X1,X2) = 0 27.20/28.24 [U121#](X) = 0 27.20/28.24 [U13#](X) = 0 27.20/28.24 [U131#](X1,X2) = 0 27.20/28.24 [U132#](X) = 0 27.20/28.24 [U141#](X1,X2) = 0 27.20/28.24 [U142#](X) = 0 27.20/28.24 [U151#](X) = 0 27.20/28.24 [U161#](X) = 0 27.20/28.24 [U171#](X1,X2) = 0 27.20/28.24 [U172#](X) = 0 27.20/28.24 [U181#](X1,X2,X3) = 0 27.20/28.24 [U182#](X1,X2,X3) = 0 27.20/28.24 [U183#](X1,X2,X3) = 0 27.20/28.24 [U184#](X1,X2,X3) = 0 27.20/28.24 [U191#](X1,X2,X3) = 0 27.20/28.24 [U192#](X1,X2,X3) = 0 27.20/28.24 [U193#](X1,X2,X3) = 0 27.20/28.24 [U194#](X1,X2,X3) = 0 27.20/28.24 [U201#](X1,X2) = 0 27.20/28.24 [U202#](X1,X2) = 0 27.20/28.24 [U21#](X1,X2,X3) = 0 27.20/28.24 [U211#](X1,X2,X3) = 0 27.20/28.24 [U212#](X1,X2,X3) = 0 27.20/28.24 [U213#](X1,X2,X3) = 0 27.20/28.24 [U214#](X1,X2,X3) = 0 27.20/28.24 [U22#](X1,X2,X3) = 0 27.20/28.24 [U221#](X1,X2,X3) = X1 + 2.X2 + 2.X3 27.20/28.24 [U222#](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.20/28.24 [U223#](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.20/28.24 [U224#](X1,X2,X3) = 2.X2 + 2.X3 + 1 27.20/28.24 [U23#](X1,X2,X3) = 0 27.20/28.24 [U231#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.24 [U232#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.24 [U233#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.24 [U234#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.24 [U24#](X1,X2,X3) = 0 27.20/28.24 [U241#](X1,X2) = 0 27.20/28.24 [U242#](X1,X2) = 0 27.20/28.24 [U25#](X1,X2) = 0 27.20/28.24 [U251#](X1,X2,X3) = 0 27.20/28.24 [U252#](X1,X2,X3) = 0 27.20/28.24 [U253#](X1,X2,X3) = 0 27.20/28.24 [U254#](X1,X2,X3) = 0 27.20/28.24 [U26#](X) = 0 27.20/28.24 [U261#](X1,X2) = 0 27.20/28.24 [U262#](X1,X2) = 0 27.20/28.24 [U271#](X1,X2,X3) = 0 27.20/28.24 [U272#](X1,X2,X3) = 0 27.20/28.24 [U273#](X1,X2,X3) = 0 27.20/28.24 [U274#](X1,X2,X3) = 0 27.20/28.24 [U31#](X) = 0 27.20/28.24 [U41#](X1,X2) = 0 27.20/28.24 [U42#](X) = 0 27.20/28.24 [U51#](X1,X2) = 0 27.20/28.24 [U52#](X1,X2) = 0 27.20/28.24 [U53#](X) = 0 27.20/28.24 [U61#](X1,X2) = 0 27.20/28.24 [U62#](X1,X2) = 0 27.20/28.24 [U63#](X) = 0 27.20/28.24 [U71#](X1,X2,X3) = 0 27.20/28.24 [U72#](X1,X2,X3) = 0 27.20/28.24 [U73#](X1,X2,X3) = 0 27.20/28.24 [U74#](X1,X2,X3) = 0 27.20/28.24 [U75#](X1,X2) = 0 27.20/28.24 [U76#](X) = 0 27.20/28.24 [U81#](X1,X2,X3) = 0 27.20/28.24 [U82#](X1,X2,X3) = 0 27.20/28.24 [U83#](X1,X2,X3) = 0 27.20/28.24 [U84#](X1,X2,X3) = 0 27.20/28.24 [U85#](X1,X2) = 0 27.20/28.24 [U86#](X) = 0 27.20/28.24 [U91#](X1,X2) = 0 27.20/28.24 [U92#](X1,X2) = 0 27.20/28.24 [U93#](X) = 0 27.20/28.24 [ISBAG](X) = 0 27.20/28.24 [ISBAGKIND](X) = 0 27.20/28.24 [ISBIN](X) = 0 27.20/28.24 [ISBINKIND](X) = 0 27.20/28.24 [MULT](X1,X2) = 0 27.20/28.24 [PLUS](X1,X2) = 2.X1 + 2.X2 27.20/28.24 [PROD](X) = 0 27.20/28.24 [SUM](X) = 0 27.20/28.24 [UNION](X1,X2) = 0 27.20/28.24 27.20/28.24 Problem 1.4: 27.20/28.24 27.20/28.24 SCC Processor: 27.20/28.24 -> FAxioms: 27.20/28.24 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.24 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.24 -> Pairs: 27.20/28.24 U222#(tt,X,Y) -> U223#(isBin(Y),X,Y) 27.20/28.24 U223#(tt,X,Y) -> U224#(isBinKind(Y),X,Y) 27.20/28.24 U224#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 U231#(tt,X,Y) -> U232#(isBinKind(X),X,Y) 27.20/28.24 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.20/28.24 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.20/28.24 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.20/28.24 U234#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 PLUS(0(X),1(Y)) -> U221#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> U221#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.24 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.20/28.24 -> EAxioms: 27.20/28.24 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.24 mult(x6,x7) = mult(x7,x6) 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.24 union(x6,x7) = union(x7,x6) 27.20/28.24 -> Rules: 27.20/28.24 0(z) -> z 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.24 U172(tt) -> z 27.20/28.24 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.24 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.24 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.24 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.24 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.24 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.24 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.24 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.24 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.24 U202(tt,X) -> X 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.24 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.24 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.24 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.24 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.24 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.24 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.24 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.24 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.24 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.24 U242(tt,X) -> X 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.24 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.24 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.24 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.24 U262(tt,X) -> X 27.20/28.24 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.24 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.24 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.24 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.24 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.24 mult(z,X) -> U171(isBin(X),X) 27.20/28.24 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.24 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.24 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.24 plus(z,X) -> U201(isBin(X),X) 27.20/28.24 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.24 prod(empty) -> 1(z) 27.20/28.24 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.24 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.24 sum(empty) -> 0(z) 27.20/28.24 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.24 union(empty,X) -> X 27.20/28.24 union(X,empty) -> X 27.20/28.24 -> SRules: 27.20/28.24 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.24 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.24 ->Strongly Connected Components: 27.20/28.24 ->->Cycle: 27.20/28.24 ->->-> Pairs: 27.20/28.24 U231#(tt,X,Y) -> U232#(isBinKind(X),X,Y) 27.20/28.24 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.20/28.24 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.20/28.24 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.20/28.24 U234#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.24 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.20/28.24 -> FAxioms: 27.20/28.24 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.24 mult(x6,x7) -> mult(x7,x6) 27.20/28.24 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) -> plus(x7,x6) 27.20/28.24 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.24 union(x6,x7) -> union(x7,x6) 27.20/28.24 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8)) 27.20/28.24 PLUS(x6,x7) -> PLUS(x7,x6) 27.20/28.24 -> EAxioms: 27.20/28.24 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.24 mult(x6,x7) = mult(x7,x6) 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.24 union(x6,x7) = union(x7,x6) 27.20/28.24 ->->-> Rules: 27.20/28.24 0(z) -> z 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.24 U172(tt) -> z 27.20/28.24 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.24 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.24 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.24 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.24 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.24 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.24 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.24 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.24 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.24 U202(tt,X) -> X 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.24 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.24 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.24 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.24 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.24 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.24 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.24 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.24 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.24 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.24 U242(tt,X) -> X 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.24 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.24 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.24 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.24 U262(tt,X) -> X 27.20/28.24 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.24 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.24 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.24 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.24 U42(tt) -> tt 27.20/28.24 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.24 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.24 U53(tt) -> tt 27.20/28.24 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.24 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.24 U63(tt) -> tt 27.20/28.24 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.24 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.24 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.24 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.24 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.24 U76(tt) -> tt 27.20/28.24 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.24 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.24 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.24 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.24 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.24 U86(tt) -> tt 27.20/28.24 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.24 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.24 U93(tt) -> tt 27.20/28.24 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.24 isBag(empty) -> tt 27.20/28.24 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.24 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.24 isBagKind(empty) -> tt 27.20/28.24 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.24 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.24 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.24 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.24 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.24 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.24 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.24 isBin(z) -> tt 27.20/28.24 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.24 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.24 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.24 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.24 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.24 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.24 isBinKind(z) -> tt 27.20/28.24 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.24 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.24 mult(z,X) -> U171(isBin(X),X) 27.20/28.24 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.24 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.24 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.24 plus(z,X) -> U201(isBin(X),X) 27.20/28.24 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.24 prod(empty) -> 1(z) 27.20/28.24 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.24 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.24 sum(empty) -> 0(z) 27.20/28.24 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.24 union(empty,X) -> X 27.20/28.24 union(X,empty) -> X 27.20/28.24 -> SRules: 27.20/28.24 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.24 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.24 27.20/28.24 Problem 1.4: 27.20/28.24 27.20/28.24 Reduction Pairs Processor: 27.20/28.24 -> FAxioms: 27.20/28.24 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.24 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.24 -> Pairs: 27.20/28.24 U231#(tt,X,Y) -> U232#(isBinKind(X),X,Y) 27.20/28.24 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.20/28.24 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.20/28.24 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.20/28.24 U234#(tt,X,Y) -> PLUS(X,Y) 27.20/28.24 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.20/28.24 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.24 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.24 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.20/28.24 -> EAxioms: 27.20/28.24 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.24 mult(x6,x7) = mult(x7,x6) 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.24 union(x6,x7) = union(x7,x6) 27.20/28.24 -> Usable Equations: 27.20/28.24 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.24 plus(x6,x7) = plus(x7,x6) 27.20/28.24 -> Rules: 27.20/28.24 0(z) -> z 27.20/28.24 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.24 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.24 U103(tt) -> tt 27.20/28.24 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.24 U111(tt) -> tt 27.20/28.24 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.24 U121(tt) -> tt 27.20/28.24 U13(tt) -> tt 27.20/28.24 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.24 U132(tt) -> tt 27.20/28.24 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.24 U142(tt) -> tt 27.20/28.24 U151(tt) -> tt 27.20/28.24 U161(tt) -> tt 27.20/28.24 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.24 U172(tt) -> z 27.20/28.24 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.24 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.24 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.24 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.24 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.24 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.24 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.24 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.24 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.24 U202(tt,X) -> X 27.20/28.24 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.24 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.24 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.24 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.24 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.24 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.24 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.24 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.24 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.24 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.24 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.24 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.24 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.24 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.24 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.24 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.24 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.24 U242(tt,X) -> X 27.20/28.24 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.24 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.24 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.24 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.24 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.24 U26(tt) -> tt 27.20/28.24 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.24 U262(tt,X) -> X 27.20/28.24 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.24 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.24 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.24 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.24 U31(tt) -> tt 27.20/28.24 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.25 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.25 mult(z,X) -> U171(isBin(X),X) 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.25 prod(empty) -> 1(z) 27.20/28.25 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.25 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.25 sum(empty) -> 0(z) 27.20/28.25 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.25 union(empty,X) -> X 27.20/28.25 union(X,empty) -> X 27.20/28.25 -> Usable Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 -> SRules: 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.25 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.25 ->Interpretation type: 27.20/28.25 Linear 27.20/28.25 ->Coefficients: 27.20/28.25 Natural Numbers 27.20/28.25 ->Dimension: 27.20/28.25 1 27.20/28.25 ->Bound: 27.20/28.25 2 27.20/28.25 ->Interpretation: 27.20/28.25 27.20/28.25 [0](X) = X + 2 27.20/28.25 [U101](X1,X2) = 2 27.20/28.25 [U102](X1,X2) = 2 27.20/28.25 [U103](X) = X 27.20/28.25 [U11](X1,X2) = 2 27.20/28.25 [U111](X) = 2 27.20/28.25 [U12](X1,X2) = 2 27.20/28.25 [U121](X) = 2 27.20/28.25 [U13](X) = 2 27.20/28.25 [U131](X1,X2) = X1 27.20/28.25 [U132](X) = 2 27.20/28.25 [U141](X1,X2) = 2 27.20/28.25 [U142](X) = X 27.20/28.25 [U151](X) = X 27.20/28.25 [U161](X) = 2 27.20/28.25 [U171](X1,X2) = 0 27.20/28.25 [U172](X) = 0 27.20/28.25 [U181](X1,X2,X3) = 0 27.20/28.25 [U182](X1,X2,X3) = 0 27.20/28.25 [U183](X1,X2,X3) = 0 27.20/28.25 [U184](X1,X2,X3) = 0 27.20/28.25 [U191](X1,X2,X3) = 0 27.20/28.25 [U192](X1,X2,X3) = 0 27.20/28.25 [U193](X1,X2,X3) = 0 27.20/28.25 [U194](X1,X2,X3) = 0 27.20/28.25 [U201](X1,X2) = X2 27.20/28.25 [U202](X1,X2) = X2 27.20/28.25 [U21](X1,X2,X3) = X1 27.20/28.25 [U211](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.25 [U212](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.25 [U213](X1,X2,X3) = 2.X1 + X2 + X3 27.20/28.25 [U214](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.25 [U22](X1,X2,X3) = X1 27.20/28.25 [U221](X1,X2,X3) = 2.X1 + X2 + X3 27.20/28.25 [U222](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.25 [U223](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.25 [U224](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.25 [U23](X1,X2,X3) = X1 27.20/28.25 [U231](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.25 [U232](X1,X2,X3) = 2.X1 + X2 + X3 27.20/28.25 [U233](X1,X2,X3) = 2.X1 + X2 + X3 27.20/28.25 [U234](X1,X2,X3) = 2.X1 + X2 + X3 27.20/28.25 [U24](X1,X2,X3) = 2 27.20/28.25 [U241](X1,X2) = 0 27.20/28.25 [U242](X1,X2) = 0 27.20/28.25 [U25](X1,X2) = 2 27.20/28.25 [U251](X1,X2,X3) = 0 27.20/28.25 [U252](X1,X2,X3) = 0 27.20/28.25 [U253](X1,X2,X3) = 0 27.20/28.25 [U254](X1,X2,X3) = 0 27.20/28.25 [U26](X) = X 27.20/28.25 [U261](X1,X2) = 0 27.20/28.25 [U262](X1,X2) = 0 27.20/28.25 [U271](X1,X2,X3) = 0 27.20/28.25 [U272](X1,X2,X3) = 0 27.20/28.25 [U273](X1,X2,X3) = 0 27.20/28.25 [U274](X1,X2,X3) = 0 27.20/28.25 [U31](X) = 2 27.20/28.25 [U41](X1,X2) = X1 27.20/28.25 [U42](X) = 2 27.20/28.25 [U51](X1,X2) = 2 27.20/28.25 [U52](X1,X2) = 2 27.20/28.25 [U53](X) = 2 27.20/28.25 [U61](X1,X2) = 2 27.20/28.25 [U62](X1,X2) = X1 27.20/28.25 [U63](X) = 2 27.20/28.25 [U71](X1,X2,X3) = X1 27.20/28.25 [U72](X1,X2,X3) = X1 27.20/28.25 [U73](X1,X2,X3) = X1 27.20/28.25 [U74](X1,X2,X3) = X1 27.20/28.25 [U75](X1,X2) = 2 27.20/28.25 [U76](X) = 2 27.20/28.25 [U81](X1,X2,X3) = X1 27.20/28.25 [U82](X1,X2,X3) = 2 27.20/28.25 [U83](X1,X2,X3) = X1 27.20/28.25 [U84](X1,X2,X3) = X1 27.20/28.25 [U85](X1,X2) = X1 27.20/28.25 [U86](X) = X 27.20/28.25 [U91](X1,X2) = X1 27.20/28.25 [U92](X1,X2) = 2 27.20/28.25 [U93](X) = 2 27.20/28.25 [isBag](X) = 2 27.20/28.25 [isBagKind](X) = 2 27.20/28.25 [isBin](X) = 2 27.20/28.25 [isBinKind](X) = 2 27.20/28.25 [mult](X1,X2) = 2.X2 27.20/28.25 [plus](X1,X2) = X1 + X2 27.20/28.25 [prod](X) = 2.X 27.20/28.25 [sum](X) = X + 1 27.20/28.25 [union](X1,X2) = 0 27.20/28.25 [1](X) = X + 2 27.20/28.25 [empty] = 2 27.20/28.25 [singl](X) = 2 27.20/28.25 [tt] = 2 27.20/28.25 [z] = 0 27.20/28.25 [0#](X) = 0 27.20/28.25 [U101#](X1,X2) = 0 27.20/28.25 [U102#](X1,X2) = 0 27.20/28.25 [U103#](X) = 0 27.20/28.25 [U11#](X1,X2) = 0 27.20/28.25 [U111#](X) = 0 27.20/28.25 [U12#](X1,X2) = 0 27.20/28.25 [U121#](X) = 0 27.20/28.25 [U13#](X) = 0 27.20/28.25 [U131#](X1,X2) = 0 27.20/28.25 [U132#](X) = 0 27.20/28.25 [U141#](X1,X2) = 0 27.20/28.25 [U142#](X) = 0 27.20/28.25 [U151#](X) = 0 27.20/28.25 [U161#](X) = 0 27.20/28.25 [U171#](X1,X2) = 0 27.20/28.25 [U172#](X) = 0 27.20/28.25 [U181#](X1,X2,X3) = 0 27.20/28.25 [U182#](X1,X2,X3) = 0 27.20/28.25 [U183#](X1,X2,X3) = 0 27.20/28.25 [U184#](X1,X2,X3) = 0 27.20/28.25 [U191#](X1,X2,X3) = 0 27.20/28.25 [U192#](X1,X2,X3) = 0 27.20/28.25 [U193#](X1,X2,X3) = 0 27.20/28.25 [U194#](X1,X2,X3) = 0 27.20/28.25 [U201#](X1,X2) = 0 27.20/28.25 [U202#](X1,X2) = 0 27.20/28.25 [U21#](X1,X2,X3) = 0 27.20/28.25 [U211#](X1,X2,X3) = 0 27.20/28.25 [U212#](X1,X2,X3) = 0 27.20/28.25 [U213#](X1,X2,X3) = 0 27.20/28.25 [U214#](X1,X2,X3) = 0 27.20/28.25 [U22#](X1,X2,X3) = 0 27.20/28.25 [U221#](X1,X2,X3) = 0 27.20/28.25 [U222#](X1,X2,X3) = 0 27.20/28.25 [U223#](X1,X2,X3) = 0 27.20/28.25 [U224#](X1,X2,X3) = 0 27.20/28.25 [U23#](X1,X2,X3) = 0 27.20/28.25 [U231#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.25 [U232#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 27.20/28.25 [U233#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.25 [U234#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.25 [U24#](X1,X2,X3) = 0 27.20/28.25 [U241#](X1,X2) = 0 27.20/28.25 [U242#](X1,X2) = 0 27.20/28.25 [U25#](X1,X2) = 0 27.20/28.25 [U251#](X1,X2,X3) = 0 27.20/28.25 [U252#](X1,X2,X3) = 0 27.20/28.25 [U253#](X1,X2,X3) = 0 27.20/28.25 [U254#](X1,X2,X3) = 0 27.20/28.25 [U26#](X) = 0 27.20/28.25 [U261#](X1,X2) = 0 27.20/28.25 [U262#](X1,X2) = 0 27.20/28.25 [U271#](X1,X2,X3) = 0 27.20/28.25 [U272#](X1,X2,X3) = 0 27.20/28.25 [U273#](X1,X2,X3) = 0 27.20/28.25 [U274#](X1,X2,X3) = 0 27.20/28.25 [U31#](X) = 0 27.20/28.25 [U41#](X1,X2) = 0 27.20/28.25 [U42#](X) = 0 27.20/28.25 [U51#](X1,X2) = 0 27.20/28.25 [U52#](X1,X2) = 0 27.20/28.25 [U53#](X) = 0 27.20/28.25 [U61#](X1,X2) = 0 27.20/28.25 [U62#](X1,X2) = 0 27.20/28.25 [U63#](X) = 0 27.20/28.25 [U71#](X1,X2,X3) = 0 27.20/28.25 [U72#](X1,X2,X3) = 0 27.20/28.25 [U73#](X1,X2,X3) = 0 27.20/28.25 [U74#](X1,X2,X3) = 0 27.20/28.25 [U75#](X1,X2) = 0 27.20/28.25 [U76#](X) = 0 27.20/28.25 [U81#](X1,X2,X3) = 0 27.20/28.25 [U82#](X1,X2,X3) = 0 27.20/28.25 [U83#](X1,X2,X3) = 0 27.20/28.25 [U84#](X1,X2,X3) = 0 27.20/28.25 [U85#](X1,X2) = 0 27.20/28.25 [U86#](X) = 0 27.20/28.25 [U91#](X1,X2) = 0 27.20/28.25 [U92#](X1,X2) = 0 27.20/28.25 [U93#](X) = 0 27.20/28.25 [ISBAG](X) = 0 27.20/28.25 [ISBAGKIND](X) = 0 27.20/28.25 [ISBIN](X) = 0 27.20/28.25 [ISBINKIND](X) = 0 27.20/28.25 [MULT](X1,X2) = 0 27.20/28.25 [PLUS](X1,X2) = 2.X1 + 2.X2 27.20/28.25 [PROD](X) = 0 27.20/28.25 [SUM](X) = 0 27.20/28.25 [UNION](X1,X2) = 0 27.20/28.25 27.20/28.25 Problem 1.4: 27.20/28.25 27.20/28.25 SCC Processor: 27.20/28.25 -> FAxioms: 27.20/28.25 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.25 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.25 -> Pairs: 27.20/28.25 U232#(tt,X,Y) -> U233#(isBin(Y),X,Y) 27.20/28.25 U233#(tt,X,Y) -> U234#(isBinKind(Y),X,Y) 27.20/28.25 U234#(tt,X,Y) -> PLUS(plus(X,Y),1(z)) 27.20/28.25 U234#(tt,X,Y) -> PLUS(X,Y) 27.20/28.25 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(1(X),1(Y)),x6) -> U231#(isBin(X),X,Y) 27.20/28.25 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.25 PLUS(1(X),1(Y)) -> U231#(isBin(X),X,Y) 27.20/28.25 -> EAxioms: 27.20/28.25 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) = mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) = union(x7,x6) 27.20/28.25 -> Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.25 U172(tt) -> z 27.20/28.25 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.25 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.25 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.25 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.25 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.25 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.25 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.25 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.25 U242(tt,X) -> X 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.25 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.25 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.25 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.25 U262(tt,X) -> X 27.20/28.25 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.25 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.25 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.25 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.25 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.25 mult(z,X) -> U171(isBin(X),X) 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.25 prod(empty) -> 1(z) 27.20/28.25 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.25 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.25 sum(empty) -> 0(z) 27.20/28.25 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.25 union(empty,X) -> X 27.20/28.25 union(X,empty) -> X 27.20/28.25 -> SRules: 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.25 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.25 ->Strongly Connected Components: 27.20/28.25 ->->Cycle: 27.20/28.25 ->->-> Pairs: 27.20/28.25 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.25 -> FAxioms: 27.20/28.25 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) -> mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) -> plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) -> union(x7,x6) 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8)) 27.20/28.25 PLUS(x6,x7) -> PLUS(x7,x6) 27.20/28.25 -> EAxioms: 27.20/28.25 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) = mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) = union(x7,x6) 27.20/28.25 ->->-> Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.25 U172(tt) -> z 27.20/28.25 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.25 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.25 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.25 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.25 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.25 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.25 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.25 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.25 U242(tt,X) -> X 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.25 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.25 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.25 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.25 U262(tt,X) -> X 27.20/28.25 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.25 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.25 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.25 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.25 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.25 mult(z,X) -> U171(isBin(X),X) 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.25 prod(empty) -> 1(z) 27.20/28.25 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.25 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.25 sum(empty) -> 0(z) 27.20/28.25 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.25 union(empty,X) -> X 27.20/28.25 union(X,empty) -> X 27.20/28.25 -> SRules: 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.25 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.25 27.20/28.25 Problem 1.4: 27.20/28.25 27.20/28.25 Reduction Pairs Processor: 27.20/28.25 -> FAxioms: 27.20/28.25 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.25 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.25 -> Pairs: 27.20/28.25 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U211(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.25 -> EAxioms: 27.20/28.25 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) = mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) = union(x7,x6) 27.20/28.25 -> Usable Equations: 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 -> Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.25 U172(tt) -> z 27.20/28.25 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.25 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.25 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.25 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.25 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.25 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.25 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.25 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.25 U242(tt,X) -> X 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.25 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.25 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.25 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.25 U262(tt,X) -> X 27.20/28.25 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.25 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.25 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.25 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.25 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.25 mult(z,X) -> U171(isBin(X),X) 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.25 prod(empty) -> 1(z) 27.20/28.25 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.25 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.25 sum(empty) -> 0(z) 27.20/28.25 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.25 union(empty,X) -> X 27.20/28.25 union(X,empty) -> X 27.20/28.25 -> Usable Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 -> SRules: 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.25 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.25 ->Interpretation type: 27.20/28.25 Linear 27.20/28.25 ->Coefficients: 27.20/28.25 Natural Numbers 27.20/28.25 ->Dimension: 27.20/28.25 1 27.20/28.25 ->Bound: 27.20/28.25 2 27.20/28.25 ->Interpretation: 27.20/28.25 27.20/28.25 [0](X) = 1 27.20/28.25 [U101](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.25 [U102](X1,X2) = X1 + 2.X2 + 2 27.20/28.25 [U103](X) = X + 2 27.20/28.25 [U11](X1,X2) = 2 27.20/28.25 [U111](X) = 2 27.20/28.25 [U12](X1,X2) = 2 27.20/28.25 [U121](X) = 2 27.20/28.25 [U13](X) = 2 27.20/28.25 [U131](X1,X2) = 2.X1 + 2.X2 27.20/28.25 [U132](X) = 2.X + 2 27.20/28.25 [U141](X1,X2) = X1 + X2 + 1 27.20/28.25 [U142](X) = X + 1 27.20/28.25 [U151](X) = 2 27.20/28.25 [U161](X) = 2 27.20/28.25 [U171](X1,X2) = 0 27.20/28.25 [U172](X) = 0 27.20/28.25 [U181](X1,X2,X3) = 0 27.20/28.25 [U182](X1,X2,X3) = 0 27.20/28.25 [U183](X1,X2,X3) = 0 27.20/28.25 [U184](X1,X2,X3) = 0 27.20/28.25 [U191](X1,X2,X3) = 0 27.20/28.25 [U192](X1,X2,X3) = 0 27.20/28.25 [U193](X1,X2,X3) = 0 27.20/28.25 [U194](X1,X2,X3) = 0 27.20/28.25 [U201](X1,X2) = X2 + 2 27.20/28.25 [U202](X1,X2) = X2 + 2 27.20/28.25 [U21](X1,X2,X3) = X1 27.20/28.25 [U211](X1,X2,X3) = 2 27.20/28.25 [U212](X1,X2,X3) = 2 27.20/28.25 [U213](X1,X2,X3) = 2 27.20/28.25 [U214](X1,X2,X3) = 2 27.20/28.25 [U22](X1,X2,X3) = X1 27.20/28.25 [U221](X1,X2,X3) = 2 27.20/28.25 [U222](X1,X2,X3) = 2 27.20/28.25 [U223](X1,X2,X3) = 2 27.20/28.25 [U224](X1,X2,X3) = 2 27.20/28.25 [U23](X1,X2,X3) = 2 27.20/28.25 [U231](X1,X2,X3) = 2 27.20/28.25 [U232](X1,X2,X3) = 2 27.20/28.25 [U233](X1,X2,X3) = 2 27.20/28.25 [U234](X1,X2,X3) = 1 27.20/28.25 [U24](X1,X2,X3) = 2 27.20/28.25 [U241](X1,X2) = 0 27.20/28.25 [U242](X1,X2) = 0 27.20/28.25 [U25](X1,X2) = X1 27.20/28.25 [U251](X1,X2,X3) = 0 27.20/28.25 [U252](X1,X2,X3) = 0 27.20/28.25 [U253](X1,X2,X3) = 0 27.20/28.25 [U254](X1,X2,X3) = 0 27.20/28.25 [U26](X) = X 27.20/28.25 [U261](X1,X2) = 0 27.20/28.25 [U262](X1,X2) = 0 27.20/28.25 [U271](X1,X2,X3) = 0 27.20/28.25 [U272](X1,X2,X3) = 0 27.20/28.25 [U273](X1,X2,X3) = 0 27.20/28.25 [U274](X1,X2,X3) = 0 27.20/28.25 [U31](X) = 2 27.20/28.25 [U41](X1,X2) = 2 27.20/28.25 [U42](X) = 2 27.20/28.25 [U51](X1,X2) = 2 27.20/28.25 [U52](X1,X2) = 2 27.20/28.25 [U53](X) = 2 27.20/28.25 [U61](X1,X2) = 2 27.20/28.25 [U62](X1,X2) = 2 27.20/28.25 [U63](X) = 2 27.20/28.25 [U71](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.25 [U72](X1,X2,X3) = 2.X1 + 2.X3 + 2 27.20/28.25 [U73](X1,X2,X3) = 2.X3 + 2 27.20/28.25 [U74](X1,X2,X3) = X1 + X3 + 1 27.20/28.25 [U75](X1,X2) = 2 27.20/28.25 [U76](X) = 2 27.20/28.25 [U81](X1,X2,X3) = X1 + X2 + 2.X3 + 1 27.20/28.25 [U82](X1,X2,X3) = X1 + 2.X3 + 2 27.20/28.25 [U83](X1,X2,X3) = X1 + X3 + 1 27.20/28.25 [U84](X1,X2,X3) = X1 27.20/28.25 [U85](X1,X2) = 2 27.20/28.25 [U86](X) = 2 27.20/28.25 [U91](X1,X2) = 2.X1 + X2 27.20/28.25 [U92](X1,X2) = X1 + X2 + 2 27.20/28.25 [U93](X) = 2.X 27.20/28.25 [isBag](X) = 2 27.20/28.25 [isBagKind](X) = 2 27.20/28.25 [isBin](X) = 2.X + 2 27.20/28.25 [isBinKind](X) = X + 1 27.20/28.25 [mult](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.25 [plus](X1,X2) = X1 + X2 + 2 27.20/28.25 [prod](X) = 2.X + 2 27.20/28.25 [sum](X) = X + 2 27.20/28.25 [union](X1,X2) = 2.X2 27.20/28.25 [1](X) = 2 27.20/28.25 [empty] = 0 27.20/28.25 [singl](X) = 2 27.20/28.25 [tt] = 2 27.20/28.25 [z] = 1 27.20/28.25 [0#](X) = 0 27.20/28.25 [U101#](X1,X2) = 0 27.20/28.25 [U102#](X1,X2) = 0 27.20/28.25 [U103#](X) = 0 27.20/28.25 [U11#](X1,X2) = 0 27.20/28.25 [U111#](X) = 0 27.20/28.25 [U12#](X1,X2) = 0 27.20/28.25 [U121#](X) = 0 27.20/28.25 [U13#](X) = 0 27.20/28.25 [U131#](X1,X2) = 0 27.20/28.25 [U132#](X) = 0 27.20/28.25 [U141#](X1,X2) = 0 27.20/28.25 [U142#](X) = 0 27.20/28.25 [U151#](X) = 0 27.20/28.25 [U161#](X) = 0 27.20/28.25 [U171#](X1,X2) = 0 27.20/28.25 [U172#](X) = 0 27.20/28.25 [U181#](X1,X2,X3) = 0 27.20/28.25 [U182#](X1,X2,X3) = 0 27.20/28.25 [U183#](X1,X2,X3) = 0 27.20/28.25 [U184#](X1,X2,X3) = 0 27.20/28.25 [U191#](X1,X2,X3) = 0 27.20/28.25 [U192#](X1,X2,X3) = 0 27.20/28.25 [U193#](X1,X2,X3) = 0 27.20/28.25 [U194#](X1,X2,X3) = 0 27.20/28.25 [U201#](X1,X2) = 0 27.20/28.25 [U202#](X1,X2) = 0 27.20/28.25 [U21#](X1,X2,X3) = 0 27.20/28.25 [U211#](X1,X2,X3) = 0 27.20/28.25 [U212#](X1,X2,X3) = 0 27.20/28.25 [U213#](X1,X2,X3) = 0 27.20/28.25 [U214#](X1,X2,X3) = 0 27.20/28.25 [U22#](X1,X2,X3) = 0 27.20/28.25 [U221#](X1,X2,X3) = 0 27.20/28.25 [U222#](X1,X2,X3) = 0 27.20/28.25 [U223#](X1,X2,X3) = 0 27.20/28.25 [U224#](X1,X2,X3) = 0 27.20/28.25 [U23#](X1,X2,X3) = 0 27.20/28.25 [U231#](X1,X2,X3) = 0 27.20/28.25 [U232#](X1,X2,X3) = 0 27.20/28.25 [U233#](X1,X2,X3) = 0 27.20/28.25 [U234#](X1,X2,X3) = 0 27.20/28.25 [U24#](X1,X2,X3) = 0 27.20/28.25 [U241#](X1,X2) = 0 27.20/28.25 [U242#](X1,X2) = 0 27.20/28.25 [U25#](X1,X2) = 0 27.20/28.25 [U251#](X1,X2,X3) = 0 27.20/28.25 [U252#](X1,X2,X3) = 0 27.20/28.25 [U253#](X1,X2,X3) = 0 27.20/28.25 [U254#](X1,X2,X3) = 0 27.20/28.25 [U26#](X) = 0 27.20/28.25 [U261#](X1,X2) = 0 27.20/28.25 [U262#](X1,X2) = 0 27.20/28.25 [U271#](X1,X2,X3) = 0 27.20/28.25 [U272#](X1,X2,X3) = 0 27.20/28.25 [U273#](X1,X2,X3) = 0 27.20/28.25 [U274#](X1,X2,X3) = 0 27.20/28.25 [U31#](X) = 0 27.20/28.25 [U41#](X1,X2) = 0 27.20/28.25 [U42#](X) = 0 27.20/28.25 [U51#](X1,X2) = 0 27.20/28.25 [U52#](X1,X2) = 0 27.20/28.25 [U53#](X) = 0 27.20/28.25 [U61#](X1,X2) = 0 27.20/28.25 [U62#](X1,X2) = 0 27.20/28.25 [U63#](X) = 0 27.20/28.25 [U71#](X1,X2,X3) = 0 27.20/28.25 [U72#](X1,X2,X3) = 0 27.20/28.25 [U73#](X1,X2,X3) = 0 27.20/28.25 [U74#](X1,X2,X3) = 0 27.20/28.25 [U75#](X1,X2) = 0 27.20/28.25 [U76#](X) = 0 27.20/28.25 [U81#](X1,X2,X3) = 0 27.20/28.25 [U82#](X1,X2,X3) = 0 27.20/28.25 [U83#](X1,X2,X3) = 0 27.20/28.25 [U84#](X1,X2,X3) = 0 27.20/28.25 [U85#](X1,X2) = 0 27.20/28.25 [U86#](X) = 0 27.20/28.25 [U91#](X1,X2) = 0 27.20/28.25 [U92#](X1,X2) = 0 27.20/28.25 [U93#](X) = 0 27.20/28.25 [ISBAG](X) = 0 27.20/28.25 [ISBAGKIND](X) = 0 27.20/28.25 [ISBIN](X) = 0 27.20/28.25 [ISBINKIND](X) = 0 27.20/28.25 [MULT](X1,X2) = 0 27.20/28.25 [PLUS](X1,X2) = 2.X1 + 2.X2 27.20/28.25 [PROD](X) = 0 27.20/28.25 [SUM](X) = 0 27.20/28.25 [UNION](X1,X2) = 0 27.20/28.25 27.20/28.25 Problem 1.4: 27.20/28.25 27.20/28.25 SCC Processor: 27.20/28.25 -> FAxioms: 27.20/28.25 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.25 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.25 -> Pairs: 27.20/28.25 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.25 -> EAxioms: 27.20/28.25 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) = mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) = union(x7,x6) 27.20/28.25 -> Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.25 U172(tt) -> z 27.20/28.25 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.25 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.25 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.25 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.25 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.25 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.25 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.25 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.25 U242(tt,X) -> X 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.25 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.25 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.25 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.25 U262(tt,X) -> X 27.20/28.25 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.25 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.25 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.25 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.25 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.25 mult(z,X) -> U171(isBin(X),X) 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.25 prod(empty) -> 1(z) 27.20/28.25 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.25 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.25 sum(empty) -> 0(z) 27.20/28.25 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.25 union(empty,X) -> X 27.20/28.25 union(X,empty) -> X 27.20/28.25 -> SRules: 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.25 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.25 ->Strongly Connected Components: 27.20/28.25 ->->Cycle: 27.20/28.25 ->->-> Pairs: 27.20/28.25 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.25 -> FAxioms: 27.20/28.25 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) -> mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) -> plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) -> union(x7,x6) 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8)) 27.20/28.25 PLUS(x6,x7) -> PLUS(x7,x6) 27.20/28.25 -> EAxioms: 27.20/28.25 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) = mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) = union(x7,x6) 27.20/28.25 ->->-> Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.25 U172(tt) -> z 27.20/28.25 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.25 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.25 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.25 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.25 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.25 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.25 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.25 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.25 U242(tt,X) -> X 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.25 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.25 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.25 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.25 U262(tt,X) -> X 27.20/28.25 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.25 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.25 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.25 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.25 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.25 mult(z,X) -> U171(isBin(X),X) 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.25 prod(empty) -> 1(z) 27.20/28.25 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.25 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.25 sum(empty) -> 0(z) 27.20/28.25 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.25 union(empty,X) -> X 27.20/28.25 union(X,empty) -> X 27.20/28.25 -> SRules: 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.25 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.25 27.20/28.25 Problem 1.4: 27.20/28.25 27.20/28.25 Reduction Pairs Processor: 27.20/28.25 -> FAxioms: 27.20/28.25 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.25 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.25 -> Pairs: 27.20/28.25 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U221(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.25 -> EAxioms: 27.20/28.25 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) = mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) = union(x7,x6) 27.20/28.25 -> Usable Equations: 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 -> Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.25 U172(tt) -> z 27.20/28.25 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.25 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.25 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.25 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.25 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.25 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.25 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.25 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.25 U242(tt,X) -> X 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.25 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.25 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.25 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.25 U262(tt,X) -> X 27.20/28.25 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.25 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.25 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.25 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.25 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.25 mult(z,X) -> U171(isBin(X),X) 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.25 prod(empty) -> 1(z) 27.20/28.25 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.25 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.25 sum(empty) -> 0(z) 27.20/28.25 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.25 union(empty,X) -> X 27.20/28.25 union(X,empty) -> X 27.20/28.25 -> Usable Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 -> SRules: 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.25 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.25 ->Interpretation type: 27.20/28.25 Linear 27.20/28.25 ->Coefficients: 27.20/28.25 Natural Numbers 27.20/28.25 ->Dimension: 27.20/28.25 1 27.20/28.25 ->Bound: 27.20/28.25 2 27.20/28.25 ->Interpretation: 27.20/28.25 27.20/28.25 [0](X) = 2 27.20/28.25 [U101](X1,X2) = X1 + X2 + 2 27.20/28.25 [U102](X1,X2) = X1 + 2 27.20/28.25 [U103](X) = 2 27.20/28.25 [U11](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.25 [U111](X) = 2 27.20/28.25 [U12](X1,X2) = X1 + X2 + 2 27.20/28.25 [U121](X) = 2 27.20/28.25 [U13](X) = X + 2 27.20/28.25 [U131](X1,X2) = 2.X1 + X2 27.20/28.25 [U132](X) = X + 2 27.20/28.25 [U141](X1,X2) = X1 + X2 + 2 27.20/28.25 [U142](X) = X 27.20/28.25 [U151](X) = 2 27.20/28.25 [U161](X) = 2.X + 2 27.20/28.25 [U171](X1,X2) = 0 27.20/28.25 [U172](X) = 0 27.20/28.25 [U181](X1,X2,X3) = 0 27.20/28.25 [U182](X1,X2,X3) = 0 27.20/28.25 [U183](X1,X2,X3) = 0 27.20/28.25 [U184](X1,X2,X3) = 0 27.20/28.25 [U191](X1,X2,X3) = 0 27.20/28.25 [U192](X1,X2,X3) = 0 27.20/28.25 [U193](X1,X2,X3) = 0 27.20/28.25 [U194](X1,X2,X3) = 0 27.20/28.25 [U201](X1,X2) = X2 27.20/28.25 [U202](X1,X2) = X2 27.20/28.25 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.25 [U211](X1,X2,X3) = 2 27.20/28.25 [U212](X1,X2,X3) = 2 27.20/28.25 [U213](X1,X2,X3) = 2 27.20/28.25 [U214](X1,X2,X3) = 2 27.20/28.25 [U22](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.25 [U221](X1,X2,X3) = 2 27.20/28.25 [U222](X1,X2,X3) = 2 27.20/28.25 [U223](X1,X2,X3) = 2 27.20/28.25 [U224](X1,X2,X3) = 2 27.20/28.25 [U23](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.25 [U231](X1,X2,X3) = 2 27.20/28.25 [U232](X1,X2,X3) = 2 27.20/28.25 [U233](X1,X2,X3) = 2 27.20/28.25 [U234](X1,X2,X3) = 2 27.20/28.25 [U24](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.25 [U241](X1,X2) = 0 27.20/28.25 [U242](X1,X2) = 0 27.20/28.25 [U25](X1,X2) = X1 + 2.X2 + 2 27.20/28.25 [U251](X1,X2,X3) = 0 27.20/28.25 [U252](X1,X2,X3) = 0 27.20/28.25 [U253](X1,X2,X3) = 0 27.20/28.25 [U254](X1,X2,X3) = 0 27.20/28.25 [U26](X) = X + 1 27.20/28.25 [U261](X1,X2) = 0 27.20/28.25 [U262](X1,X2) = 0 27.20/28.25 [U271](X1,X2,X3) = 0 27.20/28.25 [U272](X1,X2,X3) = 0 27.20/28.25 [U273](X1,X2,X3) = 0 27.20/28.25 [U274](X1,X2,X3) = 0 27.20/28.25 [U31](X) = 2.X + 1 27.20/28.25 [U41](X1,X2) = 2.X1 + 2 27.20/28.25 [U42](X) = 2 27.20/28.25 [U51](X1,X2) = 2 27.20/28.25 [U52](X1,X2) = 2 27.20/28.25 [U53](X) = 2 27.20/28.25 [U61](X1,X2) = 2 27.20/28.25 [U62](X1,X2) = 2 27.20/28.25 [U63](X) = 2 27.20/28.25 [U71](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.25 [U72](X1,X2,X3) = X1 + X2 + 2.X3 + 2 27.20/28.25 [U73](X1,X2,X3) = X2 + 2.X3 + 2 27.20/28.25 [U74](X1,X2,X3) = X1 + X2 + X3 + 2 27.20/28.25 [U75](X1,X2) = X1 + X2 + 2 27.20/28.25 [U76](X) = X + 2 27.20/28.25 [U81](X1,X2,X3) = X2 + X3 + 2 27.20/28.25 [U82](X1,X2,X3) = X1 + X3 27.20/28.25 [U83](X1,X2,X3) = X3 + 2 27.20/28.25 [U84](X1,X2,X3) = X1 + 2 27.20/28.25 [U85](X1,X2) = 2 27.20/28.25 [U86](X) = 2 27.20/28.25 [U91](X1,X2) = 2.X2 + 1 27.20/28.25 [U92](X1,X2) = 2.X2 + 1 27.20/28.25 [U93](X) = X 27.20/28.25 [isBag](X) = 2.X 27.20/28.25 [isBagKind](X) = X 27.20/28.25 [isBin](X) = X + 1 27.20/28.25 [isBinKind](X) = X 27.20/28.25 [mult](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.25 [plus](X1,X2) = X1 + X2 + 2 27.20/28.25 [prod](X) = 2.X + 2 27.20/28.25 [sum](X) = 2.X + 2 27.20/28.25 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.25 [1](X) = 2 27.20/28.25 [empty] = 2 27.20/28.25 [singl](X) = 2.X + 1 27.20/28.25 [tt] = 2 27.20/28.25 [z] = 2 27.20/28.25 [0#](X) = 0 27.20/28.25 [U101#](X1,X2) = 0 27.20/28.25 [U102#](X1,X2) = 0 27.20/28.25 [U103#](X) = 0 27.20/28.25 [U11#](X1,X2) = 0 27.20/28.25 [U111#](X) = 0 27.20/28.25 [U12#](X1,X2) = 0 27.20/28.25 [U121#](X) = 0 27.20/28.25 [U13#](X) = 0 27.20/28.25 [U131#](X1,X2) = 0 27.20/28.25 [U132#](X) = 0 27.20/28.25 [U141#](X1,X2) = 0 27.20/28.25 [U142#](X) = 0 27.20/28.25 [U151#](X) = 0 27.20/28.25 [U161#](X) = 0 27.20/28.25 [U171#](X1,X2) = 0 27.20/28.25 [U172#](X) = 0 27.20/28.25 [U181#](X1,X2,X3) = 0 27.20/28.25 [U182#](X1,X2,X3) = 0 27.20/28.25 [U183#](X1,X2,X3) = 0 27.20/28.25 [U184#](X1,X2,X3) = 0 27.20/28.25 [U191#](X1,X2,X3) = 0 27.20/28.25 [U192#](X1,X2,X3) = 0 27.20/28.25 [U193#](X1,X2,X3) = 0 27.20/28.25 [U194#](X1,X2,X3) = 0 27.20/28.25 [U201#](X1,X2) = 0 27.20/28.25 [U202#](X1,X2) = 0 27.20/28.25 [U21#](X1,X2,X3) = 0 27.20/28.25 [U211#](X1,X2,X3) = 0 27.20/28.25 [U212#](X1,X2,X3) = 0 27.20/28.25 [U213#](X1,X2,X3) = 0 27.20/28.25 [U214#](X1,X2,X3) = 0 27.20/28.25 [U22#](X1,X2,X3) = 0 27.20/28.25 [U221#](X1,X2,X3) = 0 27.20/28.25 [U222#](X1,X2,X3) = 0 27.20/28.25 [U223#](X1,X2,X3) = 0 27.20/28.25 [U224#](X1,X2,X3) = 0 27.20/28.25 [U23#](X1,X2,X3) = 0 27.20/28.25 [U231#](X1,X2,X3) = 0 27.20/28.25 [U232#](X1,X2,X3) = 0 27.20/28.25 [U233#](X1,X2,X3) = 0 27.20/28.25 [U234#](X1,X2,X3) = 0 27.20/28.25 [U24#](X1,X2,X3) = 0 27.20/28.25 [U241#](X1,X2) = 0 27.20/28.25 [U242#](X1,X2) = 0 27.20/28.25 [U25#](X1,X2) = 0 27.20/28.25 [U251#](X1,X2,X3) = 0 27.20/28.25 [U252#](X1,X2,X3) = 0 27.20/28.25 [U253#](X1,X2,X3) = 0 27.20/28.25 [U254#](X1,X2,X3) = 0 27.20/28.25 [U26#](X) = 0 27.20/28.25 [U261#](X1,X2) = 0 27.20/28.25 [U262#](X1,X2) = 0 27.20/28.25 [U271#](X1,X2,X3) = 0 27.20/28.25 [U272#](X1,X2,X3) = 0 27.20/28.25 [U273#](X1,X2,X3) = 0 27.20/28.25 [U274#](X1,X2,X3) = 0 27.20/28.25 [U31#](X) = 0 27.20/28.25 [U41#](X1,X2) = 0 27.20/28.25 [U42#](X) = 0 27.20/28.25 [U51#](X1,X2) = 0 27.20/28.25 [U52#](X1,X2) = 0 27.20/28.25 [U53#](X) = 0 27.20/28.25 [U61#](X1,X2) = 0 27.20/28.25 [U62#](X1,X2) = 0 27.20/28.25 [U63#](X) = 0 27.20/28.25 [U71#](X1,X2,X3) = 0 27.20/28.25 [U72#](X1,X2,X3) = 0 27.20/28.25 [U73#](X1,X2,X3) = 0 27.20/28.25 [U74#](X1,X2,X3) = 0 27.20/28.25 [U75#](X1,X2) = 0 27.20/28.25 [U76#](X) = 0 27.20/28.25 [U81#](X1,X2,X3) = 0 27.20/28.25 [U82#](X1,X2,X3) = 0 27.20/28.25 [U83#](X1,X2,X3) = 0 27.20/28.25 [U84#](X1,X2,X3) = 0 27.20/28.25 [U85#](X1,X2) = 0 27.20/28.25 [U86#](X) = 0 27.20/28.25 [U91#](X1,X2) = 0 27.20/28.25 [U92#](X1,X2) = 0 27.20/28.25 [U93#](X) = 0 27.20/28.25 [ISBAG](X) = 0 27.20/28.25 [ISBAGKIND](X) = 0 27.20/28.25 [ISBIN](X) = 0 27.20/28.25 [ISBINKIND](X) = 0 27.20/28.25 [MULT](X1,X2) = 0 27.20/28.25 [PLUS](X1,X2) = 2.X1 + 2.X2 27.20/28.25 [PROD](X) = 0 27.20/28.25 [SUM](X) = 0 27.20/28.25 [UNION](X1,X2) = 0 27.20/28.25 27.20/28.25 Problem 1.4: 27.20/28.25 27.20/28.25 SCC Processor: 27.20/28.25 -> FAxioms: 27.20/28.25 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.25 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.25 -> Pairs: 27.20/28.25 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.25 -> EAxioms: 27.20/28.25 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) = mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) = union(x7,x6) 27.20/28.25 -> Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.25 U172(tt) -> z 27.20/28.25 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.25 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.25 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.25 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.25 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.25 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.25 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.25 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.25 U242(tt,X) -> X 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.25 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.25 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.25 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.25 U262(tt,X) -> X 27.20/28.25 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.25 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.25 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.25 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.25 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.25 mult(z,X) -> U171(isBin(X),X) 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.25 prod(empty) -> 1(z) 27.20/28.25 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.25 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.25 sum(empty) -> 0(z) 27.20/28.25 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.25 union(empty,X) -> X 27.20/28.25 union(X,empty) -> X 27.20/28.25 -> SRules: 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.25 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.25 ->Strongly Connected Components: 27.20/28.25 ->->Cycle: 27.20/28.25 ->->-> Pairs: 27.20/28.25 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.25 -> FAxioms: 27.20/28.25 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) -> mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) -> plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) -> union(x7,x6) 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8)) 27.20/28.25 PLUS(x6,x7) -> PLUS(x7,x6) 27.20/28.25 -> EAxioms: 27.20/28.25 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) = mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) = union(x7,x6) 27.20/28.25 ->->-> Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.25 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.25 U172(tt) -> z 27.20/28.25 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.25 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.25 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.25 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.25 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.25 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.25 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.25 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.25 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.25 U202(tt,X) -> X 27.20/28.25 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.25 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.25 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.25 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.25 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.25 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.25 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.25 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.25 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.25 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.25 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.25 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.25 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.25 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.25 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.25 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.25 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.25 U242(tt,X) -> X 27.20/28.25 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.25 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.25 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.25 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.25 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.25 U26(tt) -> tt 27.20/28.25 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.25 U262(tt,X) -> X 27.20/28.25 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.25 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.25 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.25 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.25 U31(tt) -> tt 27.20/28.25 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.25 U42(tt) -> tt 27.20/28.25 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.25 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.25 U53(tt) -> tt 27.20/28.25 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.25 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.25 U63(tt) -> tt 27.20/28.25 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.25 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.25 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.25 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.25 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.25 U76(tt) -> tt 27.20/28.25 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.25 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.25 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.25 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.25 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.25 U86(tt) -> tt 27.20/28.25 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.25 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.25 U93(tt) -> tt 27.20/28.25 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.25 isBag(empty) -> tt 27.20/28.25 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.25 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.25 isBagKind(empty) -> tt 27.20/28.25 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.25 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.25 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.25 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.25 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.25 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.25 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.25 isBin(z) -> tt 27.20/28.25 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.25 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.25 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.25 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.25 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.25 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.25 isBinKind(z) -> tt 27.20/28.25 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.25 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.25 mult(z,X) -> U171(isBin(X),X) 27.20/28.25 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.25 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.25 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.25 plus(z,X) -> U201(isBin(X),X) 27.20/28.25 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.25 prod(empty) -> 1(z) 27.20/28.25 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.25 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.25 sum(empty) -> 0(z) 27.20/28.25 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.25 union(empty,X) -> X 27.20/28.25 union(X,empty) -> X 27.20/28.25 -> SRules: 27.20/28.25 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.25 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.25 27.20/28.25 Problem 1.4: 27.20/28.25 27.20/28.25 Reduction Pairs Processor: 27.20/28.25 -> FAxioms: 27.20/28.25 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.25 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.25 -> Pairs: 27.20/28.25 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U231(isBin(X),X,Y),x6) 27.20/28.25 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.25 -> EAxioms: 27.20/28.25 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.25 mult(x6,x7) = mult(x7,x6) 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.25 union(x6,x7) = union(x7,x6) 27.20/28.25 -> Usable Equations: 27.20/28.25 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.25 plus(x6,x7) = plus(x7,x6) 27.20/28.25 -> Rules: 27.20/28.25 0(z) -> z 27.20/28.25 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.25 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.25 U103(tt) -> tt 27.20/28.25 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.25 U111(tt) -> tt 27.20/28.25 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.25 U121(tt) -> tt 27.20/28.25 U13(tt) -> tt 27.20/28.25 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.25 U132(tt) -> tt 27.20/28.25 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.25 U142(tt) -> tt 27.20/28.25 U151(tt) -> tt 27.20/28.25 U161(tt) -> tt 27.20/28.26 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.26 U172(tt) -> z 27.20/28.26 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.26 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.26 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.26 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.26 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.26 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.26 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.26 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.26 U242(tt,X) -> X 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.26 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.26 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.26 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.26 U262(tt,X) -> X 27.20/28.26 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.26 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.26 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.26 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.26 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.26 mult(z,X) -> U171(isBin(X),X) 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.26 prod(empty) -> 1(z) 27.20/28.26 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.26 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.26 sum(empty) -> 0(z) 27.20/28.26 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.26 union(empty,X) -> X 27.20/28.26 union(X,empty) -> X 27.20/28.26 -> Usable Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 -> SRules: 27.20/28.26 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.26 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.26 ->Interpretation type: 27.20/28.26 Linear 27.20/28.26 ->Coefficients: 27.20/28.26 Natural Numbers 27.20/28.26 ->Dimension: 27.20/28.26 1 27.20/28.26 ->Bound: 27.20/28.26 2 27.20/28.26 ->Interpretation: 27.20/28.26 27.20/28.26 [0](X) = 2 27.20/28.26 [U101](X1,X2) = 2 27.20/28.26 [U102](X1,X2) = 2 27.20/28.26 [U103](X) = 2 27.20/28.26 [U11](X1,X2) = 2.X1 + X2 + 2 27.20/28.26 [U111](X) = X 27.20/28.26 [U12](X1,X2) = 2.X1 + 2 27.20/28.26 [U121](X) = 2 27.20/28.26 [U13](X) = X + 2 27.20/28.26 [U131](X1,X2) = 2 27.20/28.26 [U132](X) = X 27.20/28.26 [U141](X1,X2) = X1 27.20/28.26 [U142](X) = 2 27.20/28.26 [U151](X) = 2 27.20/28.26 [U161](X) = 2 27.20/28.26 [U171](X1,X2) = 0 27.20/28.26 [U172](X) = 0 27.20/28.26 [U181](X1,X2,X3) = 0 27.20/28.26 [U182](X1,X2,X3) = 0 27.20/28.26 [U183](X1,X2,X3) = 0 27.20/28.26 [U184](X1,X2,X3) = 0 27.20/28.26 [U191](X1,X2,X3) = 0 27.20/28.26 [U192](X1,X2,X3) = 0 27.20/28.26 [U193](X1,X2,X3) = 0 27.20/28.26 [U194](X1,X2,X3) = 0 27.20/28.26 [U201](X1,X2) = X1 + X2 27.20/28.26 [U202](X1,X2) = X2 + 2 27.20/28.26 [U21](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.26 [U211](X1,X2,X3) = 2.X1 + 2 27.20/28.26 [U212](X1,X2,X3) = 2.X1 + 2 27.20/28.26 [U213](X1,X2,X3) = 2.X1 + 2 27.20/28.26 [U214](X1,X2,X3) = 2 27.20/28.26 [U22](X1,X2,X3) = X1 + 2.X3 + 2 27.20/28.26 [U221](X1,X2,X3) = 2.X1 + 2 27.20/28.26 [U222](X1,X2,X3) = 2.X1 + 2 27.20/28.26 [U223](X1,X2,X3) = 2.X1 + 2 27.20/28.26 [U224](X1,X2,X3) = 2.X1 + 2 27.20/28.26 [U23](X1,X2,X3) = 2.X3 + 2 27.20/28.26 [U231](X1,X2,X3) = X1 + 2 27.20/28.26 [U232](X1,X2,X3) = X1 + 2 27.20/28.26 [U233](X1,X2,X3) = 2.X1 27.20/28.26 [U234](X1,X2,X3) = X1 + 2 27.20/28.26 [U24](X1,X2,X3) = X1 + 2 27.20/28.26 [U241](X1,X2) = 0 27.20/28.26 [U242](X1,X2) = 0 27.20/28.26 [U25](X1,X2) = 2 27.20/28.26 [U251](X1,X2,X3) = 0 27.20/28.26 [U252](X1,X2,X3) = 0 27.20/28.26 [U253](X1,X2,X3) = 0 27.20/28.26 [U254](X1,X2,X3) = 0 27.20/28.26 [U26](X) = 2 27.20/28.26 [U261](X1,X2) = 0 27.20/28.26 [U262](X1,X2) = 0 27.20/28.26 [U271](X1,X2,X3) = 0 27.20/28.26 [U272](X1,X2,X3) = 0 27.20/28.26 [U273](X1,X2,X3) = 0 27.20/28.26 [U274](X1,X2,X3) = 0 27.20/28.26 [U31](X) = X + 2 27.20/28.26 [U41](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.26 [U42](X) = X + 2 27.20/28.26 [U51](X1,X2) = 2 27.20/28.26 [U52](X1,X2) = X1 27.20/28.26 [U53](X) = X 27.20/28.26 [U61](X1,X2) = 2 27.20/28.26 [U62](X1,X2) = 2 27.20/28.26 [U63](X) = 2 27.20/28.26 [U71](X1,X2,X3) = 2 27.20/28.26 [U72](X1,X2,X3) = 2 27.20/28.26 [U73](X1,X2,X3) = X1 27.20/28.26 [U74](X1,X2,X3) = X1 27.20/28.26 [U75](X1,X2) = X1 27.20/28.26 [U76](X) = 2 27.20/28.26 [U81](X1,X2,X3) = 2 27.20/28.26 [U82](X1,X2,X3) = X1 27.20/28.26 [U83](X1,X2,X3) = X1 27.20/28.26 [U84](X1,X2,X3) = 2 27.20/28.26 [U85](X1,X2) = X1 27.20/28.26 [U86](X) = 2 27.20/28.26 [U91](X1,X2) = 2 27.20/28.26 [U92](X1,X2) = 2 27.20/28.26 [U93](X) = 2 27.20/28.26 [isBag](X) = 2.X + 2 27.20/28.26 [isBagKind](X) = 2.X 27.20/28.26 [isBin](X) = 2 27.20/28.26 [isBinKind](X) = 2 27.20/28.26 [mult](X1,X2) = 2.X1 + X2 + 2 27.20/28.26 [plus](X1,X2) = X1 + X2 + 2 27.20/28.26 [prod](X) = 0 27.20/28.26 [sum](X) = 2.X + 2 27.20/28.26 [union](X1,X2) = 2.X1 + X2 + 2 27.20/28.26 [1](X) = 2 27.20/28.26 [empty] = 2 27.20/28.26 [singl](X) = 2.X + 2 27.20/28.26 [tt] = 2 27.20/28.26 [z] = 0 27.20/28.26 [0#](X) = 0 27.20/28.26 [U101#](X1,X2) = 0 27.20/28.26 [U102#](X1,X2) = 0 27.20/28.26 [U103#](X) = 0 27.20/28.26 [U11#](X1,X2) = 0 27.20/28.26 [U111#](X) = 0 27.20/28.26 [U12#](X1,X2) = 0 27.20/28.26 [U121#](X) = 0 27.20/28.26 [U13#](X) = 0 27.20/28.26 [U131#](X1,X2) = 0 27.20/28.26 [U132#](X) = 0 27.20/28.26 [U141#](X1,X2) = 0 27.20/28.26 [U142#](X) = 0 27.20/28.26 [U151#](X) = 0 27.20/28.26 [U161#](X) = 0 27.20/28.26 [U171#](X1,X2) = 0 27.20/28.26 [U172#](X) = 0 27.20/28.26 [U181#](X1,X2,X3) = 0 27.20/28.26 [U182#](X1,X2,X3) = 0 27.20/28.26 [U183#](X1,X2,X3) = 0 27.20/28.26 [U184#](X1,X2,X3) = 0 27.20/28.26 [U191#](X1,X2,X3) = 0 27.20/28.26 [U192#](X1,X2,X3) = 0 27.20/28.26 [U193#](X1,X2,X3) = 0 27.20/28.26 [U194#](X1,X2,X3) = 0 27.20/28.26 [U201#](X1,X2) = 0 27.20/28.26 [U202#](X1,X2) = 0 27.20/28.26 [U21#](X1,X2,X3) = 0 27.20/28.26 [U211#](X1,X2,X3) = 0 27.20/28.26 [U212#](X1,X2,X3) = 0 27.20/28.26 [U213#](X1,X2,X3) = 0 27.20/28.26 [U214#](X1,X2,X3) = 0 27.20/28.26 [U22#](X1,X2,X3) = 0 27.20/28.26 [U221#](X1,X2,X3) = 0 27.20/28.26 [U222#](X1,X2,X3) = 0 27.20/28.26 [U223#](X1,X2,X3) = 0 27.20/28.26 [U224#](X1,X2,X3) = 0 27.20/28.26 [U23#](X1,X2,X3) = 0 27.20/28.26 [U231#](X1,X2,X3) = 0 27.20/28.26 [U232#](X1,X2,X3) = 0 27.20/28.26 [U233#](X1,X2,X3) = 0 27.20/28.26 [U234#](X1,X2,X3) = 0 27.20/28.26 [U24#](X1,X2,X3) = 0 27.20/28.26 [U241#](X1,X2) = 0 27.20/28.26 [U242#](X1,X2) = 0 27.20/28.26 [U25#](X1,X2) = 0 27.20/28.26 [U251#](X1,X2,X3) = 0 27.20/28.26 [U252#](X1,X2,X3) = 0 27.20/28.26 [U253#](X1,X2,X3) = 0 27.20/28.26 [U254#](X1,X2,X3) = 0 27.20/28.26 [U26#](X) = 0 27.20/28.26 [U261#](X1,X2) = 0 27.20/28.26 [U262#](X1,X2) = 0 27.20/28.26 [U271#](X1,X2,X3) = 0 27.20/28.26 [U272#](X1,X2,X3) = 0 27.20/28.26 [U273#](X1,X2,X3) = 0 27.20/28.26 [U274#](X1,X2,X3) = 0 27.20/28.26 [U31#](X) = 0 27.20/28.26 [U41#](X1,X2) = 0 27.20/28.26 [U42#](X) = 0 27.20/28.26 [U51#](X1,X2) = 0 27.20/28.26 [U52#](X1,X2) = 0 27.20/28.26 [U53#](X) = 0 27.20/28.26 [U61#](X1,X2) = 0 27.20/28.26 [U62#](X1,X2) = 0 27.20/28.26 [U63#](X) = 0 27.20/28.26 [U71#](X1,X2,X3) = 0 27.20/28.26 [U72#](X1,X2,X3) = 0 27.20/28.26 [U73#](X1,X2,X3) = 0 27.20/28.26 [U74#](X1,X2,X3) = 0 27.20/28.26 [U75#](X1,X2) = 0 27.20/28.26 [U76#](X) = 0 27.20/28.26 [U81#](X1,X2,X3) = 0 27.20/28.26 [U82#](X1,X2,X3) = 0 27.20/28.26 [U83#](X1,X2,X3) = 0 27.20/28.26 [U84#](X1,X2,X3) = 0 27.20/28.26 [U85#](X1,X2) = 0 27.20/28.26 [U86#](X) = 0 27.20/28.26 [U91#](X1,X2) = 0 27.20/28.26 [U92#](X1,X2) = 0 27.20/28.26 [U93#](X) = 0 27.20/28.26 [ISBAG](X) = 0 27.20/28.26 [ISBAGKIND](X) = 0 27.20/28.26 [ISBIN](X) = 0 27.20/28.26 [ISBINKIND](X) = 0 27.20/28.26 [MULT](X1,X2) = 0 27.20/28.26 [PLUS](X1,X2) = 2.X1 + 2.X2 27.20/28.26 [PROD](X) = 0 27.20/28.26 [SUM](X) = 0 27.20/28.26 [UNION](X1,X2) = 0 27.20/28.26 27.20/28.26 Problem 1.4: 27.20/28.26 27.20/28.26 SCC Processor: 27.20/28.26 -> FAxioms: 27.20/28.26 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.26 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.26 -> Pairs: 27.20/28.26 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.26 -> EAxioms: 27.20/28.26 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) = mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.26 union(x6,x7) = union(x7,x6) 27.20/28.26 -> Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.26 U172(tt) -> z 27.20/28.26 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.26 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.26 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.26 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.26 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.26 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.26 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.26 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.26 U242(tt,X) -> X 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.26 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.26 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.26 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.26 U262(tt,X) -> X 27.20/28.26 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.26 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.26 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.26 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.26 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.26 mult(z,X) -> U171(isBin(X),X) 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.26 prod(empty) -> 1(z) 27.20/28.26 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.26 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.26 sum(empty) -> 0(z) 27.20/28.26 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.26 union(empty,X) -> X 27.20/28.26 union(X,empty) -> X 27.20/28.26 -> SRules: 27.20/28.26 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.26 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.26 ->Strongly Connected Components: 27.20/28.26 ->->Cycle: 27.20/28.26 ->->-> Pairs: 27.20/28.26 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.26 -> FAxioms: 27.20/28.26 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) -> mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) -> plus(x7,x6) 27.20/28.26 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.26 union(x6,x7) -> union(x7,x6) 27.20/28.26 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8)) 27.20/28.26 PLUS(x6,x7) -> PLUS(x7,x6) 27.20/28.26 -> EAxioms: 27.20/28.26 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) = mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.26 union(x6,x7) = union(x7,x6) 27.20/28.26 ->->-> Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.26 U172(tt) -> z 27.20/28.26 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.26 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.26 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.26 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.26 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.26 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.26 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.26 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.26 U242(tt,X) -> X 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.26 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.26 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.26 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.26 U262(tt,X) -> X 27.20/28.26 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.26 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.26 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.26 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.26 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.26 mult(z,X) -> U171(isBin(X),X) 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.26 prod(empty) -> 1(z) 27.20/28.26 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.26 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.26 sum(empty) -> 0(z) 27.20/28.26 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.26 union(empty,X) -> X 27.20/28.26 union(X,empty) -> X 27.20/28.26 -> SRules: 27.20/28.26 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.26 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.26 27.20/28.26 Problem 1.4: 27.20/28.26 27.20/28.26 Reduction Pairs Processor: 27.20/28.26 -> FAxioms: 27.20/28.26 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.26 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.26 -> Pairs: 27.20/28.26 PLUS(plus(z,X),x6) -> PLUS(U201(isBin(X),X),x6) 27.20/28.26 -> EAxioms: 27.20/28.26 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) = mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.26 union(x6,x7) = union(x7,x6) 27.20/28.26 -> Usable Equations: 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 -> Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.26 U172(tt) -> z 27.20/28.26 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.26 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.26 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.26 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.26 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.26 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.26 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.26 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.26 U242(tt,X) -> X 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.26 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.26 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.26 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.26 U262(tt,X) -> X 27.20/28.26 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.26 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.26 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.26 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.26 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.26 mult(z,X) -> U171(isBin(X),X) 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.26 prod(empty) -> 1(z) 27.20/28.26 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.26 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.26 sum(empty) -> 0(z) 27.20/28.26 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.26 union(empty,X) -> X 27.20/28.26 union(X,empty) -> X 27.20/28.26 -> Usable Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 -> SRules: 27.20/28.26 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.26 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.26 ->Interpretation type: 27.20/28.26 Linear 27.20/28.26 ->Coefficients: 27.20/28.26 Natural Numbers 27.20/28.26 ->Dimension: 27.20/28.26 1 27.20/28.26 ->Bound: 27.20/28.26 2 27.20/28.26 ->Interpretation: 27.20/28.26 27.20/28.26 [0](X) = 2 27.20/28.26 [U101](X1,X2) = 2.X2 + 2 27.20/28.26 [U102](X1,X2) = 2.X2 + 2 27.20/28.26 [U103](X) = X 27.20/28.26 [U11](X1,X2) = X1 + X2 + 2 27.20/28.26 [U111](X) = 2 27.20/28.26 [U12](X1,X2) = X1 + 2 27.20/28.26 [U121](X) = 2 27.20/28.26 [U13](X) = 2 27.20/28.26 [U131](X1,X2) = X1 + X2 + 2 27.20/28.26 [U132](X) = X + 1 27.20/28.26 [U141](X1,X2) = X1 + X2 + 2 27.20/28.26 [U142](X) = X + 2 27.20/28.26 [U151](X) = 2.X 27.20/28.26 [U161](X) = X + 1 27.20/28.26 [U171](X1,X2) = 0 27.20/28.26 [U172](X) = 0 27.20/28.26 [U181](X1,X2,X3) = 0 27.20/28.26 [U182](X1,X2,X3) = 0 27.20/28.26 [U183](X1,X2,X3) = 0 27.20/28.26 [U184](X1,X2,X3) = 0 27.20/28.26 [U191](X1,X2,X3) = 0 27.20/28.26 [U192](X1,X2,X3) = 0 27.20/28.26 [U193](X1,X2,X3) = 0 27.20/28.26 [U194](X1,X2,X3) = 0 27.20/28.26 [U201](X1,X2) = X2 27.20/28.26 [U202](X1,X2) = X2 27.20/28.26 [U21](X1,X2,X3) = X1 + 2.X2 + 2 27.20/28.26 [U211](X1,X2,X3) = 2 27.20/28.26 [U212](X1,X2,X3) = 2 27.20/28.26 [U213](X1,X2,X3) = 2 27.20/28.26 [U214](X1,X2,X3) = 2 27.20/28.26 [U22](X1,X2,X3) = 2.X1 + 2 27.20/28.26 [U221](X1,X2,X3) = 2 27.20/28.26 [U222](X1,X2,X3) = 2 27.20/28.26 [U223](X1,X2,X3) = 2 27.20/28.26 [U224](X1,X2,X3) = 2 27.20/28.26 [U23](X1,X2,X3) = 2 27.20/28.26 [U231](X1,X2,X3) = 2 27.20/28.26 [U232](X1,X2,X3) = 2 27.20/28.26 [U233](X1,X2,X3) = 2 27.20/28.26 [U234](X1,X2,X3) = 2 27.20/28.26 [U24](X1,X2,X3) = 2 27.20/28.26 [U241](X1,X2) = 0 27.20/28.26 [U242](X1,X2) = 0 27.20/28.26 [U25](X1,X2) = 2 27.20/28.26 [U251](X1,X2,X3) = 0 27.20/28.26 [U252](X1,X2,X3) = 0 27.20/28.26 [U253](X1,X2,X3) = 0 27.20/28.26 [U254](X1,X2,X3) = 0 27.20/28.26 [U26](X) = 2 27.20/28.26 [U261](X1,X2) = 0 27.20/28.26 [U262](X1,X2) = 0 27.20/28.26 [U271](X1,X2,X3) = 0 27.20/28.26 [U272](X1,X2,X3) = 0 27.20/28.26 [U273](X1,X2,X3) = 0 27.20/28.26 [U274](X1,X2,X3) = 0 27.20/28.26 [U31](X) = X + 2 27.20/28.26 [U41](X1,X2) = X1 + 1 27.20/28.26 [U42](X) = 2 27.20/28.26 [U51](X1,X2) = 2 27.20/28.26 [U52](X1,X2) = 2 27.20/28.26 [U53](X) = 2 27.20/28.26 [U61](X1,X2) = 2 27.20/28.26 [U62](X1,X2) = 2 27.20/28.26 [U63](X) = 2 27.20/28.26 [U71](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.26 [U72](X1,X2,X3) = 2.X2 + 2.X3 27.20/28.26 [U73](X1,X2,X3) = X1 + 2.X2 + X3 27.20/28.26 [U74](X1,X2,X3) = X1 + 2.X2 + 1 27.20/28.26 [U75](X1,X2) = X1 + 1 27.20/28.26 [U76](X) = 2 27.20/28.26 [U81](X1,X2,X3) = X1 + X2 + 2.X3 + 2 27.20/28.26 [U82](X1,X2,X3) = X1 + 2.X3 + 2 27.20/28.26 [U83](X1,X2,X3) = 2.X3 + 2 27.20/28.26 [U84](X1,X2,X3) = 2.X3 + 2 27.20/28.26 [U85](X1,X2) = 2.X2 + 2 27.20/28.26 [U86](X) = X 27.20/28.26 [U91](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.26 [U92](X1,X2) = 2.X1 + 2 27.20/28.26 [U93](X) = 2 27.20/28.26 [isBag](X) = 2.X + 2 27.20/28.26 [isBagKind](X) = X + 1 27.20/28.26 [isBin](X) = 2.X + 2 27.20/28.26 [isBinKind](X) = X 27.20/28.26 [mult](X1,X2) = 2.X1 + X2 + 2 27.20/28.26 [plus](X1,X2) = X1 + X2 + 2 27.20/28.26 [prod](X) = 2.X + 2 27.20/28.26 [sum](X) = 2.X + 2 27.20/28.26 [union](X1,X2) = 2.X1 + 1 27.20/28.26 [1](X) = 2 27.20/28.26 [empty] = 2 27.20/28.26 [singl](X) = X + 1 27.20/28.26 [tt] = 2 27.20/28.26 [z] = 2 27.20/28.26 [0#](X) = 0 27.20/28.26 [U101#](X1,X2) = 0 27.20/28.26 [U102#](X1,X2) = 0 27.20/28.26 [U103#](X) = 0 27.20/28.26 [U11#](X1,X2) = 0 27.20/28.26 [U111#](X) = 0 27.20/28.26 [U12#](X1,X2) = 0 27.20/28.26 [U121#](X) = 0 27.20/28.26 [U13#](X) = 0 27.20/28.26 [U131#](X1,X2) = 0 27.20/28.26 [U132#](X) = 0 27.20/28.26 [U141#](X1,X2) = 0 27.20/28.26 [U142#](X) = 0 27.20/28.26 [U151#](X) = 0 27.20/28.26 [U161#](X) = 0 27.20/28.26 [U171#](X1,X2) = 0 27.20/28.26 [U172#](X) = 0 27.20/28.26 [U181#](X1,X2,X3) = 0 27.20/28.26 [U182#](X1,X2,X3) = 0 27.20/28.26 [U183#](X1,X2,X3) = 0 27.20/28.26 [U184#](X1,X2,X3) = 0 27.20/28.26 [U191#](X1,X2,X3) = 0 27.20/28.26 [U192#](X1,X2,X3) = 0 27.20/28.26 [U193#](X1,X2,X3) = 0 27.20/28.26 [U194#](X1,X2,X3) = 0 27.20/28.26 [U201#](X1,X2) = 0 27.20/28.26 [U202#](X1,X2) = 0 27.20/28.26 [U21#](X1,X2,X3) = 0 27.20/28.26 [U211#](X1,X2,X3) = 0 27.20/28.26 [U212#](X1,X2,X3) = 0 27.20/28.26 [U213#](X1,X2,X3) = 0 27.20/28.26 [U214#](X1,X2,X3) = 0 27.20/28.26 [U22#](X1,X2,X3) = 0 27.20/28.26 [U221#](X1,X2,X3) = 0 27.20/28.26 [U222#](X1,X2,X3) = 0 27.20/28.26 [U223#](X1,X2,X3) = 0 27.20/28.26 [U224#](X1,X2,X3) = 0 27.20/28.26 [U23#](X1,X2,X3) = 0 27.20/28.26 [U231#](X1,X2,X3) = 0 27.20/28.26 [U232#](X1,X2,X3) = 0 27.20/28.26 [U233#](X1,X2,X3) = 0 27.20/28.26 [U234#](X1,X2,X3) = 0 27.20/28.26 [U24#](X1,X2,X3) = 0 27.20/28.26 [U241#](X1,X2) = 0 27.20/28.26 [U242#](X1,X2) = 0 27.20/28.26 [U25#](X1,X2) = 0 27.20/28.26 [U251#](X1,X2,X3) = 0 27.20/28.26 [U252#](X1,X2,X3) = 0 27.20/28.26 [U253#](X1,X2,X3) = 0 27.20/28.26 [U254#](X1,X2,X3) = 0 27.20/28.26 [U26#](X) = 0 27.20/28.26 [U261#](X1,X2) = 0 27.20/28.26 [U262#](X1,X2) = 0 27.20/28.26 [U271#](X1,X2,X3) = 0 27.20/28.26 [U272#](X1,X2,X3) = 0 27.20/28.26 [U273#](X1,X2,X3) = 0 27.20/28.26 [U274#](X1,X2,X3) = 0 27.20/28.26 [U31#](X) = 0 27.20/28.26 [U41#](X1,X2) = 0 27.20/28.26 [U42#](X) = 0 27.20/28.26 [U51#](X1,X2) = 0 27.20/28.26 [U52#](X1,X2) = 0 27.20/28.26 [U53#](X) = 0 27.20/28.26 [U61#](X1,X2) = 0 27.20/28.26 [U62#](X1,X2) = 0 27.20/28.26 [U63#](X) = 0 27.20/28.26 [U71#](X1,X2,X3) = 0 27.20/28.26 [U72#](X1,X2,X3) = 0 27.20/28.26 [U73#](X1,X2,X3) = 0 27.20/28.26 [U74#](X1,X2,X3) = 0 27.20/28.26 [U75#](X1,X2) = 0 27.20/28.26 [U76#](X) = 0 27.20/28.26 [U81#](X1,X2,X3) = 0 27.20/28.26 [U82#](X1,X2,X3) = 0 27.20/28.26 [U83#](X1,X2,X3) = 0 27.20/28.26 [U84#](X1,X2,X3) = 0 27.20/28.26 [U85#](X1,X2) = 0 27.20/28.26 [U86#](X) = 0 27.20/28.26 [U91#](X1,X2) = 0 27.20/28.26 [U92#](X1,X2) = 0 27.20/28.26 [U93#](X) = 0 27.20/28.26 [ISBAG](X) = 0 27.20/28.26 [ISBAGKIND](X) = 0 27.20/28.26 [ISBIN](X) = 0 27.20/28.26 [ISBINKIND](X) = 0 27.20/28.26 [MULT](X1,X2) = 0 27.20/28.26 [PLUS](X1,X2) = 2.X1 + 2.X2 27.20/28.26 [PROD](X) = 0 27.20/28.26 [SUM](X) = 0 27.20/28.26 [UNION](X1,X2) = 0 27.20/28.26 27.20/28.26 Problem 1.4: 27.20/28.26 27.20/28.26 SCC Processor: 27.20/28.26 -> FAxioms: 27.20/28.26 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8)) 27.20/28.26 PLUS(x6,x7) = PLUS(x7,x6) 27.20/28.26 -> Pairs: 27.20/28.26 Empty 27.20/28.26 -> EAxioms: 27.20/28.26 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) = mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.26 union(x6,x7) = union(x7,x6) 27.20/28.26 -> Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.26 U172(tt) -> z 27.20/28.26 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.26 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.26 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.26 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.26 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.26 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.26 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.26 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.26 U242(tt,X) -> X 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.26 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.26 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.26 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.26 U262(tt,X) -> X 27.20/28.26 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.26 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.26 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.26 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.26 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.26 mult(z,X) -> U171(isBin(X),X) 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.26 prod(empty) -> 1(z) 27.20/28.26 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.26 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.26 sum(empty) -> 0(z) 27.20/28.26 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.26 union(empty,X) -> X 27.20/28.26 union(X,empty) -> X 27.20/28.26 -> SRules: 27.20/28.26 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7) 27.20/28.26 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8) 27.20/28.26 ->Strongly Connected Components: 27.20/28.26 There is no strongly connected component 27.20/28.26 27.20/28.26 The problem is finite. 27.20/28.26 27.20/28.26 Problem 1.5: 27.20/28.26 27.20/28.26 Reduction Pairs Processor: 27.20/28.26 -> FAxioms: 27.20/28.26 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.20/28.26 MULT(x6,x7) = MULT(x7,x6) 27.20/28.26 -> Pairs: 27.20/28.26 U181#(tt,X,Y) -> U182#(isBinKind(X),X,Y) 27.20/28.26 U182#(tt,X,Y) -> U183#(isBin(Y),X,Y) 27.20/28.26 U183#(tt,X,Y) -> U184#(isBinKind(Y),X,Y) 27.20/28.26 U184#(tt,X,Y) -> MULT(X,Y) 27.20/28.26 U191#(tt,X,Y) -> U192#(isBinKind(X),X,Y) 27.20/28.26 U192#(tt,X,Y) -> U193#(isBin(Y),X,Y) 27.20/28.26 U193#(tt,X,Y) -> U194#(isBinKind(Y),X,Y) 27.20/28.26 U194#(tt,X,Y) -> MULT(X,Y) 27.20/28.26 MULT(0(X),Y) -> U181#(isBin(X),X,Y) 27.20/28.26 MULT(mult(0(X),Y),x6) -> U181#(isBin(X),X,Y) 27.20/28.26 MULT(mult(0(X),Y),x6) -> MULT(U181(isBin(X),X,Y),x6) 27.20/28.26 MULT(mult(1(X),Y),x6) -> U191#(isBin(X),X,Y) 27.20/28.26 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.20/28.26 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.26 MULT(1(X),Y) -> U191#(isBin(X),X,Y) 27.20/28.26 -> EAxioms: 27.20/28.26 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) = mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.26 union(x6,x7) = union(x7,x6) 27.20/28.26 -> Usable Equations: 27.20/28.26 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) = mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 -> Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.26 U172(tt) -> z 27.20/28.26 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.26 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.26 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.26 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.26 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.26 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.26 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.26 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.26 U242(tt,X) -> X 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.26 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.26 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.26 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.26 U262(tt,X) -> X 27.20/28.26 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.26 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.26 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.26 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.26 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.26 mult(z,X) -> U171(isBin(X),X) 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.26 prod(empty) -> 1(z) 27.20/28.26 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.26 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.26 sum(empty) -> 0(z) 27.20/28.26 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.26 union(empty,X) -> X 27.20/28.26 union(X,empty) -> X 27.20/28.26 -> Usable Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.26 U172(tt) -> z 27.20/28.26 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.26 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.26 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.26 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.26 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.26 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.26 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.26 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.26 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.26 mult(z,X) -> U171(isBin(X),X) 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 -> SRules: 27.20/28.26 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.26 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.26 ->Interpretation type: 27.20/28.26 Simple mixed 27.20/28.26 ->Coefficients: 27.20/28.26 Natural Numbers 27.20/28.26 ->Dimension: 27.20/28.26 1 27.20/28.26 ->Bound: 27.20/28.26 1 27.20/28.26 ->Interpretation: 27.20/28.26 27.20/28.26 [0](X) = X + 1 27.20/28.26 [U101](X1,X2) = 1 27.20/28.26 [U102](X1,X2) = 1 27.20/28.26 [U103](X) = 1 27.20/28.26 [U11](X1,X2) = X1.X2 + X1 + X2 + 1 27.20/28.26 [U111](X) = X.X 27.20/28.26 [U12](X1,X2) = X1.X2 + X1 + X2 + 1 27.20/28.26 [U121](X) = X 27.20/28.26 [U13](X) = X.X + 1 27.20/28.26 [U131](X1,X2) = X1 27.20/28.26 [U132](X) = 1 27.20/28.26 [U141](X1,X2) = 1 27.20/28.26 [U142](X) = 1 27.20/28.26 [U151](X) = 1 27.20/28.26 [U161](X) = 1 27.20/28.26 [U171](X1,X2) = X1.X2 27.20/28.26 [U172](X) = 0 27.20/28.26 [U181](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 27.20/28.26 [U182](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 27.20/28.26 [U183](X1,X2,X3) = X2.X3 + X1 + X2 + X3 27.20/28.26 [U184](X1,X2,X3) = X1.X2.X3 + X1.X3 + X2 + 1 27.20/28.26 [U191](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 27.20/28.26 [U192](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 27.20/28.26 [U193](X1,X2,X3) = X1.X2.X3 + X1.X3 + X2 + X3 + 1 27.20/28.26 [U194](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 27.20/28.26 [U201](X1,X2) = X1.X2 27.20/28.26 [U202](X1,X2) = X1.X2 27.20/28.26 [U21](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X2.X3 + X1 + X2 + X3 + 1 27.20/28.26 [U211](X1,X2,X3) = X1.X3 + X1 + X2 + 1 27.20/28.26 [U212](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 27.20/28.26 [U213](X1,X2,X3) = X1.X2 + X1 + X3 + 1 27.20/28.26 [U214](X1,X2,X3) = X1 + X2 + X3 + 1 27.20/28.26 [U22](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 27.20/28.26 [U221](X1,X2,X3) = X1.X3 + X1 + X2 + 1 27.20/28.26 [U222](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 27.20/28.26 [U223](X1,X2,X3) = X1.X2 + X1 + X3 + 1 27.20/28.26 [U224](X1,X2,X3) = X1.X2 + X1.X3 + X1 27.20/28.26 [U23](X1,X2,X3) = X2.X3 + X1 + X2 + 1 27.20/28.26 [U231](X1,X2,X3) = X1 + X2 + X3 + 1 27.20/28.26 [U232](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 27.20/28.26 [U233](X1,X2,X3) = X1 + X2 + X3 + 1 27.20/28.26 [U234](X1,X2,X3) = X1.X2 + X1 + X3 + 1 27.20/28.26 [U24](X1,X2,X3) = X1.X2 + 1 27.20/28.26 [U241](X1,X2) = 0 27.20/28.26 [U242](X1,X2) = 0 27.20/28.26 [U25](X1,X2) = 1 27.20/28.26 [U251](X1,X2,X3) = 0 27.20/28.26 [U252](X1,X2,X3) = 0 27.20/28.26 [U253](X1,X2,X3) = 0 27.20/28.26 [U254](X1,X2,X3) = 0 27.20/28.26 [U26](X) = 1 27.20/28.26 [U261](X1,X2) = 0 27.20/28.26 [U262](X1,X2) = 0 27.20/28.26 [U271](X1,X2,X3) = 0 27.20/28.26 [U272](X1,X2,X3) = 0 27.20/28.26 [U273](X1,X2,X3) = 0 27.20/28.26 [U274](X1,X2,X3) = 0 27.20/28.26 [U31](X) = X.X + X 27.20/28.26 [U41](X1,X2) = X1.X2 + X1 + 1 27.20/28.26 [U42](X) = X + 1 27.20/28.26 [U51](X1,X2) = 1 27.20/28.26 [U52](X1,X2) = 1 27.20/28.26 [U53](X) = X 27.20/28.26 [U61](X1,X2) = X1 27.20/28.26 [U62](X1,X2) = X1 27.20/28.26 [U63](X) = 1 27.20/28.26 [U71](X1,X2,X3) = 1 27.20/28.26 [U72](X1,X2,X3) = 1 27.20/28.26 [U73](X1,X2,X3) = 1 27.20/28.26 [U74](X1,X2,X3) = 1 27.20/28.26 [U75](X1,X2) = X1 27.20/28.26 [U76](X) = X.X 27.20/28.26 [U81](X1,X2,X3) = X1 27.20/28.26 [U82](X1,X2,X3) = 1 27.20/28.26 [U83](X1,X2,X3) = X1 27.20/28.26 [U84](X1,X2,X3) = 1 27.20/28.26 [U85](X1,X2) = 1 27.20/28.26 [U86](X) = 1 27.20/28.26 [U91](X1,X2) = 1 27.20/28.26 [U92](X1,X2) = 1 27.20/28.26 [U93](X) = 1 27.20/28.26 [isBag](X) = X.X + X + 1 27.20/28.26 [isBagKind](X) = X + 1 27.20/28.26 [isBin](X) = 1 27.20/28.26 [isBinKind](X) = 1 27.20/28.26 [mult](X1,X2) = X1.X2 + X1 + X2 27.20/28.26 [plus](X1,X2) = X1 + X2 27.20/28.26 [prod](X) = 0 27.20/28.26 [sum](X) = X.X + X + 1 27.20/28.26 [union](X1,X2) = X1.X2 + X1 + X2 + 1 27.20/28.26 [1](X) = X + 1 27.20/28.26 [empty] = 1 27.20/28.26 [singl](X) = X.X + X + 1 27.20/28.26 [tt] = 1 27.20/28.26 [z] = 0 27.20/28.26 [0#](X) = 0 27.20/28.26 [U101#](X1,X2) = 0 27.20/28.26 [U102#](X1,X2) = 0 27.20/28.26 [U103#](X) = 0 27.20/28.26 [U11#](X1,X2) = 0 27.20/28.26 [U111#](X) = 0 27.20/28.26 [U12#](X1,X2) = 0 27.20/28.26 [U121#](X) = 0 27.20/28.26 [U13#](X) = 0 27.20/28.26 [U131#](X1,X2) = 0 27.20/28.26 [U132#](X) = 0 27.20/28.26 [U141#](X1,X2) = 0 27.20/28.26 [U142#](X) = 0 27.20/28.26 [U151#](X) = 0 27.20/28.26 [U161#](X) = 0 27.20/28.26 [U171#](X1,X2) = 0 27.20/28.26 [U172#](X) = 0 27.20/28.26 [U181#](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X1 + X3 + 1 27.20/28.26 [U182#](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X1 + X3 27.20/28.26 [U183#](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1 + X3 27.20/28.26 [U184#](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 27.20/28.26 [U191#](X1,X2,X3) = X1.X2.X3 + X1.X3 + X2 + X3 + 1 27.20/28.26 [U192#](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X3 + 1 27.20/28.26 [U193#](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X3 + 1 27.20/28.26 [U194#](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X3 + 1 27.20/28.26 [U201#](X1,X2) = 0 27.20/28.26 [U202#](X1,X2) = 0 27.20/28.26 [U21#](X1,X2,X3) = 0 27.20/28.26 [U211#](X1,X2,X3) = 0 27.20/28.26 [U212#](X1,X2,X3) = 0 27.20/28.26 [U213#](X1,X2,X3) = 0 27.20/28.26 [U214#](X1,X2,X3) = 0 27.20/28.26 [U22#](X1,X2,X3) = 0 27.20/28.26 [U221#](X1,X2,X3) = 0 27.20/28.26 [U222#](X1,X2,X3) = 0 27.20/28.26 [U223#](X1,X2,X3) = 0 27.20/28.26 [U224#](X1,X2,X3) = 0 27.20/28.26 [U23#](X1,X2,X3) = 0 27.20/28.26 [U231#](X1,X2,X3) = 0 27.20/28.26 [U232#](X1,X2,X3) = 0 27.20/28.26 [U233#](X1,X2,X3) = 0 27.20/28.26 [U234#](X1,X2,X3) = 0 27.20/28.26 [U24#](X1,X2,X3) = 0 27.20/28.26 [U241#](X1,X2) = 0 27.20/28.26 [U242#](X1,X2) = 0 27.20/28.26 [U25#](X1,X2) = 0 27.20/28.26 [U251#](X1,X2,X3) = 0 27.20/28.26 [U252#](X1,X2,X3) = 0 27.20/28.26 [U253#](X1,X2,X3) = 0 27.20/28.26 [U254#](X1,X2,X3) = 0 27.20/28.26 [U26#](X) = 0 27.20/28.26 [U261#](X1,X2) = 0 27.20/28.26 [U262#](X1,X2) = 0 27.20/28.26 [U271#](X1,X2,X3) = 0 27.20/28.26 [U272#](X1,X2,X3) = 0 27.20/28.26 [U273#](X1,X2,X3) = 0 27.20/28.26 [U274#](X1,X2,X3) = 0 27.20/28.26 [U31#](X) = 0 27.20/28.26 [U41#](X1,X2) = 0 27.20/28.26 [U42#](X) = 0 27.20/28.26 [U51#](X1,X2) = 0 27.20/28.26 [U52#](X1,X2) = 0 27.20/28.26 [U53#](X) = 0 27.20/28.26 [U61#](X1,X2) = 0 27.20/28.26 [U62#](X1,X2) = 0 27.20/28.26 [U63#](X) = 0 27.20/28.26 [U71#](X1,X2,X3) = 0 27.20/28.26 [U72#](X1,X2,X3) = 0 27.20/28.26 [U73#](X1,X2,X3) = 0 27.20/28.26 [U74#](X1,X2,X3) = 0 27.20/28.26 [U75#](X1,X2) = 0 27.20/28.26 [U76#](X) = 0 27.20/28.26 [U81#](X1,X2,X3) = 0 27.20/28.26 [U82#](X1,X2,X3) = 0 27.20/28.26 [U83#](X1,X2,X3) = 0 27.20/28.26 [U84#](X1,X2,X3) = 0 27.20/28.26 [U85#](X1,X2) = 0 27.20/28.26 [U86#](X) = 0 27.20/28.26 [U91#](X1,X2) = 0 27.20/28.26 [U92#](X1,X2) = 0 27.20/28.26 [U93#](X) = 0 27.20/28.26 [ISBAG](X) = 0 27.20/28.26 [ISBAGKIND](X) = 0 27.20/28.26 [ISBIN](X) = 0 27.20/28.26 [ISBINKIND](X) = 0 27.20/28.26 [MULT](X1,X2) = X1.X2 + X1 + X2 + 1 27.20/28.26 [PLUS](X1,X2) = 0 27.20/28.26 [PROD](X) = 0 27.20/28.26 [SUM](X) = 0 27.20/28.26 [UNION](X1,X2) = 0 27.20/28.26 27.20/28.26 Problem 1.5: 27.20/28.26 27.20/28.26 SCC Processor: 27.20/28.26 -> FAxioms: 27.20/28.26 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.20/28.26 MULT(x6,x7) = MULT(x7,x6) 27.20/28.26 -> Pairs: 27.20/28.26 U182#(tt,X,Y) -> U183#(isBin(Y),X,Y) 27.20/28.26 U183#(tt,X,Y) -> U184#(isBinKind(Y),X,Y) 27.20/28.26 U184#(tt,X,Y) -> MULT(X,Y) 27.20/28.26 U191#(tt,X,Y) -> U192#(isBinKind(X),X,Y) 27.20/28.26 U192#(tt,X,Y) -> U193#(isBin(Y),X,Y) 27.20/28.26 U193#(tt,X,Y) -> U194#(isBinKind(Y),X,Y) 27.20/28.26 U194#(tt,X,Y) -> MULT(X,Y) 27.20/28.26 MULT(0(X),Y) -> U181#(isBin(X),X,Y) 27.20/28.26 MULT(mult(0(X),Y),x6) -> U181#(isBin(X),X,Y) 27.20/28.26 MULT(mult(0(X),Y),x6) -> MULT(U181(isBin(X),X,Y),x6) 27.20/28.26 MULT(mult(1(X),Y),x6) -> U191#(isBin(X),X,Y) 27.20/28.26 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.20/28.26 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.26 MULT(1(X),Y) -> U191#(isBin(X),X,Y) 27.20/28.26 -> EAxioms: 27.20/28.26 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) = mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.26 union(x6,x7) = union(x7,x6) 27.20/28.26 -> Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.26 U172(tt) -> z 27.20/28.26 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.26 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.26 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.26 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.26 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.26 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.26 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.26 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.26 U242(tt,X) -> X 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.26 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.26 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.26 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.26 U262(tt,X) -> X 27.20/28.26 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.26 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.26 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.26 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.26 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.26 mult(z,X) -> U171(isBin(X),X) 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.26 prod(empty) -> 1(z) 27.20/28.26 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.26 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.26 sum(empty) -> 0(z) 27.20/28.26 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.26 union(empty,X) -> X 27.20/28.26 union(X,empty) -> X 27.20/28.26 -> SRules: 27.20/28.26 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.26 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.26 ->Strongly Connected Components: 27.20/28.26 ->->Cycle: 27.20/28.26 ->->-> Pairs: 27.20/28.26 U191#(tt,X,Y) -> U192#(isBinKind(X),X,Y) 27.20/28.26 U192#(tt,X,Y) -> U193#(isBin(Y),X,Y) 27.20/28.26 U193#(tt,X,Y) -> U194#(isBinKind(Y),X,Y) 27.20/28.26 U194#(tt,X,Y) -> MULT(X,Y) 27.20/28.26 MULT(mult(0(X),Y),x6) -> MULT(U181(isBin(X),X,Y),x6) 27.20/28.26 MULT(mult(1(X),Y),x6) -> U191#(isBin(X),X,Y) 27.20/28.26 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.20/28.26 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.26 MULT(1(X),Y) -> U191#(isBin(X),X,Y) 27.20/28.26 -> FAxioms: 27.20/28.26 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) -> mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) -> plus(x7,x6) 27.20/28.26 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.26 union(x6,x7) -> union(x7,x6) 27.20/28.26 MULT(mult(x6,x7),x8) -> MULT(x6,mult(x7,x8)) 27.20/28.26 MULT(x6,x7) -> MULT(x7,x6) 27.20/28.26 -> EAxioms: 27.20/28.26 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) = mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.26 union(x6,x7) = union(x7,x6) 27.20/28.26 ->->-> Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.26 U172(tt) -> z 27.20/28.26 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.26 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.26 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.26 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.26 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.26 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.26 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.26 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.26 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.26 U202(tt,X) -> X 27.20/28.26 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.26 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.26 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.26 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.26 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.26 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.26 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.26 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.26 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.26 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.26 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.26 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.26 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.26 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.26 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.26 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.26 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.26 U242(tt,X) -> X 27.20/28.26 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.26 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.26 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.26 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.26 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.26 U26(tt) -> tt 27.20/28.26 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.26 U262(tt,X) -> X 27.20/28.26 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.26 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.26 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.26 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.26 U31(tt) -> tt 27.20/28.26 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.26 U42(tt) -> tt 27.20/28.26 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.26 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.26 U53(tt) -> tt 27.20/28.26 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.26 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.26 U63(tt) -> tt 27.20/28.26 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.26 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.26 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.26 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.26 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.26 U76(tt) -> tt 27.20/28.26 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.26 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.26 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.26 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.26 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.26 U86(tt) -> tt 27.20/28.26 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.26 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.26 U93(tt) -> tt 27.20/28.26 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.26 isBag(empty) -> tt 27.20/28.26 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.26 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.26 isBagKind(empty) -> tt 27.20/28.26 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.26 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.26 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.26 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.26 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.26 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.26 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.26 isBin(z) -> tt 27.20/28.26 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.26 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.26 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.26 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.26 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.26 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.26 isBinKind(z) -> tt 27.20/28.26 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.26 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.26 mult(z,X) -> U171(isBin(X),X) 27.20/28.26 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.26 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.26 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.26 plus(z,X) -> U201(isBin(X),X) 27.20/28.26 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.26 prod(empty) -> 1(z) 27.20/28.26 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.26 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.26 sum(empty) -> 0(z) 27.20/28.26 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.26 union(empty,X) -> X 27.20/28.26 union(X,empty) -> X 27.20/28.26 -> SRules: 27.20/28.26 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.26 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.26 27.20/28.26 Problem 1.5: 27.20/28.26 27.20/28.26 Reduction Pairs Processor: 27.20/28.26 -> FAxioms: 27.20/28.26 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.20/28.26 MULT(x6,x7) = MULT(x7,x6) 27.20/28.26 -> Pairs: 27.20/28.26 U191#(tt,X,Y) -> U192#(isBinKind(X),X,Y) 27.20/28.26 U192#(tt,X,Y) -> U193#(isBin(Y),X,Y) 27.20/28.26 U193#(tt,X,Y) -> U194#(isBinKind(Y),X,Y) 27.20/28.26 U194#(tt,X,Y) -> MULT(X,Y) 27.20/28.26 MULT(mult(0(X),Y),x6) -> MULT(U181(isBin(X),X,Y),x6) 27.20/28.26 MULT(mult(1(X),Y),x6) -> U191#(isBin(X),X,Y) 27.20/28.26 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.20/28.26 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.26 MULT(1(X),Y) -> U191#(isBin(X),X,Y) 27.20/28.26 -> EAxioms: 27.20/28.26 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) = mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.26 union(x6,x7) = union(x7,x6) 27.20/28.26 -> Usable Equations: 27.20/28.26 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.26 mult(x6,x7) = mult(x7,x6) 27.20/28.26 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.26 plus(x6,x7) = plus(x7,x6) 27.20/28.26 -> Rules: 27.20/28.26 0(z) -> z 27.20/28.26 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.26 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.26 U103(tt) -> tt 27.20/28.26 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.26 U111(tt) -> tt 27.20/28.26 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.26 U121(tt) -> tt 27.20/28.26 U13(tt) -> tt 27.20/28.26 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.26 U132(tt) -> tt 27.20/28.26 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.26 U142(tt) -> tt 27.20/28.26 U151(tt) -> tt 27.20/28.26 U161(tt) -> tt 27.20/28.26 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.26 U172(tt) -> z 27.20/28.26 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.26 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.26 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.26 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.26 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.26 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.27 U242(tt,X) -> X 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.27 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.27 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.27 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.27 U262(tt,X) -> X 27.20/28.27 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.27 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.27 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.27 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.27 prod(empty) -> 1(z) 27.20/28.27 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.27 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.27 sum(empty) -> 0(z) 27.20/28.27 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.27 union(empty,X) -> X 27.20/28.27 union(X,empty) -> X 27.20/28.27 -> Usable Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 -> SRules: 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.27 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.27 ->Interpretation type: 27.20/28.27 Simple mixed 27.20/28.27 ->Coefficients: 27.20/28.27 Natural Numbers 27.20/28.27 ->Dimension: 27.20/28.27 1 27.20/28.27 ->Bound: 27.20/28.27 1 27.20/28.27 ->Interpretation: 27.20/28.27 27.20/28.27 [0](X) = X + 1 27.20/28.27 [U101](X1,X2) = 1 27.20/28.27 [U102](X1,X2) = 1 27.20/28.27 [U103](X) = 1 27.20/28.27 [U11](X1,X2) = X1.X2 + X1 + X2 + 1 27.20/28.27 [U111](X) = X 27.20/28.27 [U12](X1,X2) = X1.X2 + X1 + X2 + 1 27.20/28.27 [U121](X) = 1 27.20/28.27 [U13](X) = X + 1 27.20/28.27 [U131](X1,X2) = 1 27.20/28.27 [U132](X) = X.X 27.20/28.27 [U141](X1,X2) = 1 27.20/28.27 [U142](X) = 1 27.20/28.27 [U151](X) = 1 27.20/28.27 [U161](X) = 1 27.20/28.27 [U171](X1,X2) = X1.X2 27.20/28.27 [U172](X) = 0 27.20/28.27 [U181](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 27.20/28.27 [U182](X1,X2,X3) = X1.X3 + X2.X3 + X2 + X3 + 1 27.20/28.27 [U183](X1,X2,X3) = X1.X3 + X2.X3 + X2 + X3 + 1 27.20/28.27 [U184](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X1 + X3 27.20/28.27 [U191](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X1 + X3 27.20/28.27 [U192](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X1 + X3 27.20/28.27 [U193](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 27.20/28.27 [U194](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X1 + X3 27.20/28.27 [U201](X1,X2) = X1.X2 27.20/28.27 [U202](X1,X2) = X2 27.20/28.27 [U21](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 27.20/28.27 [U211](X1,X2,X3) = X1.X3 + X1 + X2 + 1 27.20/28.27 [U212](X1,X2,X3) = X1.X2 + X1 + X3 + 1 27.20/28.27 [U213](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 27.20/28.27 [U214](X1,X2,X3) = X1.X2 + X1.X3 + X1 27.20/28.27 [U22](X1,X2,X3) = X2.X3 + X2 + X3 + 1 27.20/28.27 [U221](X1,X2,X3) = X1 + X2 + X3 + 1 27.20/28.27 [U222](X1,X2,X3) = X2 + X3 + 1 27.20/28.27 [U223](X1,X2,X3) = X1.X2 + X1.X3 + 1 27.20/28.27 [U224](X1,X2,X3) = X1.X2 + X3 + 1 27.20/28.27 [U23](X1,X2,X3) = X2.X3 + X2 + X3 + 1 27.20/28.27 [U231](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 27.20/28.27 [U232](X1,X2,X3) = X1.X3 + X1 + X2 + 1 27.20/28.27 [U233](X1,X2,X3) = X1 + X2 + X3 + 1 27.20/28.27 [U234](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 27.20/28.27 [U24](X1,X2,X3) = X2 + 1 27.20/28.27 [U241](X1,X2) = 0 27.20/28.27 [U242](X1,X2) = 0 27.20/28.27 [U25](X1,X2) = 1 27.20/28.27 [U251](X1,X2,X3) = 0 27.20/28.27 [U252](X1,X2,X3) = 0 27.20/28.27 [U253](X1,X2,X3) = 0 27.20/28.27 [U254](X1,X2,X3) = 0 27.20/28.27 [U26](X) = 1 27.20/28.27 [U261](X1,X2) = 0 27.20/28.27 [U262](X1,X2) = 0 27.20/28.27 [U271](X1,X2,X3) = 0 27.20/28.27 [U272](X1,X2,X3) = 0 27.20/28.27 [U273](X1,X2,X3) = 0 27.20/28.27 [U274](X1,X2,X3) = 0 27.20/28.27 [U31](X) = X + 1 27.20/28.27 [U41](X1,X2) = X1 + X2 + 1 27.20/28.27 [U42](X) = 1 27.20/28.27 [U51](X1,X2) = X1 27.20/28.27 [U52](X1,X2) = X1 27.20/28.27 [U53](X) = 1 27.20/28.27 [U61](X1,X2) = 1 27.20/28.27 [U62](X1,X2) = X1 27.20/28.27 [U63](X) = X.X 27.20/28.27 [U71](X1,X2,X3) = X1 27.20/28.27 [U72](X1,X2,X3) = 1 27.20/28.27 [U73](X1,X2,X3) = 1 27.20/28.27 [U74](X1,X2,X3) = X1 27.20/28.27 [U75](X1,X2) = 1 27.20/28.27 [U76](X) = 1 27.20/28.27 [U81](X1,X2,X3) = 1 27.20/28.27 [U82](X1,X2,X3) = X1 27.20/28.27 [U83](X1,X2,X3) = X1 27.20/28.27 [U84](X1,X2,X3) = X1 27.20/28.27 [U85](X1,X2) = 1 27.20/28.27 [U86](X) = 1 27.20/28.27 [U91](X1,X2) = 1 27.20/28.27 [U92](X1,X2) = 1 27.20/28.27 [U93](X) = 1 27.20/28.27 [isBag](X) = X.X + X + 1 27.20/28.27 [isBagKind](X) = X.X + X 27.20/28.27 [isBin](X) = 1 27.20/28.27 [isBinKind](X) = 1 27.20/28.27 [mult](X1,X2) = X1.X2 + X1 + X2 27.20/28.27 [plus](X1,X2) = X1 + X2 27.20/28.27 [prod](X) = X.X + X + 1 27.20/28.27 [sum](X) = X + 1 27.20/28.27 [union](X1,X2) = X1.X2 + X1 + X2 + 1 27.20/28.27 [1](X) = X + 1 27.20/28.27 [empty] = 1 27.20/28.27 [singl](X) = X.X + X + 1 27.20/28.27 [tt] = 1 27.20/28.27 [z] = 0 27.20/28.27 [0#](X) = 0 27.20/28.27 [U101#](X1,X2) = 0 27.20/28.27 [U102#](X1,X2) = 0 27.20/28.27 [U103#](X) = 0 27.20/28.27 [U11#](X1,X2) = 0 27.20/28.27 [U111#](X) = 0 27.20/28.27 [U12#](X1,X2) = 0 27.20/28.27 [U121#](X) = 0 27.20/28.27 [U13#](X) = 0 27.20/28.27 [U131#](X1,X2) = 0 27.20/28.27 [U132#](X) = 0 27.20/28.27 [U141#](X1,X2) = 0 27.20/28.27 [U142#](X) = 0 27.20/28.27 [U151#](X) = 0 27.20/28.27 [U161#](X) = 0 27.20/28.27 [U171#](X1,X2) = 0 27.20/28.27 [U172#](X) = 0 27.20/28.27 [U181#](X1,X2,X3) = 0 27.20/28.27 [U182#](X1,X2,X3) = 0 27.20/28.27 [U183#](X1,X2,X3) = 0 27.20/28.27 [U184#](X1,X2,X3) = 0 27.20/28.27 [U191#](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 + 1 27.20/28.27 [U192#](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 27.20/28.27 [U193#](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 27.20/28.27 [U194#](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X3 + 1 27.20/28.27 [U201#](X1,X2) = 0 27.20/28.27 [U202#](X1,X2) = 0 27.20/28.27 [U21#](X1,X2,X3) = 0 27.20/28.27 [U211#](X1,X2,X3) = 0 27.20/28.27 [U212#](X1,X2,X3) = 0 27.20/28.27 [U213#](X1,X2,X3) = 0 27.20/28.27 [U214#](X1,X2,X3) = 0 27.20/28.27 [U22#](X1,X2,X3) = 0 27.20/28.27 [U221#](X1,X2,X3) = 0 27.20/28.27 [U222#](X1,X2,X3) = 0 27.20/28.27 [U223#](X1,X2,X3) = 0 27.20/28.27 [U224#](X1,X2,X3) = 0 27.20/28.27 [U23#](X1,X2,X3) = 0 27.20/28.27 [U231#](X1,X2,X3) = 0 27.20/28.27 [U232#](X1,X2,X3) = 0 27.20/28.27 [U233#](X1,X2,X3) = 0 27.20/28.27 [U234#](X1,X2,X3) = 0 27.20/28.27 [U24#](X1,X2,X3) = 0 27.20/28.27 [U241#](X1,X2) = 0 27.20/28.27 [U242#](X1,X2) = 0 27.20/28.27 [U25#](X1,X2) = 0 27.20/28.27 [U251#](X1,X2,X3) = 0 27.20/28.27 [U252#](X1,X2,X3) = 0 27.20/28.27 [U253#](X1,X2,X3) = 0 27.20/28.27 [U254#](X1,X2,X3) = 0 27.20/28.27 [U26#](X) = 0 27.20/28.27 [U261#](X1,X2) = 0 27.20/28.27 [U262#](X1,X2) = 0 27.20/28.27 [U271#](X1,X2,X3) = 0 27.20/28.27 [U272#](X1,X2,X3) = 0 27.20/28.27 [U273#](X1,X2,X3) = 0 27.20/28.27 [U274#](X1,X2,X3) = 0 27.20/28.27 [U31#](X) = 0 27.20/28.27 [U41#](X1,X2) = 0 27.20/28.27 [U42#](X) = 0 27.20/28.27 [U51#](X1,X2) = 0 27.20/28.27 [U52#](X1,X2) = 0 27.20/28.27 [U53#](X) = 0 27.20/28.27 [U61#](X1,X2) = 0 27.20/28.27 [U62#](X1,X2) = 0 27.20/28.27 [U63#](X) = 0 27.20/28.27 [U71#](X1,X2,X3) = 0 27.20/28.27 [U72#](X1,X2,X3) = 0 27.20/28.27 [U73#](X1,X2,X3) = 0 27.20/28.27 [U74#](X1,X2,X3) = 0 27.20/28.27 [U75#](X1,X2) = 0 27.20/28.27 [U76#](X) = 0 27.20/28.27 [U81#](X1,X2,X3) = 0 27.20/28.27 [U82#](X1,X2,X3) = 0 27.20/28.27 [U83#](X1,X2,X3) = 0 27.20/28.27 [U84#](X1,X2,X3) = 0 27.20/28.27 [U85#](X1,X2) = 0 27.20/28.27 [U86#](X) = 0 27.20/28.27 [U91#](X1,X2) = 0 27.20/28.27 [U92#](X1,X2) = 0 27.20/28.27 [U93#](X) = 0 27.20/28.27 [ISBAG](X) = 0 27.20/28.27 [ISBAGKIND](X) = 0 27.20/28.27 [ISBIN](X) = 0 27.20/28.27 [ISBINKIND](X) = 0 27.20/28.27 [MULT](X1,X2) = X1.X2 + X1 + X2 + 1 27.20/28.27 [PLUS](X1,X2) = 0 27.20/28.27 [PROD](X) = 0 27.20/28.27 [SUM](X) = 0 27.20/28.27 [UNION](X1,X2) = 0 27.20/28.27 27.20/28.27 Problem 1.5: 27.20/28.27 27.20/28.27 SCC Processor: 27.20/28.27 -> FAxioms: 27.20/28.27 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.20/28.27 MULT(x6,x7) = MULT(x7,x6) 27.20/28.27 -> Pairs: 27.20/28.27 U192#(tt,X,Y) -> U193#(isBin(Y),X,Y) 27.20/28.27 U193#(tt,X,Y) -> U194#(isBinKind(Y),X,Y) 27.20/28.27 U194#(tt,X,Y) -> MULT(X,Y) 27.20/28.27 MULT(mult(0(X),Y),x6) -> MULT(U181(isBin(X),X,Y),x6) 27.20/28.27 MULT(mult(1(X),Y),x6) -> U191#(isBin(X),X,Y) 27.20/28.27 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.20/28.27 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.27 MULT(1(X),Y) -> U191#(isBin(X),X,Y) 27.20/28.27 -> EAxioms: 27.20/28.27 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) = mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) = plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) = union(x7,x6) 27.20/28.27 -> Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.27 U242(tt,X) -> X 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.27 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.27 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.27 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.27 U262(tt,X) -> X 27.20/28.27 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.27 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.27 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.27 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.27 prod(empty) -> 1(z) 27.20/28.27 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.27 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.27 sum(empty) -> 0(z) 27.20/28.27 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.27 union(empty,X) -> X 27.20/28.27 union(X,empty) -> X 27.20/28.27 -> SRules: 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.27 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.27 ->Strongly Connected Components: 27.20/28.27 ->->Cycle: 27.20/28.27 ->->-> Pairs: 27.20/28.27 MULT(mult(0(X),Y),x6) -> MULT(U181(isBin(X),X,Y),x6) 27.20/28.27 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.20/28.27 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.27 -> FAxioms: 27.20/28.27 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) -> mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) -> plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) -> union(x7,x6) 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,mult(x7,x8)) 27.20/28.27 MULT(x6,x7) -> MULT(x7,x6) 27.20/28.27 -> EAxioms: 27.20/28.27 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) = mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) = plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) = union(x7,x6) 27.20/28.27 ->->-> Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.27 U242(tt,X) -> X 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.27 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.27 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.27 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.27 U262(tt,X) -> X 27.20/28.27 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.27 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.27 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.27 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.27 prod(empty) -> 1(z) 27.20/28.27 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.27 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.27 sum(empty) -> 0(z) 27.20/28.27 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.27 union(empty,X) -> X 27.20/28.27 union(X,empty) -> X 27.20/28.27 -> SRules: 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.27 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.27 27.20/28.27 Problem 1.5: 27.20/28.27 27.20/28.27 Reduction Pairs Processor: 27.20/28.27 -> FAxioms: 27.20/28.27 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.20/28.27 MULT(x6,x7) = MULT(x7,x6) 27.20/28.27 -> Pairs: 27.20/28.27 MULT(mult(0(X),Y),x6) -> MULT(U181(isBin(X),X,Y),x6) 27.20/28.27 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.20/28.27 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.27 -> EAxioms: 27.20/28.27 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) = mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) = plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) = union(x7,x6) 27.20/28.27 -> Usable Equations: 27.20/28.27 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) = mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) = plus(x7,x6) 27.20/28.27 -> Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.27 U242(tt,X) -> X 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.27 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.27 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.27 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.27 U262(tt,X) -> X 27.20/28.27 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.27 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.27 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.27 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.27 prod(empty) -> 1(z) 27.20/28.27 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.27 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.27 sum(empty) -> 0(z) 27.20/28.27 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.27 union(empty,X) -> X 27.20/28.27 union(X,empty) -> X 27.20/28.27 -> Usable Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 -> SRules: 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.27 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.27 ->Interpretation type: 27.20/28.27 Linear 27.20/28.27 ->Coefficients: 27.20/28.27 Natural Numbers 27.20/28.27 ->Dimension: 27.20/28.27 1 27.20/28.27 ->Bound: 27.20/28.27 2 27.20/28.27 ->Interpretation: 27.20/28.27 27.20/28.27 [0](X) = 2 27.20/28.27 [U101](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.27 [U102](X1,X2) = 2.X1 + 2 27.20/28.27 [U103](X) = 2 27.20/28.27 [U11](X1,X2) = 2.X2 + 2 27.20/28.27 [U111](X) = 2 27.20/28.27 [U12](X1,X2) = 2.X2 + 2 27.20/28.27 [U121](X) = 2 27.20/28.27 [U13](X) = X + 2 27.20/28.27 [U131](X1,X2) = X1 + X2 + 2 27.20/28.27 [U132](X) = X + 2 27.20/28.27 [U141](X1,X2) = X1 + X2 27.20/28.27 [U142](X) = X + 1 27.20/28.27 [U151](X) = X + 1 27.20/28.27 [U161](X) = 2.X 27.20/28.27 [U171](X1,X2) = X2 + 2 27.20/28.27 [U172](X) = X + 2 27.20/28.27 [U181](X1,X2,X3) = X3 + 2 27.20/28.27 [U182](X1,X2,X3) = X3 + 2 27.20/28.27 [U183](X1,X2,X3) = X3 + 2 27.20/28.27 [U184](X1,X2,X3) = X1 + 2 27.20/28.27 [U191](X1,X2,X3) = X3 + 2 27.20/28.27 [U192](X1,X2,X3) = X3 + 2 27.20/28.27 [U193](X1,X2,X3) = X3 + 2 27.20/28.27 [U194](X1,X2,X3) = X3 + 2 27.20/28.27 [U201](X1,X2) = X2 + 2 27.20/28.27 [U202](X1,X2) = X2 + 2 27.20/28.27 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 27.20/28.27 [U211](X1,X2,X3) = 2 27.20/28.27 [U212](X1,X2,X3) = 2 27.20/28.27 [U213](X1,X2,X3) = 2 27.20/28.27 [U214](X1,X2,X3) = 2 27.20/28.27 [U22](X1,X2,X3) = X1 + X2 + 2.X3 + 2 27.20/28.27 [U221](X1,X2,X3) = 2 27.20/28.27 [U222](X1,X2,X3) = 2 27.20/28.27 [U223](X1,X2,X3) = 2 27.20/28.27 [U224](X1,X2,X3) = 2 27.20/28.27 [U23](X1,X2,X3) = 2.X1 + X2 + 2 27.20/28.27 [U231](X1,X2,X3) = 2 27.20/28.27 [U232](X1,X2,X3) = 2 27.20/28.27 [U233](X1,X2,X3) = 2 27.20/28.27 [U234](X1,X2,X3) = 2 27.20/28.27 [U24](X1,X2,X3) = X2 + 2 27.20/28.27 [U241](X1,X2) = 0 27.20/28.27 [U242](X1,X2) = 0 27.20/28.27 [U25](X1,X2) = 2 27.20/28.27 [U251](X1,X2,X3) = 0 27.20/28.27 [U252](X1,X2,X3) = 0 27.20/28.27 [U253](X1,X2,X3) = 0 27.20/28.27 [U254](X1,X2,X3) = 0 27.20/28.27 [U26](X) = 2 27.20/28.27 [U261](X1,X2) = 0 27.20/28.27 [U262](X1,X2) = 0 27.20/28.27 [U271](X1,X2,X3) = 0 27.20/28.27 [U272](X1,X2,X3) = 0 27.20/28.27 [U273](X1,X2,X3) = 0 27.20/28.27 [U274](X1,X2,X3) = 0 27.20/28.27 [U31](X) = X + 1 27.20/28.27 [U41](X1,X2) = X1 + 2 27.20/28.27 [U42](X) = 2 27.20/28.27 [U51](X1,X2) = 2 27.20/28.27 [U52](X1,X2) = 2 27.20/28.27 [U53](X) = 2 27.20/28.27 [U61](X1,X2) = 2 27.20/28.27 [U62](X1,X2) = 2 27.20/28.27 [U63](X) = 2 27.20/28.27 [U71](X1,X2,X3) = X1 + X2 + 2.X3 + 2 27.20/28.27 [U72](X1,X2,X3) = X1 + 2.X3 + 2 27.20/28.27 [U73](X1,X2,X3) = 2.X3 + 2 27.20/28.27 [U74](X1,X2,X3) = 2.X1 + 2 27.20/28.27 [U75](X1,X2) = 2 27.20/28.27 [U76](X) = 2 27.20/28.27 [U81](X1,X2,X3) = 2.X1 + 2.X3 27.20/28.27 [U82](X1,X2,X3) = 2.X3 + 2 27.20/28.27 [U83](X1,X2,X3) = X1 + X3 + 1 27.20/28.27 [U84](X1,X2,X3) = X1 + 2 27.20/28.27 [U85](X1,X2) = 2 27.20/28.27 [U86](X) = 2 27.20/28.27 [U91](X1,X2) = X1 + X2 + 2 27.20/28.27 [U92](X1,X2) = X1 + 2 27.20/28.27 [U93](X) = 2 27.20/28.27 [isBag](X) = 2.X + 2 27.20/28.27 [isBagKind](X) = X + 1 27.20/28.27 [isBin](X) = 2.X 27.20/28.27 [isBinKind](X) = X 27.20/28.27 [mult](X1,X2) = X1 + X2 + 2 27.20/28.27 [plus](X1,X2) = X1 + X2 27.20/28.27 [prod](X) = X + 2 27.20/28.27 [sum](X) = 2.X + 2 27.20/28.27 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.27 [1](X) = 2 27.20/28.27 [empty] = 2 27.20/28.27 [singl](X) = 2.X 27.20/28.27 [tt] = 2 27.20/28.27 [z] = 2 27.20/28.27 [0#](X) = 0 27.20/28.27 [U101#](X1,X2) = 0 27.20/28.27 [U102#](X1,X2) = 0 27.20/28.27 [U103#](X) = 0 27.20/28.27 [U11#](X1,X2) = 0 27.20/28.27 [U111#](X) = 0 27.20/28.27 [U12#](X1,X2) = 0 27.20/28.27 [U121#](X) = 0 27.20/28.27 [U13#](X) = 0 27.20/28.27 [U131#](X1,X2) = 0 27.20/28.27 [U132#](X) = 0 27.20/28.27 [U141#](X1,X2) = 0 27.20/28.27 [U142#](X) = 0 27.20/28.27 [U151#](X) = 0 27.20/28.27 [U161#](X) = 0 27.20/28.27 [U171#](X1,X2) = 0 27.20/28.27 [U172#](X) = 0 27.20/28.27 [U181#](X1,X2,X3) = 0 27.20/28.27 [U182#](X1,X2,X3) = 0 27.20/28.27 [U183#](X1,X2,X3) = 0 27.20/28.27 [U184#](X1,X2,X3) = 0 27.20/28.27 [U191#](X1,X2,X3) = 0 27.20/28.27 [U192#](X1,X2,X3) = 0 27.20/28.27 [U193#](X1,X2,X3) = 0 27.20/28.27 [U194#](X1,X2,X3) = 0 27.20/28.27 [U201#](X1,X2) = 0 27.20/28.27 [U202#](X1,X2) = 0 27.20/28.27 [U21#](X1,X2,X3) = 0 27.20/28.27 [U211#](X1,X2,X3) = 0 27.20/28.27 [U212#](X1,X2,X3) = 0 27.20/28.27 [U213#](X1,X2,X3) = 0 27.20/28.27 [U214#](X1,X2,X3) = 0 27.20/28.27 [U22#](X1,X2,X3) = 0 27.20/28.27 [U221#](X1,X2,X3) = 0 27.20/28.27 [U222#](X1,X2,X3) = 0 27.20/28.27 [U223#](X1,X2,X3) = 0 27.20/28.27 [U224#](X1,X2,X3) = 0 27.20/28.27 [U23#](X1,X2,X3) = 0 27.20/28.27 [U231#](X1,X2,X3) = 0 27.20/28.27 [U232#](X1,X2,X3) = 0 27.20/28.27 [U233#](X1,X2,X3) = 0 27.20/28.27 [U234#](X1,X2,X3) = 0 27.20/28.27 [U24#](X1,X2,X3) = 0 27.20/28.27 [U241#](X1,X2) = 0 27.20/28.27 [U242#](X1,X2) = 0 27.20/28.27 [U25#](X1,X2) = 0 27.20/28.27 [U251#](X1,X2,X3) = 0 27.20/28.27 [U252#](X1,X2,X3) = 0 27.20/28.27 [U253#](X1,X2,X3) = 0 27.20/28.27 [U254#](X1,X2,X3) = 0 27.20/28.27 [U26#](X) = 0 27.20/28.27 [U261#](X1,X2) = 0 27.20/28.27 [U262#](X1,X2) = 0 27.20/28.27 [U271#](X1,X2,X3) = 0 27.20/28.27 [U272#](X1,X2,X3) = 0 27.20/28.27 [U273#](X1,X2,X3) = 0 27.20/28.27 [U274#](X1,X2,X3) = 0 27.20/28.27 [U31#](X) = 0 27.20/28.27 [U41#](X1,X2) = 0 27.20/28.27 [U42#](X) = 0 27.20/28.27 [U51#](X1,X2) = 0 27.20/28.27 [U52#](X1,X2) = 0 27.20/28.27 [U53#](X) = 0 27.20/28.27 [U61#](X1,X2) = 0 27.20/28.27 [U62#](X1,X2) = 0 27.20/28.27 [U63#](X) = 0 27.20/28.27 [U71#](X1,X2,X3) = 0 27.20/28.27 [U72#](X1,X2,X3) = 0 27.20/28.27 [U73#](X1,X2,X3) = 0 27.20/28.27 [U74#](X1,X2,X3) = 0 27.20/28.27 [U75#](X1,X2) = 0 27.20/28.27 [U76#](X) = 0 27.20/28.27 [U81#](X1,X2,X3) = 0 27.20/28.27 [U82#](X1,X2,X3) = 0 27.20/28.27 [U83#](X1,X2,X3) = 0 27.20/28.27 [U84#](X1,X2,X3) = 0 27.20/28.27 [U85#](X1,X2) = 0 27.20/28.27 [U86#](X) = 0 27.20/28.27 [U91#](X1,X2) = 0 27.20/28.27 [U92#](X1,X2) = 0 27.20/28.27 [U93#](X) = 0 27.20/28.27 [ISBAG](X) = 0 27.20/28.27 [ISBAGKIND](X) = 0 27.20/28.27 [ISBIN](X) = 0 27.20/28.27 [ISBINKIND](X) = 0 27.20/28.27 [MULT](X1,X2) = 2.X1 + 2.X2 27.20/28.27 [PLUS](X1,X2) = 0 27.20/28.27 [PROD](X) = 0 27.20/28.27 [SUM](X) = 0 27.20/28.27 [UNION](X1,X2) = 0 27.20/28.27 27.20/28.27 Problem 1.5: 27.20/28.27 27.20/28.27 SCC Processor: 27.20/28.27 -> FAxioms: 27.20/28.27 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.20/28.27 MULT(x6,x7) = MULT(x7,x6) 27.20/28.27 -> Pairs: 27.20/28.27 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.20/28.27 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.27 -> EAxioms: 27.20/28.27 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) = mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) = plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) = union(x7,x6) 27.20/28.27 -> Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.27 U242(tt,X) -> X 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.27 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.27 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.27 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.27 U262(tt,X) -> X 27.20/28.27 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.27 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.27 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.27 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.27 prod(empty) -> 1(z) 27.20/28.27 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.27 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.27 sum(empty) -> 0(z) 27.20/28.27 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.27 union(empty,X) -> X 27.20/28.27 union(X,empty) -> X 27.20/28.27 -> SRules: 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.27 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.27 ->Strongly Connected Components: 27.20/28.27 ->->Cycle: 27.20/28.27 ->->-> Pairs: 27.20/28.27 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.20/28.27 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.27 -> FAxioms: 27.20/28.27 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) -> mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) -> plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) -> union(x7,x6) 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,mult(x7,x8)) 27.20/28.27 MULT(x6,x7) -> MULT(x7,x6) 27.20/28.27 -> EAxioms: 27.20/28.27 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) = mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) = plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) = union(x7,x6) 27.20/28.27 ->->-> Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.27 U242(tt,X) -> X 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.27 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.27 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.27 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.27 U262(tt,X) -> X 27.20/28.27 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.27 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.27 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.27 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.27 prod(empty) -> 1(z) 27.20/28.27 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.27 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.27 sum(empty) -> 0(z) 27.20/28.27 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.27 union(empty,X) -> X 27.20/28.27 union(X,empty) -> X 27.20/28.27 -> SRules: 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.27 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.27 27.20/28.27 Problem 1.5: 27.20/28.27 27.20/28.27 Reduction Pairs Processor: 27.20/28.27 -> FAxioms: 27.20/28.27 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.20/28.27 MULT(x6,x7) = MULT(x7,x6) 27.20/28.27 -> Pairs: 27.20/28.27 MULT(mult(1(X),Y),x6) -> MULT(U191(isBin(X),X,Y),x6) 27.20/28.27 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.27 -> EAxioms: 27.20/28.27 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) = mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) = plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) = union(x7,x6) 27.20/28.27 -> Usable Equations: 27.20/28.27 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) = mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) = plus(x7,x6) 27.20/28.27 -> Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.27 U242(tt,X) -> X 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.27 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.27 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.27 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.27 U262(tt,X) -> X 27.20/28.27 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.27 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.27 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.27 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.27 prod(empty) -> 1(z) 27.20/28.27 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.27 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.27 sum(empty) -> 0(z) 27.20/28.27 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.27 union(empty,X) -> X 27.20/28.27 union(X,empty) -> X 27.20/28.27 -> Usable Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 -> SRules: 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.27 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.27 ->Interpretation type: 27.20/28.27 Linear 27.20/28.27 ->Coefficients: 27.20/28.27 Natural Numbers 27.20/28.27 ->Dimension: 27.20/28.27 1 27.20/28.27 ->Bound: 27.20/28.27 2 27.20/28.27 ->Interpretation: 27.20/28.27 27.20/28.27 [0](X) = 0 27.20/28.27 [U101](X1,X2) = X1 + 2.X2 + 1 27.20/28.27 [U102](X1,X2) = 2.X2 + 2 27.20/28.27 [U103](X) = X + 2 27.20/28.27 [U11](X1,X2) = X1 + 2.X2 + 2 27.20/28.27 [U111](X) = 2 27.20/28.27 [U12](X1,X2) = 2.X2 + 2 27.20/28.27 [U121](X) = 2 27.20/28.27 [U13](X) = X 27.20/28.27 [U131](X1,X2) = X1 + 2.X2 + 2 27.20/28.27 [U132](X) = X + 2 27.20/28.27 [U141](X1,X2) = X1 + 2.X2 + 2 27.20/28.27 [U142](X) = X + 2 27.20/28.27 [U151](X) = X + 1 27.20/28.27 [U161](X) = 2.X + 2 27.20/28.27 [U171](X1,X2) = 2 27.20/28.27 [U172](X) = 2 27.20/28.27 [U181](X1,X2,X3) = X3 + 2 27.20/28.27 [U182](X1,X2,X3) = X3 + 2 27.20/28.27 [U183](X1,X2,X3) = X3 + 2 27.20/28.27 [U184](X1,X2,X3) = X3 + 2 27.20/28.27 [U191](X1,X2,X3) = X3 + 2 27.20/28.27 [U192](X1,X2,X3) = X3 + 2 27.20/28.27 [U193](X1,X2,X3) = X3 + 2 27.20/28.27 [U194](X1,X2,X3) = X3 + 2 27.20/28.27 [U201](X1,X2) = X2 27.20/28.27 [U202](X1,X2) = X2 27.20/28.27 [U21](X1,X2,X3) = X1 + 2.X2 + 2.X3 27.20/28.27 [U211](X1,X2,X3) = 2 27.20/28.27 [U212](X1,X2,X3) = 2 27.20/28.27 [U213](X1,X2,X3) = 2 27.20/28.27 [U214](X1,X2,X3) = 2 27.20/28.27 [U22](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.27 [U221](X1,X2,X3) = 2 27.20/28.27 [U222](X1,X2,X3) = 2 27.20/28.27 [U223](X1,X2,X3) = 2 27.20/28.27 [U224](X1,X2,X3) = 2 27.20/28.27 [U23](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.27 [U231](X1,X2,X3) = 2 27.20/28.27 [U232](X1,X2,X3) = 2 27.20/28.27 [U233](X1,X2,X3) = 2 27.20/28.27 [U234](X1,X2,X3) = 2 27.20/28.27 [U24](X1,X2,X3) = X1 + 2.X2 27.20/28.27 [U241](X1,X2) = 0 27.20/28.27 [U242](X1,X2) = 0 27.20/28.27 [U25](X1,X2) = X1 + 2 27.20/28.27 [U251](X1,X2,X3) = 0 27.20/28.27 [U252](X1,X2,X3) = 0 27.20/28.27 [U253](X1,X2,X3) = 0 27.20/28.27 [U254](X1,X2,X3) = 0 27.20/28.27 [U26](X) = 2 27.20/28.27 [U261](X1,X2) = 0 27.20/28.27 [U262](X1,X2) = 0 27.20/28.27 [U271](X1,X2,X3) = 0 27.20/28.27 [U272](X1,X2,X3) = 0 27.20/28.27 [U273](X1,X2,X3) = 0 27.20/28.27 [U274](X1,X2,X3) = 0 27.20/28.27 [U31](X) = X + 2 27.20/28.27 [U41](X1,X2) = 2.X1 + 2.X2 + 1 27.20/28.27 [U42](X) = X + 2 27.20/28.27 [U51](X1,X2) = 2 27.20/28.27 [U52](X1,X2) = 2 27.20/28.27 [U53](X) = 2 27.20/28.27 [U61](X1,X2) = 2 27.20/28.27 [U62](X1,X2) = 2 27.20/28.27 [U63](X) = 2 27.20/28.27 [U71](X1,X2,X3) = X1 + 2.X3 27.20/28.27 [U72](X1,X2,X3) = 2.X3 + 2 27.20/28.27 [U73](X1,X2,X3) = 2.X3 + 2 27.20/28.27 [U74](X1,X2,X3) = X1 27.20/28.27 [U75](X1,X2) = 2 27.20/28.27 [U76](X) = 2 27.20/28.27 [U81](X1,X2,X3) = X1 + 2.X3 + 2 27.20/28.27 [U82](X1,X2,X3) = 2.X3 + 2 27.20/28.27 [U83](X1,X2,X3) = 2.X3 + 2 27.20/28.27 [U84](X1,X2,X3) = 2.X3 + 2 27.20/28.27 [U85](X1,X2) = 2.X2 + 2 27.20/28.27 [U86](X) = X 27.20/28.27 [U91](X1,X2) = 2.X1 + 2 27.20/28.27 [U92](X1,X2) = 2 27.20/28.27 [U93](X) = 2 27.20/28.27 [isBag](X) = 2.X 27.20/28.27 [isBagKind](X) = 2.X + 2 27.20/28.27 [isBin](X) = 2.X + 2 27.20/28.27 [isBinKind](X) = 2.X + 2 27.20/28.27 [mult](X1,X2) = X1 + X2 + 2 27.20/28.27 [plus](X1,X2) = X1 + X2 + 2 27.20/28.27 [prod](X) = 2.X + 2 27.20/28.27 [sum](X) = 2.X + 2 27.20/28.27 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.27 [1](X) = 2 27.20/28.27 [empty] = 2 27.20/28.27 [singl](X) = 2.X + 2 27.20/28.27 [tt] = 2 27.20/28.27 [z] = 0 27.20/28.27 [0#](X) = 0 27.20/28.27 [U101#](X1,X2) = 0 27.20/28.27 [U102#](X1,X2) = 0 27.20/28.27 [U103#](X) = 0 27.20/28.27 [U11#](X1,X2) = 0 27.20/28.27 [U111#](X) = 0 27.20/28.27 [U12#](X1,X2) = 0 27.20/28.27 [U121#](X) = 0 27.20/28.27 [U13#](X) = 0 27.20/28.27 [U131#](X1,X2) = 0 27.20/28.27 [U132#](X) = 0 27.20/28.27 [U141#](X1,X2) = 0 27.20/28.27 [U142#](X) = 0 27.20/28.27 [U151#](X) = 0 27.20/28.27 [U161#](X) = 0 27.20/28.27 [U171#](X1,X2) = 0 27.20/28.27 [U172#](X) = 0 27.20/28.27 [U181#](X1,X2,X3) = 0 27.20/28.27 [U182#](X1,X2,X3) = 0 27.20/28.27 [U183#](X1,X2,X3) = 0 27.20/28.27 [U184#](X1,X2,X3) = 0 27.20/28.27 [U191#](X1,X2,X3) = 0 27.20/28.27 [U192#](X1,X2,X3) = 0 27.20/28.27 [U193#](X1,X2,X3) = 0 27.20/28.27 [U194#](X1,X2,X3) = 0 27.20/28.27 [U201#](X1,X2) = 0 27.20/28.27 [U202#](X1,X2) = 0 27.20/28.27 [U21#](X1,X2,X3) = 0 27.20/28.27 [U211#](X1,X2,X3) = 0 27.20/28.27 [U212#](X1,X2,X3) = 0 27.20/28.27 [U213#](X1,X2,X3) = 0 27.20/28.27 [U214#](X1,X2,X3) = 0 27.20/28.27 [U22#](X1,X2,X3) = 0 27.20/28.27 [U221#](X1,X2,X3) = 0 27.20/28.27 [U222#](X1,X2,X3) = 0 27.20/28.27 [U223#](X1,X2,X3) = 0 27.20/28.27 [U224#](X1,X2,X3) = 0 27.20/28.27 [U23#](X1,X2,X3) = 0 27.20/28.27 [U231#](X1,X2,X3) = 0 27.20/28.27 [U232#](X1,X2,X3) = 0 27.20/28.27 [U233#](X1,X2,X3) = 0 27.20/28.27 [U234#](X1,X2,X3) = 0 27.20/28.27 [U24#](X1,X2,X3) = 0 27.20/28.27 [U241#](X1,X2) = 0 27.20/28.27 [U242#](X1,X2) = 0 27.20/28.27 [U25#](X1,X2) = 0 27.20/28.27 [U251#](X1,X2,X3) = 0 27.20/28.27 [U252#](X1,X2,X3) = 0 27.20/28.27 [U253#](X1,X2,X3) = 0 27.20/28.27 [U254#](X1,X2,X3) = 0 27.20/28.27 [U26#](X) = 0 27.20/28.27 [U261#](X1,X2) = 0 27.20/28.27 [U262#](X1,X2) = 0 27.20/28.27 [U271#](X1,X2,X3) = 0 27.20/28.27 [U272#](X1,X2,X3) = 0 27.20/28.27 [U273#](X1,X2,X3) = 0 27.20/28.27 [U274#](X1,X2,X3) = 0 27.20/28.27 [U31#](X) = 0 27.20/28.27 [U41#](X1,X2) = 0 27.20/28.27 [U42#](X) = 0 27.20/28.27 [U51#](X1,X2) = 0 27.20/28.27 [U52#](X1,X2) = 0 27.20/28.27 [U53#](X) = 0 27.20/28.27 [U61#](X1,X2) = 0 27.20/28.27 [U62#](X1,X2) = 0 27.20/28.27 [U63#](X) = 0 27.20/28.27 [U71#](X1,X2,X3) = 0 27.20/28.27 [U72#](X1,X2,X3) = 0 27.20/28.27 [U73#](X1,X2,X3) = 0 27.20/28.27 [U74#](X1,X2,X3) = 0 27.20/28.27 [U75#](X1,X2) = 0 27.20/28.27 [U76#](X) = 0 27.20/28.27 [U81#](X1,X2,X3) = 0 27.20/28.27 [U82#](X1,X2,X3) = 0 27.20/28.27 [U83#](X1,X2,X3) = 0 27.20/28.27 [U84#](X1,X2,X3) = 0 27.20/28.27 [U85#](X1,X2) = 0 27.20/28.27 [U86#](X) = 0 27.20/28.27 [U91#](X1,X2) = 0 27.20/28.27 [U92#](X1,X2) = 0 27.20/28.27 [U93#](X) = 0 27.20/28.27 [ISBAG](X) = 0 27.20/28.27 [ISBAGKIND](X) = 0 27.20/28.27 [ISBIN](X) = 0 27.20/28.27 [ISBINKIND](X) = 0 27.20/28.27 [MULT](X1,X2) = 2.X1 + 2.X2 27.20/28.27 [PLUS](X1,X2) = 0 27.20/28.27 [PROD](X) = 0 27.20/28.27 [SUM](X) = 0 27.20/28.27 [UNION](X1,X2) = 0 27.20/28.27 27.20/28.27 Problem 1.5: 27.20/28.27 27.20/28.27 SCC Processor: 27.20/28.27 -> FAxioms: 27.20/28.27 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.20/28.27 MULT(x6,x7) = MULT(x7,x6) 27.20/28.27 -> Pairs: 27.20/28.27 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.27 -> EAxioms: 27.20/28.27 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) = mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) = plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) = union(x7,x6) 27.20/28.27 -> Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.27 U242(tt,X) -> X 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.27 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.27 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.27 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.27 U262(tt,X) -> X 27.20/28.27 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.27 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.27 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.27 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.27 U53(tt) -> tt 27.20/28.27 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.27 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.27 U63(tt) -> tt 27.20/28.27 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.27 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.27 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.27 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.27 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.27 U76(tt) -> tt 27.20/28.27 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.27 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.27 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.27 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.27 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.27 U86(tt) -> tt 27.20/28.27 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.27 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.27 U93(tt) -> tt 27.20/28.27 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.27 isBag(empty) -> tt 27.20/28.27 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.27 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.27 isBagKind(empty) -> tt 27.20/28.27 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.27 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.27 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.27 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.27 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.27 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.27 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.27 isBin(z) -> tt 27.20/28.27 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.27 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.27 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.27 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.27 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.27 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.27 isBinKind(z) -> tt 27.20/28.27 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.27 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.27 mult(z,X) -> U171(isBin(X),X) 27.20/28.27 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.27 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.27 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.27 plus(z,X) -> U201(isBin(X),X) 27.20/28.27 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.27 prod(empty) -> 1(z) 27.20/28.27 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.27 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.27 sum(empty) -> 0(z) 27.20/28.27 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.27 union(empty,X) -> X 27.20/28.27 union(X,empty) -> X 27.20/28.27 -> SRules: 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.27 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.27 ->Strongly Connected Components: 27.20/28.27 ->->Cycle: 27.20/28.27 ->->-> Pairs: 27.20/28.27 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.27 -> FAxioms: 27.20/28.27 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) -> mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) -> plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) -> union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) -> union(x7,x6) 27.20/28.27 MULT(mult(x6,x7),x8) -> MULT(x6,mult(x7,x8)) 27.20/28.27 MULT(x6,x7) -> MULT(x7,x6) 27.20/28.27 -> EAxioms: 27.20/28.27 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.27 mult(x6,x7) = mult(x7,x6) 27.20/28.27 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.27 plus(x6,x7) = plus(x7,x6) 27.20/28.27 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.27 union(x6,x7) = union(x7,x6) 27.20/28.27 ->->-> Rules: 27.20/28.27 0(z) -> z 27.20/28.27 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.27 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.27 U103(tt) -> tt 27.20/28.27 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.27 U111(tt) -> tt 27.20/28.27 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.27 U121(tt) -> tt 27.20/28.27 U13(tt) -> tt 27.20/28.27 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.27 U132(tt) -> tt 27.20/28.27 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.27 U142(tt) -> tt 27.20/28.27 U151(tt) -> tt 27.20/28.27 U161(tt) -> tt 27.20/28.27 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.27 U172(tt) -> z 27.20/28.27 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.27 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.27 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.27 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.27 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.27 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.27 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.27 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.27 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.27 U202(tt,X) -> X 27.20/28.27 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.27 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.27 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.27 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.27 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.27 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.27 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.27 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.27 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.27 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.27 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.27 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.27 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.27 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.27 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.27 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.27 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.27 U242(tt,X) -> X 27.20/28.27 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.27 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.27 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.27 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.27 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.27 U26(tt) -> tt 27.20/28.27 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.27 U262(tt,X) -> X 27.20/28.27 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.27 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.27 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.27 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.27 U31(tt) -> tt 27.20/28.27 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.27 U42(tt) -> tt 27.20/28.27 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.27 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.28 U53(tt) -> tt 27.20/28.28 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.28 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.28 U63(tt) -> tt 27.20/28.28 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.28 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.28 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.28 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.28 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.28 U76(tt) -> tt 27.20/28.28 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.28 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.28 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.28 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.28 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.28 U86(tt) -> tt 27.20/28.28 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.28 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.28 U93(tt) -> tt 27.20/28.28 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.28 isBag(empty) -> tt 27.20/28.28 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.28 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.28 isBagKind(empty) -> tt 27.20/28.28 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.28 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.28 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.28 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.28 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.28 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.28 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.28 isBin(z) -> tt 27.20/28.28 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.28 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.28 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.28 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.28 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.28 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.28 isBinKind(z) -> tt 27.20/28.28 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.28 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.28 mult(z,X) -> U171(isBin(X),X) 27.20/28.28 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.28 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.28 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.28 plus(z,X) -> U201(isBin(X),X) 27.20/28.28 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.28 prod(empty) -> 1(z) 27.20/28.28 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.28 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.28 sum(empty) -> 0(z) 27.20/28.28 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.28 union(empty,X) -> X 27.20/28.28 union(X,empty) -> X 27.20/28.28 -> SRules: 27.20/28.28 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.28 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.28 27.20/28.28 Problem 1.5: 27.20/28.28 27.20/28.28 Reduction Pairs Processor: 27.20/28.28 -> FAxioms: 27.20/28.28 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.20/28.28 MULT(x6,x7) = MULT(x7,x6) 27.20/28.28 -> Pairs: 27.20/28.28 MULT(mult(z,X),x6) -> MULT(U171(isBin(X),X),x6) 27.20/28.28 -> EAxioms: 27.20/28.28 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.28 mult(x6,x7) = mult(x7,x6) 27.20/28.28 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.28 plus(x6,x7) = plus(x7,x6) 27.20/28.28 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.28 union(x6,x7) = union(x7,x6) 27.20/28.28 -> Usable Equations: 27.20/28.28 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.28 mult(x6,x7) = mult(x7,x6) 27.20/28.28 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.28 plus(x6,x7) = plus(x7,x6) 27.20/28.28 -> Rules: 27.20/28.28 0(z) -> z 27.20/28.28 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.28 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.28 U103(tt) -> tt 27.20/28.28 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.28 U111(tt) -> tt 27.20/28.28 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.28 U121(tt) -> tt 27.20/28.28 U13(tt) -> tt 27.20/28.28 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.28 U132(tt) -> tt 27.20/28.28 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.28 U142(tt) -> tt 27.20/28.28 U151(tt) -> tt 27.20/28.28 U161(tt) -> tt 27.20/28.28 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.28 U172(tt) -> z 27.20/28.28 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.28 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.28 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.28 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.28 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.28 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.28 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.28 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.28 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.28 U202(tt,X) -> X 27.20/28.28 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.28 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.28 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.28 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.28 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.28 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.28 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.28 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.28 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.28 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.28 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.28 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.28 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.28 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.28 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.28 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.28 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.28 U242(tt,X) -> X 27.20/28.28 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.28 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.28 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.28 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.28 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.28 U26(tt) -> tt 27.20/28.28 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.28 U262(tt,X) -> X 27.20/28.28 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.28 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.28 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.28 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.28 U31(tt) -> tt 27.20/28.28 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.28 U42(tt) -> tt 27.20/28.28 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.28 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.28 U53(tt) -> tt 27.20/28.28 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.28 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.28 U63(tt) -> tt 27.20/28.28 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.28 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.28 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.28 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.28 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.28 U76(tt) -> tt 27.20/28.28 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.28 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.28 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.28 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.28 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.28 U86(tt) -> tt 27.20/28.28 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.28 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.28 U93(tt) -> tt 27.20/28.28 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.28 isBag(empty) -> tt 27.20/28.28 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.28 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.28 isBagKind(empty) -> tt 27.20/28.28 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.28 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.28 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.28 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.28 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.28 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.28 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.28 isBin(z) -> tt 27.20/28.28 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.28 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.28 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.28 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.28 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.28 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.28 isBinKind(z) -> tt 27.20/28.28 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.28 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.28 mult(z,X) -> U171(isBin(X),X) 27.20/28.28 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.28 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.28 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.28 plus(z,X) -> U201(isBin(X),X) 27.20/28.28 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.28 prod(empty) -> 1(z) 27.20/28.28 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.28 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.28 sum(empty) -> 0(z) 27.20/28.28 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.28 union(empty,X) -> X 27.20/28.28 union(X,empty) -> X 27.20/28.28 -> Usable Rules: 27.20/28.28 0(z) -> z 27.20/28.28 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.28 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.28 U103(tt) -> tt 27.20/28.28 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.28 U111(tt) -> tt 27.20/28.28 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.28 U121(tt) -> tt 27.20/28.28 U13(tt) -> tt 27.20/28.28 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.28 U132(tt) -> tt 27.20/28.28 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.28 U142(tt) -> tt 27.20/28.28 U151(tt) -> tt 27.20/28.28 U161(tt) -> tt 27.20/28.28 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.28 U172(tt) -> z 27.20/28.28 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.28 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.28 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.28 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.28 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.28 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.28 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.28 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.28 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.28 U202(tt,X) -> X 27.20/28.28 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.28 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.28 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.28 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.28 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.28 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.28 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.28 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.28 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.28 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.28 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.28 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.28 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.28 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.28 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.28 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.28 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.28 U26(tt) -> tt 27.20/28.28 U31(tt) -> tt 27.20/28.28 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.28 U42(tt) -> tt 27.20/28.28 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.28 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.28 U53(tt) -> tt 27.20/28.28 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.28 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.28 U63(tt) -> tt 27.20/28.28 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.28 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.28 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.28 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.28 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.28 U76(tt) -> tt 27.20/28.28 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.28 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.28 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.28 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.28 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.28 U86(tt) -> tt 27.20/28.28 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.28 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.28 U93(tt) -> tt 27.20/28.28 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.28 isBag(empty) -> tt 27.20/28.28 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.28 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.28 isBagKind(empty) -> tt 27.20/28.28 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.28 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.28 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.28 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.28 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.28 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.28 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.28 isBin(z) -> tt 27.20/28.28 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.28 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.28 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.28 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.28 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.28 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.28 isBinKind(z) -> tt 27.20/28.28 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.28 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.28 mult(z,X) -> U171(isBin(X),X) 27.20/28.28 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.28 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.28 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.28 plus(z,X) -> U201(isBin(X),X) 27.20/28.28 -> SRules: 27.20/28.28 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.28 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.28 ->Interpretation type: 27.20/28.28 Linear 27.20/28.28 ->Coefficients: 27.20/28.28 Natural Numbers 27.20/28.28 ->Dimension: 27.20/28.28 1 27.20/28.28 ->Bound: 27.20/28.28 2 27.20/28.28 ->Interpretation: 27.20/28.28 27.20/28.28 [0](X) = 2 27.20/28.28 [U101](X1,X2) = 2 27.20/28.28 [U102](X1,X2) = 2 27.20/28.28 [U103](X) = 2 27.20/28.28 [U11](X1,X2) = 2.X1 + 2.X2 + 1 27.20/28.28 [U111](X) = 2 27.20/28.28 [U12](X1,X2) = 2.X1 + 2 27.20/28.28 [U121](X) = 2 27.20/28.28 [U13](X) = 2 27.20/28.28 [U131](X1,X2) = X1 + X2 + 2 27.20/28.28 [U132](X) = X + 2 27.20/28.28 [U141](X1,X2) = X1 + X2 27.20/28.28 [U142](X) = X + 2 27.20/28.28 [U151](X) = 2 27.20/28.28 [U161](X) = 2 27.20/28.28 [U171](X1,X2) = X2 + 2 27.20/28.28 [U172](X) = X + 2 27.20/28.28 [U181](X1,X2,X3) = X3 + 2 27.20/28.28 [U182](X1,X2,X3) = X3 + 2 27.20/28.28 [U183](X1,X2,X3) = X3 + 2 27.20/28.28 [U184](X1,X2,X3) = X1 + 2 27.20/28.28 [U191](X1,X2,X3) = X3 + 2 27.20/28.28 [U192](X1,X2,X3) = X3 + 2 27.20/28.28 [U193](X1,X2,X3) = X3 + 2 27.20/28.28 [U194](X1,X2,X3) = X3 + 2 27.20/28.28 [U201](X1,X2) = X2 + 2 27.20/28.28 [U202](X1,X2) = X2 + 2 27.20/28.28 [U21](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.28 [U211](X1,X2,X3) = 2 27.20/28.28 [U212](X1,X2,X3) = 2 27.20/28.28 [U213](X1,X2,X3) = 2 27.20/28.28 [U214](X1,X2,X3) = 2 27.20/28.28 [U22](X1,X2,X3) = X1 + 2.X3 + 2 27.20/28.28 [U221](X1,X2,X3) = 2 27.20/28.28 [U222](X1,X2,X3) = 2 27.20/28.28 [U223](X1,X2,X3) = 2 27.20/28.28 [U224](X1,X2,X3) = 2 27.20/28.28 [U23](X1,X2,X3) = 2.X3 + 2 27.20/28.28 [U231](X1,X2,X3) = 2 27.20/28.28 [U232](X1,X2,X3) = 2 27.20/28.28 [U233](X1,X2,X3) = 2 27.20/28.28 [U234](X1,X2,X3) = 2 27.20/28.28 [U24](X1,X2,X3) = X1 27.20/28.28 [U241](X1,X2) = 0 27.20/28.28 [U242](X1,X2) = 0 27.20/28.28 [U25](X1,X2) = 2 27.20/28.28 [U251](X1,X2,X3) = 0 27.20/28.28 [U252](X1,X2,X3) = 0 27.20/28.28 [U253](X1,X2,X3) = 0 27.20/28.28 [U254](X1,X2,X3) = 0 27.20/28.28 [U26](X) = 2 27.20/28.28 [U261](X1,X2) = 0 27.20/28.28 [U262](X1,X2) = 0 27.20/28.28 [U271](X1,X2,X3) = 0 27.20/28.28 [U272](X1,X2,X3) = 0 27.20/28.28 [U273](X1,X2,X3) = 0 27.20/28.28 [U274](X1,X2,X3) = 0 27.20/28.28 [U31](X) = 2.X + 1 27.20/28.28 [U41](X1,X2) = 2.X2 + 2 27.20/28.28 [U42](X) = X + 2 27.20/28.28 [U51](X1,X2) = 2 27.20/28.28 [U52](X1,X2) = 2 27.20/28.28 [U53](X) = 2 27.20/28.28 [U61](X1,X2) = 2 27.20/28.28 [U62](X1,X2) = 2 27.20/28.28 [U63](X) = 2 27.20/28.28 [U71](X1,X2,X3) = 2.X1 + 2.X3 + 2 27.20/28.28 [U72](X1,X2,X3) = 2.X3 + 2 27.20/28.28 [U73](X1,X2,X3) = 2.X1 + 2 27.20/28.28 [U74](X1,X2,X3) = 2 27.20/28.28 [U75](X1,X2) = 2 27.20/28.28 [U76](X) = 2 27.20/28.28 [U81](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U82](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U83](X1,X2,X3) = X1 + 2.X2 + X3 + 2 27.20/28.28 [U84](X1,X2,X3) = X1 + 2.X2 + 2 27.20/28.28 [U85](X1,X2) = X1 27.20/28.28 [U86](X) = 2 27.20/28.28 [U91](X1,X2) = 2.X1 + 2 27.20/28.28 [U92](X1,X2) = 2 27.20/28.28 [U93](X) = 2 27.20/28.28 [isBag](X) = 2.X + 2 27.20/28.28 [isBagKind](X) = 2.X 27.20/28.28 [isBin](X) = 2.X + 2 27.20/28.28 [isBinKind](X) = X 27.20/28.28 [mult](X1,X2) = X1 + X2 + 2 27.20/28.28 [plus](X1,X2) = X1 + X2 27.20/28.28 [prod](X) = 2.X + 2 27.20/28.28 [sum](X) = 2 27.20/28.28 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [1](X) = 2 27.20/28.28 [empty] = 1 27.20/28.28 [singl](X) = 2.X + 2 27.20/28.28 [tt] = 2 27.20/28.28 [z] = 2 27.20/28.28 [0#](X) = 0 27.20/28.28 [U101#](X1,X2) = 0 27.20/28.28 [U102#](X1,X2) = 0 27.20/28.28 [U103#](X) = 0 27.20/28.28 [U11#](X1,X2) = 0 27.20/28.28 [U111#](X) = 0 27.20/28.28 [U12#](X1,X2) = 0 27.20/28.28 [U121#](X) = 0 27.20/28.28 [U13#](X) = 0 27.20/28.28 [U131#](X1,X2) = 0 27.20/28.28 [U132#](X) = 0 27.20/28.28 [U141#](X1,X2) = 0 27.20/28.28 [U142#](X) = 0 27.20/28.28 [U151#](X) = 0 27.20/28.28 [U161#](X) = 0 27.20/28.28 [U171#](X1,X2) = 0 27.20/28.28 [U172#](X) = 0 27.20/28.28 [U181#](X1,X2,X3) = 0 27.20/28.28 [U182#](X1,X2,X3) = 0 27.20/28.28 [U183#](X1,X2,X3) = 0 27.20/28.28 [U184#](X1,X2,X3) = 0 27.20/28.28 [U191#](X1,X2,X3) = 0 27.20/28.28 [U192#](X1,X2,X3) = 0 27.20/28.28 [U193#](X1,X2,X3) = 0 27.20/28.28 [U194#](X1,X2,X3) = 0 27.20/28.28 [U201#](X1,X2) = 0 27.20/28.28 [U202#](X1,X2) = 0 27.20/28.28 [U21#](X1,X2,X3) = 0 27.20/28.28 [U211#](X1,X2,X3) = 0 27.20/28.28 [U212#](X1,X2,X3) = 0 27.20/28.28 [U213#](X1,X2,X3) = 0 27.20/28.28 [U214#](X1,X2,X3) = 0 27.20/28.28 [U22#](X1,X2,X3) = 0 27.20/28.28 [U221#](X1,X2,X3) = 0 27.20/28.28 [U222#](X1,X2,X3) = 0 27.20/28.28 [U223#](X1,X2,X3) = 0 27.20/28.28 [U224#](X1,X2,X3) = 0 27.20/28.28 [U23#](X1,X2,X3) = 0 27.20/28.28 [U231#](X1,X2,X3) = 0 27.20/28.28 [U232#](X1,X2,X3) = 0 27.20/28.28 [U233#](X1,X2,X3) = 0 27.20/28.28 [U234#](X1,X2,X3) = 0 27.20/28.28 [U24#](X1,X2,X3) = 0 27.20/28.28 [U241#](X1,X2) = 0 27.20/28.28 [U242#](X1,X2) = 0 27.20/28.28 [U25#](X1,X2) = 0 27.20/28.28 [U251#](X1,X2,X3) = 0 27.20/28.28 [U252#](X1,X2,X3) = 0 27.20/28.28 [U253#](X1,X2,X3) = 0 27.20/28.28 [U254#](X1,X2,X3) = 0 27.20/28.28 [U26#](X) = 0 27.20/28.28 [U261#](X1,X2) = 0 27.20/28.28 [U262#](X1,X2) = 0 27.20/28.28 [U271#](X1,X2,X3) = 0 27.20/28.28 [U272#](X1,X2,X3) = 0 27.20/28.28 [U273#](X1,X2,X3) = 0 27.20/28.28 [U274#](X1,X2,X3) = 0 27.20/28.28 [U31#](X) = 0 27.20/28.28 [U41#](X1,X2) = 0 27.20/28.28 [U42#](X) = 0 27.20/28.28 [U51#](X1,X2) = 0 27.20/28.28 [U52#](X1,X2) = 0 27.20/28.28 [U53#](X) = 0 27.20/28.28 [U61#](X1,X2) = 0 27.20/28.28 [U62#](X1,X2) = 0 27.20/28.28 [U63#](X) = 0 27.20/28.28 [U71#](X1,X2,X3) = 0 27.20/28.28 [U72#](X1,X2,X3) = 0 27.20/28.28 [U73#](X1,X2,X3) = 0 27.20/28.28 [U74#](X1,X2,X3) = 0 27.20/28.28 [U75#](X1,X2) = 0 27.20/28.28 [U76#](X) = 0 27.20/28.28 [U81#](X1,X2,X3) = 0 27.20/28.28 [U82#](X1,X2,X3) = 0 27.20/28.28 [U83#](X1,X2,X3) = 0 27.20/28.28 [U84#](X1,X2,X3) = 0 27.20/28.28 [U85#](X1,X2) = 0 27.20/28.28 [U86#](X) = 0 27.20/28.28 [U91#](X1,X2) = 0 27.20/28.28 [U92#](X1,X2) = 0 27.20/28.28 [U93#](X) = 0 27.20/28.28 [ISBAG](X) = 0 27.20/28.28 [ISBAGKIND](X) = 0 27.20/28.28 [ISBIN](X) = 0 27.20/28.28 [ISBINKIND](X) = 0 27.20/28.28 [MULT](X1,X2) = 2.X1 + 2.X2 27.20/28.28 [PLUS](X1,X2) = 0 27.20/28.28 [PROD](X) = 0 27.20/28.28 [SUM](X) = 0 27.20/28.28 [UNION](X1,X2) = 0 27.20/28.28 27.20/28.28 Problem 1.5: 27.20/28.28 27.20/28.28 SCC Processor: 27.20/28.28 -> FAxioms: 27.20/28.28 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8)) 27.20/28.28 MULT(x6,x7) = MULT(x7,x6) 27.20/28.28 -> Pairs: 27.20/28.28 Empty 27.20/28.28 -> EAxioms: 27.20/28.28 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.28 mult(x6,x7) = mult(x7,x6) 27.20/28.28 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.28 plus(x6,x7) = plus(x7,x6) 27.20/28.28 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.28 union(x6,x7) = union(x7,x6) 27.20/28.28 -> Rules: 27.20/28.28 0(z) -> z 27.20/28.28 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.28 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.28 U103(tt) -> tt 27.20/28.28 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.28 U111(tt) -> tt 27.20/28.28 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.28 U121(tt) -> tt 27.20/28.28 U13(tt) -> tt 27.20/28.28 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.28 U132(tt) -> tt 27.20/28.28 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.28 U142(tt) -> tt 27.20/28.28 U151(tt) -> tt 27.20/28.28 U161(tt) -> tt 27.20/28.28 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.28 U172(tt) -> z 27.20/28.28 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.28 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.28 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.28 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.28 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.28 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.28 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.28 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.28 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.28 U202(tt,X) -> X 27.20/28.28 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.28 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.28 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.28 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.28 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.28 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.28 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.28 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.28 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.28 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.28 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.28 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.28 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.28 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.28 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.28 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.28 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.28 U242(tt,X) -> X 27.20/28.28 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.28 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.28 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.28 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.28 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.28 U26(tt) -> tt 27.20/28.28 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.28 U262(tt,X) -> X 27.20/28.28 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.28 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.28 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.28 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.28 U31(tt) -> tt 27.20/28.28 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.28 U42(tt) -> tt 27.20/28.28 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.28 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.28 U53(tt) -> tt 27.20/28.28 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.28 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.28 U63(tt) -> tt 27.20/28.28 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.28 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.28 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.28 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.28 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.28 U76(tt) -> tt 27.20/28.28 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.28 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.28 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.28 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.28 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.28 U86(tt) -> tt 27.20/28.28 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.28 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.28 U93(tt) -> tt 27.20/28.28 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.28 isBag(empty) -> tt 27.20/28.28 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.28 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.28 isBagKind(empty) -> tt 27.20/28.28 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.28 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.28 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.28 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.28 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.28 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.28 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.28 isBin(z) -> tt 27.20/28.28 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.28 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.28 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.28 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.28 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.28 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.28 isBinKind(z) -> tt 27.20/28.28 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.28 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.28 mult(z,X) -> U171(isBin(X),X) 27.20/28.28 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.28 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.28 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.28 plus(z,X) -> U201(isBin(X),X) 27.20/28.28 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.28 prod(empty) -> 1(z) 27.20/28.28 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.28 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.28 sum(empty) -> 0(z) 27.20/28.28 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.28 union(empty,X) -> X 27.20/28.28 union(X,empty) -> X 27.20/28.28 -> SRules: 27.20/28.28 MULT(mult(x6,x7),x8) -> MULT(x6,x7) 27.20/28.28 MULT(x6,mult(x7,x8)) -> MULT(x7,x8) 27.20/28.28 ->Strongly Connected Components: 27.20/28.28 There is no strongly connected component 27.20/28.28 27.20/28.28 The problem is finite. 27.20/28.28 27.20/28.28 Problem 1.6: 27.20/28.28 27.20/28.28 Reduction Pairs Processor: 27.20/28.28 -> FAxioms: 27.20/28.28 Empty 27.20/28.28 -> Pairs: 27.20/28.28 U251#(tt,A,B) -> U252#(isBagKind(A),A,B) 27.20/28.28 U252#(tt,A,B) -> U253#(isBag(B),A,B) 27.20/28.28 U253#(tt,A,B) -> U254#(isBagKind(B),A,B) 27.20/28.28 U254#(tt,A,B) -> PROD(A) 27.20/28.28 U254#(tt,A,B) -> PROD(B) 27.20/28.28 PROD(union(A,B)) -> U251#(isBag(A),A,B) 27.20/28.28 -> EAxioms: 27.20/28.28 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.28 mult(x6,x7) = mult(x7,x6) 27.20/28.28 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.28 plus(x6,x7) = plus(x7,x6) 27.20/28.28 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.28 union(x6,x7) = union(x7,x6) 27.20/28.28 -> Usable Equations: 27.20/28.28 Empty 27.20/28.28 -> Rules: 27.20/28.28 0(z) -> z 27.20/28.28 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.28 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.28 U103(tt) -> tt 27.20/28.28 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.28 U111(tt) -> tt 27.20/28.28 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.28 U121(tt) -> tt 27.20/28.28 U13(tt) -> tt 27.20/28.28 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.28 U132(tt) -> tt 27.20/28.28 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.28 U142(tt) -> tt 27.20/28.28 U151(tt) -> tt 27.20/28.28 U161(tt) -> tt 27.20/28.28 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.28 U172(tt) -> z 27.20/28.28 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.28 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.28 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.28 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.28 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.28 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.28 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.28 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.28 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.28 U202(tt,X) -> X 27.20/28.28 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.28 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.28 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.28 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.28 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.28 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.28 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.28 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.28 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.28 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.28 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.28 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.28 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.28 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.28 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.28 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.28 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.28 U242(tt,X) -> X 27.20/28.28 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.28 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.28 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.28 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.28 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.28 U26(tt) -> tt 27.20/28.28 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.28 U262(tt,X) -> X 27.20/28.28 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.28 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.28 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.28 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.28 U31(tt) -> tt 27.20/28.28 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.28 U42(tt) -> tt 27.20/28.28 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.28 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.28 U53(tt) -> tt 27.20/28.28 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.28 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.28 U63(tt) -> tt 27.20/28.28 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.28 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.28 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.28 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.28 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.28 U76(tt) -> tt 27.20/28.28 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.28 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.28 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.28 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.28 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.28 U86(tt) -> tt 27.20/28.28 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.28 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.28 U93(tt) -> tt 27.20/28.28 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.28 isBag(empty) -> tt 27.20/28.28 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.28 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.28 isBagKind(empty) -> tt 27.20/28.28 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.28 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.28 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.28 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.28 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.28 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.28 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.28 isBin(z) -> tt 27.20/28.28 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.28 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.28 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.28 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.28 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.28 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.28 isBinKind(z) -> tt 27.20/28.28 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.28 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.28 mult(z,X) -> U171(isBin(X),X) 27.20/28.28 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.28 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.28 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.28 plus(z,X) -> U201(isBin(X),X) 27.20/28.28 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.28 prod(empty) -> 1(z) 27.20/28.28 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.28 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.28 sum(empty) -> 0(z) 27.20/28.28 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.28 union(empty,X) -> X 27.20/28.28 union(X,empty) -> X 27.20/28.28 -> Usable Rules: 27.20/28.28 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.28 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.28 U103(tt) -> tt 27.20/28.28 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.28 U111(tt) -> tt 27.20/28.28 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.28 U121(tt) -> tt 27.20/28.28 U13(tt) -> tt 27.20/28.28 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.28 U132(tt) -> tt 27.20/28.28 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.28 U142(tt) -> tt 27.20/28.28 U151(tt) -> tt 27.20/28.28 U161(tt) -> tt 27.20/28.28 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.28 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.28 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.28 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.28 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.28 U26(tt) -> tt 27.20/28.28 U31(tt) -> tt 27.20/28.28 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.28 U42(tt) -> tt 27.20/28.28 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.28 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.28 U53(tt) -> tt 27.20/28.28 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.28 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.28 U63(tt) -> tt 27.20/28.28 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.28 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.28 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.28 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.28 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.28 U76(tt) -> tt 27.20/28.28 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.28 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.28 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.28 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.28 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.28 U86(tt) -> tt 27.20/28.28 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.28 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.28 U93(tt) -> tt 27.20/28.28 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.28 isBag(empty) -> tt 27.20/28.28 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.28 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.28 isBagKind(empty) -> tt 27.20/28.28 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.28 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.28 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.28 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.28 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.28 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.28 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.28 isBin(z) -> tt 27.20/28.28 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.28 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.28 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.28 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.28 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.28 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.28 isBinKind(z) -> tt 27.20/28.28 -> SRules: 27.20/28.28 Empty 27.20/28.28 ->Interpretation type: 27.20/28.28 Linear 27.20/28.28 ->Coefficients: 27.20/28.28 Natural Numbers 27.20/28.28 ->Dimension: 27.20/28.28 1 27.20/28.28 ->Bound: 27.20/28.28 2 27.20/28.28 ->Interpretation: 27.20/28.28 27.20/28.28 [0](X) = 2.X + 2 27.20/28.28 [U101](X1,X2) = 2 27.20/28.28 [U102](X1,X2) = 2 27.20/28.28 [U103](X) = 2 27.20/28.28 [U11](X1,X2) = X1 + 2.X2 + 2 27.20/28.28 [U111](X) = 2.X + 1 27.20/28.28 [U12](X1,X2) = X1 + 2 27.20/28.28 [U121](X) = 2.X 27.20/28.28 [U13](X) = 2 27.20/28.28 [U131](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [U132](X) = X + 2 27.20/28.28 [U141](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [U142](X) = X + 1 27.20/28.28 [U151](X) = X + 2 27.20/28.28 [U161](X) = 2 27.20/28.28 [U171](X1,X2) = 0 27.20/28.28 [U172](X) = 0 27.20/28.28 [U181](X1,X2,X3) = 0 27.20/28.28 [U182](X1,X2,X3) = 0 27.20/28.28 [U183](X1,X2,X3) = 0 27.20/28.28 [U184](X1,X2,X3) = 0 27.20/28.28 [U191](X1,X2,X3) = 0 27.20/28.28 [U192](X1,X2,X3) = 0 27.20/28.28 [U193](X1,X2,X3) = 0 27.20/28.28 [U194](X1,X2,X3) = 0 27.20/28.28 [U201](X1,X2) = 0 27.20/28.28 [U202](X1,X2) = 0 27.20/28.28 [U21](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.28 [U211](X1,X2,X3) = 0 27.20/28.28 [U212](X1,X2,X3) = 0 27.20/28.28 [U213](X1,X2,X3) = 0 27.20/28.28 [U214](X1,X2,X3) = 0 27.20/28.28 [U22](X1,X2,X3) = X1 + 2.X3 + 1 27.20/28.28 [U221](X1,X2,X3) = 0 27.20/28.28 [U222](X1,X2,X3) = 0 27.20/28.28 [U223](X1,X2,X3) = 0 27.20/28.28 [U224](X1,X2,X3) = 0 27.20/28.28 [U23](X1,X2,X3) = 2.X3 + 2 27.20/28.28 [U231](X1,X2,X3) = 0 27.20/28.28 [U232](X1,X2,X3) = 0 27.20/28.28 [U233](X1,X2,X3) = 0 27.20/28.28 [U234](X1,X2,X3) = 0 27.20/28.28 [U24](X1,X2,X3) = 2.X3 + 2 27.20/28.28 [U241](X1,X2) = 0 27.20/28.28 [U242](X1,X2) = 0 27.20/28.28 [U25](X1,X2) = 2.X2 + 2 27.20/28.28 [U251](X1,X2,X3) = 0 27.20/28.28 [U252](X1,X2,X3) = 0 27.20/28.28 [U253](X1,X2,X3) = 0 27.20/28.28 [U254](X1,X2,X3) = 0 27.20/28.28 [U26](X) = X + 1 27.20/28.28 [U261](X1,X2) = 0 27.20/28.28 [U262](X1,X2) = 0 27.20/28.28 [U271](X1,X2,X3) = 0 27.20/28.28 [U272](X1,X2,X3) = 0 27.20/28.28 [U273](X1,X2,X3) = 0 27.20/28.28 [U274](X1,X2,X3) = 0 27.20/28.28 [U31](X) = X + 2 27.20/28.28 [U41](X1,X2) = 2.X1 + 2.X2 + 1 27.20/28.28 [U42](X) = X + 2 27.20/28.28 [U51](X1,X2) = 2.X1 27.20/28.28 [U52](X1,X2) = 2 27.20/28.28 [U53](X) = 2 27.20/28.28 [U61](X1,X2) = 2.X1 27.20/28.28 [U62](X1,X2) = 2 27.20/28.28 [U63](X) = 2 27.20/28.28 [U71](X1,X2,X3) = 2.X1 + 2 27.20/28.28 [U72](X1,X2,X3) = 2 27.20/28.28 [U73](X1,X2,X3) = 2 27.20/28.28 [U74](X1,X2,X3) = 2 27.20/28.28 [U75](X1,X2) = 2 27.20/28.28 [U76](X) = 2 27.20/28.28 [U81](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.28 [U82](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U83](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U84](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U85](X1,X2) = X1 + 2.X2 + 2 27.20/28.28 [U86](X) = X + 1 27.20/28.28 [U91](X1,X2) = X1 + 2.X2 + 2 27.20/28.28 [U92](X1,X2) = 2.X2 + 2 27.20/28.28 [U93](X) = 2 27.20/28.28 [isBag](X) = 2.X + 1 27.20/28.28 [isBagKind](X) = 2.X + 2 27.20/28.28 [isBin](X) = 2.X 27.20/28.28 [isBinKind](X) = 2.X 27.20/28.28 [mult](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [plus](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [prod](X) = 2.X + 2 27.20/28.28 [sum](X) = 2 27.20/28.28 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [1](X) = 2.X 27.20/28.28 [empty] = 1 27.20/28.28 [singl](X) = 2.X + 2 27.20/28.28 [tt] = 2 27.20/28.28 [z] = 2 27.20/28.28 [0#](X) = 0 27.20/28.28 [U101#](X1,X2) = 0 27.20/28.28 [U102#](X1,X2) = 0 27.20/28.28 [U103#](X) = 0 27.20/28.28 [U11#](X1,X2) = 0 27.20/28.28 [U111#](X) = 0 27.20/28.28 [U12#](X1,X2) = 0 27.20/28.28 [U121#](X) = 0 27.20/28.28 [U13#](X) = 0 27.20/28.28 [U131#](X1,X2) = 0 27.20/28.28 [U132#](X) = 0 27.20/28.28 [U141#](X1,X2) = 0 27.20/28.28 [U142#](X) = 0 27.20/28.28 [U151#](X) = 0 27.20/28.28 [U161#](X) = 0 27.20/28.28 [U171#](X1,X2) = 0 27.20/28.28 [U172#](X) = 0 27.20/28.28 [U181#](X1,X2,X3) = 0 27.20/28.28 [U182#](X1,X2,X3) = 0 27.20/28.28 [U183#](X1,X2,X3) = 0 27.20/28.28 [U184#](X1,X2,X3) = 0 27.20/28.28 [U191#](X1,X2,X3) = 0 27.20/28.28 [U192#](X1,X2,X3) = 0 27.20/28.28 [U193#](X1,X2,X3) = 0 27.20/28.28 [U194#](X1,X2,X3) = 0 27.20/28.28 [U201#](X1,X2) = 0 27.20/28.28 [U202#](X1,X2) = 0 27.20/28.28 [U21#](X1,X2,X3) = 0 27.20/28.28 [U211#](X1,X2,X3) = 0 27.20/28.28 [U212#](X1,X2,X3) = 0 27.20/28.28 [U213#](X1,X2,X3) = 0 27.20/28.28 [U214#](X1,X2,X3) = 0 27.20/28.28 [U22#](X1,X2,X3) = 0 27.20/28.28 [U221#](X1,X2,X3) = 0 27.20/28.28 [U222#](X1,X2,X3) = 0 27.20/28.28 [U223#](X1,X2,X3) = 0 27.20/28.28 [U224#](X1,X2,X3) = 0 27.20/28.28 [U23#](X1,X2,X3) = 0 27.20/28.28 [U231#](X1,X2,X3) = 0 27.20/28.28 [U232#](X1,X2,X3) = 0 27.20/28.28 [U233#](X1,X2,X3) = 0 27.20/28.28 [U234#](X1,X2,X3) = 0 27.20/28.28 [U24#](X1,X2,X3) = 0 27.20/28.28 [U241#](X1,X2) = 0 27.20/28.28 [U242#](X1,X2) = 0 27.20/28.28 [U25#](X1,X2) = 0 27.20/28.28 [U251#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 27.20/28.28 [U252#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U253#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U254#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U26#](X) = 0 27.20/28.28 [U261#](X1,X2) = 0 27.20/28.28 [U262#](X1,X2) = 0 27.20/28.28 [U271#](X1,X2,X3) = 0 27.20/28.28 [U272#](X1,X2,X3) = 0 27.20/28.28 [U273#](X1,X2,X3) = 0 27.20/28.28 [U274#](X1,X2,X3) = 0 27.20/28.28 [U31#](X) = 0 27.20/28.28 [U41#](X1,X2) = 0 27.20/28.28 [U42#](X) = 0 27.20/28.28 [U51#](X1,X2) = 0 27.20/28.28 [U52#](X1,X2) = 0 27.20/28.28 [U53#](X) = 0 27.20/28.28 [U61#](X1,X2) = 0 27.20/28.28 [U62#](X1,X2) = 0 27.20/28.28 [U63#](X) = 0 27.20/28.28 [U71#](X1,X2,X3) = 0 27.20/28.28 [U72#](X1,X2,X3) = 0 27.20/28.28 [U73#](X1,X2,X3) = 0 27.20/28.28 [U74#](X1,X2,X3) = 0 27.20/28.28 [U75#](X1,X2) = 0 27.20/28.28 [U76#](X) = 0 27.20/28.28 [U81#](X1,X2,X3) = 0 27.20/28.28 [U82#](X1,X2,X3) = 0 27.20/28.28 [U83#](X1,X2,X3) = 0 27.20/28.28 [U84#](X1,X2,X3) = 0 27.20/28.28 [U85#](X1,X2) = 0 27.20/28.28 [U86#](X) = 0 27.20/28.28 [U91#](X1,X2) = 0 27.20/28.28 [U92#](X1,X2) = 0 27.20/28.28 [U93#](X) = 0 27.20/28.28 [ISBAG](X) = 0 27.20/28.28 [ISBAGKIND](X) = 0 27.20/28.28 [ISBIN](X) = 0 27.20/28.28 [ISBINKIND](X) = 0 27.20/28.28 [MULT](X1,X2) = 0 27.20/28.28 [PLUS](X1,X2) = 0 27.20/28.28 [PROD](X) = 2.X + 2 27.20/28.28 [SUM](X) = 0 27.20/28.28 [UNION](X1,X2) = 0 27.20/28.28 27.20/28.28 Problem 1.6: 27.20/28.28 27.20/28.28 SCC Processor: 27.20/28.28 -> FAxioms: 27.20/28.28 Empty 27.20/28.28 -> Pairs: 27.20/28.28 U252#(tt,A,B) -> U253#(isBag(B),A,B) 27.20/28.28 U253#(tt,A,B) -> U254#(isBagKind(B),A,B) 27.20/28.28 U254#(tt,A,B) -> PROD(A) 27.20/28.28 U254#(tt,A,B) -> PROD(B) 27.20/28.28 PROD(union(A,B)) -> U251#(isBag(A),A,B) 27.20/28.28 -> EAxioms: 27.20/28.28 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.28 mult(x6,x7) = mult(x7,x6) 27.20/28.28 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.28 plus(x6,x7) = plus(x7,x6) 27.20/28.28 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.28 union(x6,x7) = union(x7,x6) 27.20/28.28 -> Rules: 27.20/28.28 0(z) -> z 27.20/28.28 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.28 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.28 U103(tt) -> tt 27.20/28.28 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.28 U111(tt) -> tt 27.20/28.28 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.28 U121(tt) -> tt 27.20/28.28 U13(tt) -> tt 27.20/28.28 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.28 U132(tt) -> tt 27.20/28.28 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.28 U142(tt) -> tt 27.20/28.28 U151(tt) -> tt 27.20/28.28 U161(tt) -> tt 27.20/28.28 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.28 U172(tt) -> z 27.20/28.28 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.28 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.28 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.28 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.28 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.28 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.28 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.28 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.28 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.28 U202(tt,X) -> X 27.20/28.28 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.28 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.28 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.28 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.28 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.28 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.28 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.28 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.28 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.28 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.28 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.28 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.28 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.28 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.28 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.28 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.28 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.28 U242(tt,X) -> X 27.20/28.28 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.28 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.28 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.28 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.28 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.28 U26(tt) -> tt 27.20/28.28 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.28 U262(tt,X) -> X 27.20/28.28 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.28 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.28 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.28 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.28 U31(tt) -> tt 27.20/28.28 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.28 U42(tt) -> tt 27.20/28.28 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.28 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.28 U53(tt) -> tt 27.20/28.28 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.28 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.28 U63(tt) -> tt 27.20/28.28 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.28 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.28 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.28 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.28 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.28 U76(tt) -> tt 27.20/28.28 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.28 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.28 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.28 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.28 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.28 U86(tt) -> tt 27.20/28.28 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.28 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.28 U93(tt) -> tt 27.20/28.28 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.28 isBag(empty) -> tt 27.20/28.28 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.28 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.28 isBagKind(empty) -> tt 27.20/28.28 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.28 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.28 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.28 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.28 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.28 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.28 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.28 isBin(z) -> tt 27.20/28.28 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.28 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.28 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.28 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.28 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.28 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.28 isBinKind(z) -> tt 27.20/28.28 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.28 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.28 mult(z,X) -> U171(isBin(X),X) 27.20/28.28 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.28 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.28 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.28 plus(z,X) -> U201(isBin(X),X) 27.20/28.28 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.28 prod(empty) -> 1(z) 27.20/28.28 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.28 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.28 sum(empty) -> 0(z) 27.20/28.28 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.28 union(empty,X) -> X 27.20/28.28 union(X,empty) -> X 27.20/28.28 -> SRules: 27.20/28.28 Empty 27.20/28.28 ->Strongly Connected Components: 27.20/28.28 There is no strongly connected component 27.20/28.28 27.20/28.28 The problem is finite. 27.20/28.28 27.20/28.28 Problem 1.7: 27.20/28.28 27.20/28.28 Reduction Pairs Processor: 27.20/28.28 -> FAxioms: 27.20/28.28 Empty 27.20/28.28 -> Pairs: 27.20/28.28 U271#(tt,A,B) -> U272#(isBagKind(A),A,B) 27.20/28.28 U272#(tt,A,B) -> U273#(isBag(B),A,B) 27.20/28.28 U273#(tt,A,B) -> U274#(isBagKind(B),A,B) 27.20/28.28 U274#(tt,A,B) -> SUM(A) 27.20/28.28 U274#(tt,A,B) -> SUM(B) 27.20/28.28 SUM(union(A,B)) -> U271#(isBag(A),A,B) 27.20/28.28 -> EAxioms: 27.20/28.28 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.28 mult(x6,x7) = mult(x7,x6) 27.20/28.28 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.28 plus(x6,x7) = plus(x7,x6) 27.20/28.28 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.28 union(x6,x7) = union(x7,x6) 27.20/28.28 -> Usable Equations: 27.20/28.28 Empty 27.20/28.28 -> Rules: 27.20/28.28 0(z) -> z 27.20/28.28 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.28 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.28 U103(tt) -> tt 27.20/28.28 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.28 U111(tt) -> tt 27.20/28.28 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.28 U121(tt) -> tt 27.20/28.28 U13(tt) -> tt 27.20/28.28 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.28 U132(tt) -> tt 27.20/28.28 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.28 U142(tt) -> tt 27.20/28.28 U151(tt) -> tt 27.20/28.28 U161(tt) -> tt 27.20/28.28 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.28 U172(tt) -> z 27.20/28.28 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.28 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.28 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.28 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.28 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.28 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.28 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.28 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.28 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.28 U202(tt,X) -> X 27.20/28.28 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.28 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.28 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.28 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.28 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.28 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.28 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.28 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.28 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.28 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.28 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.28 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.28 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.28 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.28 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.28 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.28 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.28 U242(tt,X) -> X 27.20/28.28 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.28 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.28 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.28 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.28 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.28 U26(tt) -> tt 27.20/28.28 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.28 U262(tt,X) -> X 27.20/28.28 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.28 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.28 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.28 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.28 U31(tt) -> tt 27.20/28.28 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.28 U42(tt) -> tt 27.20/28.28 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.28 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.28 U53(tt) -> tt 27.20/28.28 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.28 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.28 U63(tt) -> tt 27.20/28.28 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.28 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.28 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.28 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.28 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.28 U76(tt) -> tt 27.20/28.28 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.28 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.28 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.28 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.28 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.28 U86(tt) -> tt 27.20/28.28 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.28 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.28 U93(tt) -> tt 27.20/28.28 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.28 isBag(empty) -> tt 27.20/28.28 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.28 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.28 isBagKind(empty) -> tt 27.20/28.28 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.28 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.28 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.28 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.28 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.28 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.28 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.28 isBin(z) -> tt 27.20/28.28 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.28 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.28 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.28 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.28 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.28 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.28 isBinKind(z) -> tt 27.20/28.28 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.28 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.28 mult(z,X) -> U171(isBin(X),X) 27.20/28.28 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.28 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.28 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.28 plus(z,X) -> U201(isBin(X),X) 27.20/28.28 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.28 prod(empty) -> 1(z) 27.20/28.28 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.28 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.28 sum(empty) -> 0(z) 27.20/28.28 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.28 union(empty,X) -> X 27.20/28.28 union(X,empty) -> X 27.20/28.28 -> Usable Rules: 27.20/28.28 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.28 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.28 U103(tt) -> tt 27.20/28.28 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.28 U111(tt) -> tt 27.20/28.28 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.28 U121(tt) -> tt 27.20/28.28 U13(tt) -> tt 27.20/28.28 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.28 U132(tt) -> tt 27.20/28.28 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.28 U142(tt) -> tt 27.20/28.28 U151(tt) -> tt 27.20/28.28 U161(tt) -> tt 27.20/28.28 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.28 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.28 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.28 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.28 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.28 U26(tt) -> tt 27.20/28.28 U31(tt) -> tt 27.20/28.28 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.28 U42(tt) -> tt 27.20/28.28 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.28 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.28 U53(tt) -> tt 27.20/28.28 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.28 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.28 U63(tt) -> tt 27.20/28.28 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.28 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.28 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.28 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.28 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.28 U76(tt) -> tt 27.20/28.28 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.28 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.28 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.28 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.28 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.28 U86(tt) -> tt 27.20/28.28 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.28 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.28 U93(tt) -> tt 27.20/28.28 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.28 isBag(empty) -> tt 27.20/28.28 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.28 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.28 isBagKind(empty) -> tt 27.20/28.28 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.28 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.28 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.28 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.28 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.28 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.28 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.28 isBin(z) -> tt 27.20/28.28 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.28 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.28 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.28 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.28 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.28 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.28 isBinKind(z) -> tt 27.20/28.28 -> SRules: 27.20/28.28 Empty 27.20/28.28 ->Interpretation type: 27.20/28.28 Linear 27.20/28.28 ->Coefficients: 27.20/28.28 Natural Numbers 27.20/28.28 ->Dimension: 27.20/28.28 1 27.20/28.28 ->Bound: 27.20/28.28 2 27.20/28.28 ->Interpretation: 27.20/28.28 27.20/28.28 [0](X) = 2.X + 2 27.20/28.28 [U101](X1,X2) = 2 27.20/28.28 [U102](X1,X2) = 2 27.20/28.28 [U103](X) = 2 27.20/28.28 [U11](X1,X2) = X1 + 2.X2 + 2 27.20/28.28 [U111](X) = 2.X + 1 27.20/28.28 [U12](X1,X2) = X1 + 2 27.20/28.28 [U121](X) = 2.X 27.20/28.28 [U13](X) = 2 27.20/28.28 [U131](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [U132](X) = X + 2 27.20/28.28 [U141](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [U142](X) = X + 1 27.20/28.28 [U151](X) = X + 2 27.20/28.28 [U161](X) = 2 27.20/28.28 [U171](X1,X2) = 0 27.20/28.28 [U172](X) = 0 27.20/28.28 [U181](X1,X2,X3) = 0 27.20/28.28 [U182](X1,X2,X3) = 0 27.20/28.28 [U183](X1,X2,X3) = 0 27.20/28.28 [U184](X1,X2,X3) = 0 27.20/28.28 [U191](X1,X2,X3) = 0 27.20/28.28 [U192](X1,X2,X3) = 0 27.20/28.28 [U193](X1,X2,X3) = 0 27.20/28.28 [U194](X1,X2,X3) = 0 27.20/28.28 [U201](X1,X2) = 0 27.20/28.28 [U202](X1,X2) = 0 27.20/28.28 [U21](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.28 [U211](X1,X2,X3) = 0 27.20/28.28 [U212](X1,X2,X3) = 0 27.20/28.28 [U213](X1,X2,X3) = 0 27.20/28.28 [U214](X1,X2,X3) = 0 27.20/28.28 [U22](X1,X2,X3) = X1 + 2.X3 + 1 27.20/28.28 [U221](X1,X2,X3) = 0 27.20/28.28 [U222](X1,X2,X3) = 0 27.20/28.28 [U223](X1,X2,X3) = 0 27.20/28.28 [U224](X1,X2,X3) = 0 27.20/28.28 [U23](X1,X2,X3) = 2.X3 + 2 27.20/28.28 [U231](X1,X2,X3) = 0 27.20/28.28 [U232](X1,X2,X3) = 0 27.20/28.28 [U233](X1,X2,X3) = 0 27.20/28.28 [U234](X1,X2,X3) = 0 27.20/28.28 [U24](X1,X2,X3) = 2.X3 + 2 27.20/28.28 [U241](X1,X2) = 0 27.20/28.28 [U242](X1,X2) = 0 27.20/28.28 [U25](X1,X2) = 2.X2 + 2 27.20/28.28 [U251](X1,X2,X3) = 0 27.20/28.28 [U252](X1,X2,X3) = 0 27.20/28.28 [U253](X1,X2,X3) = 0 27.20/28.28 [U254](X1,X2,X3) = 0 27.20/28.28 [U26](X) = X + 1 27.20/28.28 [U261](X1,X2) = 0 27.20/28.28 [U262](X1,X2) = 0 27.20/28.28 [U271](X1,X2,X3) = 0 27.20/28.28 [U272](X1,X2,X3) = 0 27.20/28.28 [U273](X1,X2,X3) = 0 27.20/28.28 [U274](X1,X2,X3) = 0 27.20/28.28 [U31](X) = X + 2 27.20/28.28 [U41](X1,X2) = 2.X1 + 2.X2 + 1 27.20/28.28 [U42](X) = X + 2 27.20/28.28 [U51](X1,X2) = 2.X1 27.20/28.28 [U52](X1,X2) = 2 27.20/28.28 [U53](X) = 2 27.20/28.28 [U61](X1,X2) = 2.X1 27.20/28.28 [U62](X1,X2) = 2 27.20/28.28 [U63](X) = 2 27.20/28.28 [U71](X1,X2,X3) = 2.X1 + 2 27.20/28.28 [U72](X1,X2,X3) = 2 27.20/28.28 [U73](X1,X2,X3) = 2 27.20/28.28 [U74](X1,X2,X3) = 2 27.20/28.28 [U75](X1,X2) = 2 27.20/28.28 [U76](X) = 2 27.20/28.28 [U81](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 27.20/28.28 [U82](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U83](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U84](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U85](X1,X2) = X1 + 2.X2 + 2 27.20/28.28 [U86](X) = X + 1 27.20/28.28 [U91](X1,X2) = X1 + 2.X2 + 2 27.20/28.28 [U92](X1,X2) = 2.X2 + 2 27.20/28.28 [U93](X) = 2 27.20/28.28 [isBag](X) = 2.X + 1 27.20/28.28 [isBagKind](X) = 2.X + 2 27.20/28.28 [isBin](X) = 2.X 27.20/28.28 [isBinKind](X) = 2.X 27.20/28.28 [mult](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [plus](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [prod](X) = 2.X + 2 27.20/28.28 [sum](X) = 2 27.20/28.28 [union](X1,X2) = 2.X1 + 2.X2 + 2 27.20/28.28 [1](X) = 2.X 27.20/28.28 [empty] = 1 27.20/28.28 [singl](X) = 2.X + 2 27.20/28.28 [tt] = 2 27.20/28.28 [z] = 2 27.20/28.28 [0#](X) = 0 27.20/28.28 [U101#](X1,X2) = 0 27.20/28.28 [U102#](X1,X2) = 0 27.20/28.28 [U103#](X) = 0 27.20/28.28 [U11#](X1,X2) = 0 27.20/28.28 [U111#](X) = 0 27.20/28.28 [U12#](X1,X2) = 0 27.20/28.28 [U121#](X) = 0 27.20/28.28 [U13#](X) = 0 27.20/28.28 [U131#](X1,X2) = 0 27.20/28.28 [U132#](X) = 0 27.20/28.28 [U141#](X1,X2) = 0 27.20/28.28 [U142#](X) = 0 27.20/28.28 [U151#](X) = 0 27.20/28.28 [U161#](X) = 0 27.20/28.28 [U171#](X1,X2) = 0 27.20/28.28 [U172#](X) = 0 27.20/28.28 [U181#](X1,X2,X3) = 0 27.20/28.28 [U182#](X1,X2,X3) = 0 27.20/28.28 [U183#](X1,X2,X3) = 0 27.20/28.28 [U184#](X1,X2,X3) = 0 27.20/28.28 [U191#](X1,X2,X3) = 0 27.20/28.28 [U192#](X1,X2,X3) = 0 27.20/28.28 [U193#](X1,X2,X3) = 0 27.20/28.28 [U194#](X1,X2,X3) = 0 27.20/28.28 [U201#](X1,X2) = 0 27.20/28.28 [U202#](X1,X2) = 0 27.20/28.28 [U21#](X1,X2,X3) = 0 27.20/28.28 [U211#](X1,X2,X3) = 0 27.20/28.28 [U212#](X1,X2,X3) = 0 27.20/28.28 [U213#](X1,X2,X3) = 0 27.20/28.28 [U214#](X1,X2,X3) = 0 27.20/28.28 [U22#](X1,X2,X3) = 0 27.20/28.28 [U221#](X1,X2,X3) = 0 27.20/28.28 [U222#](X1,X2,X3) = 0 27.20/28.28 [U223#](X1,X2,X3) = 0 27.20/28.28 [U224#](X1,X2,X3) = 0 27.20/28.28 [U23#](X1,X2,X3) = 0 27.20/28.28 [U231#](X1,X2,X3) = 0 27.20/28.28 [U232#](X1,X2,X3) = 0 27.20/28.28 [U233#](X1,X2,X3) = 0 27.20/28.28 [U234#](X1,X2,X3) = 0 27.20/28.28 [U24#](X1,X2,X3) = 0 27.20/28.28 [U241#](X1,X2) = 0 27.20/28.28 [U242#](X1,X2) = 0 27.20/28.28 [U25#](X1,X2) = 0 27.20/28.28 [U251#](X1,X2,X3) = 0 27.20/28.28 [U252#](X1,X2,X3) = 0 27.20/28.28 [U253#](X1,X2,X3) = 0 27.20/28.28 [U254#](X1,X2,X3) = 0 27.20/28.28 [U26#](X) = 0 27.20/28.28 [U261#](X1,X2) = 0 27.20/28.28 [U262#](X1,X2) = 0 27.20/28.28 [U271#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 27.20/28.28 [U272#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U273#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U274#](X1,X2,X3) = 2.X2 + 2.X3 + 2 27.20/28.28 [U31#](X) = 0 27.20/28.28 [U41#](X1,X2) = 0 27.20/28.28 [U42#](X) = 0 27.20/28.28 [U51#](X1,X2) = 0 27.20/28.28 [U52#](X1,X2) = 0 27.20/28.28 [U53#](X) = 0 27.20/28.28 [U61#](X1,X2) = 0 27.20/28.28 [U62#](X1,X2) = 0 27.20/28.28 [U63#](X) = 0 27.20/28.28 [U71#](X1,X2,X3) = 0 27.20/28.28 [U72#](X1,X2,X3) = 0 27.20/28.28 [U73#](X1,X2,X3) = 0 27.20/28.28 [U74#](X1,X2,X3) = 0 27.20/28.28 [U75#](X1,X2) = 0 27.20/28.28 [U76#](X) = 0 27.20/28.28 [U81#](X1,X2,X3) = 0 27.20/28.28 [U82#](X1,X2,X3) = 0 27.20/28.28 [U83#](X1,X2,X3) = 0 27.20/28.28 [U84#](X1,X2,X3) = 0 27.20/28.28 [U85#](X1,X2) = 0 27.20/28.28 [U86#](X) = 0 27.20/28.28 [U91#](X1,X2) = 0 27.20/28.28 [U92#](X1,X2) = 0 27.20/28.28 [U93#](X) = 0 27.20/28.28 [ISBAG](X) = 0 27.20/28.28 [ISBAGKIND](X) = 0 27.20/28.28 [ISBIN](X) = 0 27.20/28.28 [ISBINKIND](X) = 0 27.20/28.28 [MULT](X1,X2) = 0 27.20/28.28 [PLUS](X1,X2) = 0 27.20/28.28 [PROD](X) = 0 27.20/28.28 [SUM](X) = 2.X + 2 27.20/28.28 [UNION](X1,X2) = 0 27.20/28.28 27.20/28.28 Problem 1.7: 27.20/28.28 27.20/28.28 SCC Processor: 27.20/28.28 -> FAxioms: 27.20/28.28 Empty 27.20/28.28 -> Pairs: 27.20/28.28 U272#(tt,A,B) -> U273#(isBag(B),A,B) 27.20/28.28 U273#(tt,A,B) -> U274#(isBagKind(B),A,B) 27.20/28.28 U274#(tt,A,B) -> SUM(A) 27.20/28.28 U274#(tt,A,B) -> SUM(B) 27.20/28.28 SUM(union(A,B)) -> U271#(isBag(A),A,B) 27.20/28.28 -> EAxioms: 27.20/28.28 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8)) 27.20/28.28 mult(x6,x7) = mult(x7,x6) 27.20/28.28 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8)) 27.20/28.28 plus(x6,x7) = plus(x7,x6) 27.20/28.28 union(union(x6,x7),x8) = union(x6,union(x7,x8)) 27.20/28.28 union(x6,x7) = union(x7,x6) 27.20/28.28 -> Rules: 27.20/28.28 0(z) -> z 27.20/28.28 U101(tt,V1) -> U102(isBagKind(V1),V1) 27.20/28.28 U102(tt,V1) -> U103(isBag(V1)) 27.20/28.28 U103(tt) -> tt 27.20/28.28 U11(tt,V1) -> U12(isBinKind(V1),V1) 27.20/28.28 U111(tt) -> tt 27.20/28.28 U12(tt,V1) -> U13(isBin(V1)) 27.20/28.28 U121(tt) -> tt 27.20/28.28 U13(tt) -> tt 27.20/28.28 U131(tt,V2) -> U132(isBinKind(V2)) 27.20/28.28 U132(tt) -> tt 27.20/28.28 U141(tt,V2) -> U142(isBinKind(V2)) 27.20/28.28 U142(tt) -> tt 27.20/28.28 U151(tt) -> tt 27.20/28.28 U161(tt) -> tt 27.20/28.28 U171(tt,X) -> U172(isBinKind(X)) 27.20/28.28 U172(tt) -> z 27.20/28.28 U181(tt,X,Y) -> U182(isBinKind(X),X,Y) 27.20/28.28 U182(tt,X,Y) -> U183(isBin(Y),X,Y) 27.20/28.28 U183(tt,X,Y) -> U184(isBinKind(Y),X,Y) 27.20/28.28 U184(tt,X,Y) -> 0(mult(X,Y)) 27.20/28.28 U191(tt,X,Y) -> U192(isBinKind(X),X,Y) 27.20/28.28 U192(tt,X,Y) -> U193(isBin(Y),X,Y) 27.20/28.28 U193(tt,X,Y) -> U194(isBinKind(Y),X,Y) 27.20/28.28 U194(tt,X,Y) -> plus(0(mult(X,Y)),Y) 27.20/28.28 U201(tt,X) -> U202(isBinKind(X),X) 27.20/28.28 U202(tt,X) -> X 27.20/28.28 U21(tt,V1,V2) -> U22(isBagKind(V1),V1,V2) 27.20/28.28 U211(tt,X,Y) -> U212(isBinKind(X),X,Y) 27.20/28.28 U212(tt,X,Y) -> U213(isBin(Y),X,Y) 27.20/28.28 U213(tt,X,Y) -> U214(isBinKind(Y),X,Y) 27.20/28.28 U214(tt,X,Y) -> 0(plus(X,Y)) 27.20/28.28 U22(tt,V1,V2) -> U23(isBagKind(V2),V1,V2) 27.20/28.28 U221(tt,X,Y) -> U222(isBinKind(X),X,Y) 27.20/28.28 U222(tt,X,Y) -> U223(isBin(Y),X,Y) 27.20/28.28 U223(tt,X,Y) -> U224(isBinKind(Y),X,Y) 27.20/28.28 U224(tt,X,Y) -> 1(plus(X,Y)) 27.20/28.28 U23(tt,V1,V2) -> U24(isBagKind(V2),V1,V2) 27.20/28.28 U231(tt,X,Y) -> U232(isBinKind(X),X,Y) 27.20/28.28 U232(tt,X,Y) -> U233(isBin(Y),X,Y) 27.20/28.28 U233(tt,X,Y) -> U234(isBinKind(Y),X,Y) 27.20/28.28 U234(tt,X,Y) -> 0(plus(plus(X,Y),1(z))) 27.20/28.28 U24(tt,V1,V2) -> U25(isBag(V1),V2) 27.20/28.28 U241(tt,X) -> U242(isBinKind(X),X) 27.20/28.28 U242(tt,X) -> X 27.20/28.28 U25(tt,V2) -> U26(isBag(V2)) 27.20/28.28 U251(tt,A,B) -> U252(isBagKind(A),A,B) 27.20/28.28 U252(tt,A,B) -> U253(isBag(B),A,B) 27.20/28.28 U253(tt,A,B) -> U254(isBagKind(B),A,B) 27.20/28.28 U254(tt,A,B) -> mult(prod(A),prod(B)) 27.20/28.28 U26(tt) -> tt 27.20/28.28 U261(tt,X) -> U262(isBinKind(X),X) 27.20/28.28 U262(tt,X) -> X 27.20/28.28 U271(tt,A,B) -> U272(isBagKind(A),A,B) 27.20/28.28 U272(tt,A,B) -> U273(isBag(B),A,B) 27.20/28.28 U273(tt,A,B) -> U274(isBagKind(B),A,B) 27.20/28.28 U274(tt,A,B) -> plus(sum(A),sum(B)) 27.20/28.28 U31(tt) -> tt 27.20/28.28 U41(tt,V2) -> U42(isBagKind(V2)) 27.20/28.28 U42(tt) -> tt 27.20/28.28 U51(tt,V1) -> U52(isBinKind(V1),V1) 27.20/28.28 U52(tt,V1) -> U53(isBin(V1)) 27.20/28.28 U53(tt) -> tt 27.20/28.28 U61(tt,V1) -> U62(isBinKind(V1),V1) 27.20/28.28 U62(tt,V1) -> U63(isBin(V1)) 27.20/28.28 U63(tt) -> tt 27.20/28.28 U71(tt,V1,V2) -> U72(isBinKind(V1),V1,V2) 27.20/28.28 U72(tt,V1,V2) -> U73(isBinKind(V2),V1,V2) 27.20/28.28 U73(tt,V1,V2) -> U74(isBinKind(V2),V1,V2) 27.20/28.28 U74(tt,V1,V2) -> U75(isBin(V1),V2) 27.20/28.28 U75(tt,V2) -> U76(isBin(V2)) 27.20/28.28 U76(tt) -> tt 27.20/28.28 U81(tt,V1,V2) -> U82(isBinKind(V1),V1,V2) 27.20/28.28 U82(tt,V1,V2) -> U83(isBinKind(V2),V1,V2) 27.20/28.28 U83(tt,V1,V2) -> U84(isBinKind(V2),V1,V2) 27.20/28.28 U84(tt,V1,V2) -> U85(isBin(V1),V2) 27.20/28.28 U85(tt,V2) -> U86(isBin(V2)) 27.20/28.28 U86(tt) -> tt 27.20/28.28 U91(tt,V1) -> U92(isBagKind(V1),V1) 27.20/28.28 U92(tt,V1) -> U93(isBag(V1)) 27.20/28.28 U93(tt) -> tt 27.20/28.28 isBag(union(V1,V2)) -> U21(isBagKind(V1),V1,V2) 27.20/28.28 isBag(empty) -> tt 27.20/28.28 isBag(singl(V1)) -> U11(isBinKind(V1),V1) 27.20/28.28 isBagKind(union(V1,V2)) -> U41(isBagKind(V1),V2) 27.20/28.28 isBagKind(empty) -> tt 27.20/28.28 isBagKind(singl(V1)) -> U31(isBinKind(V1)) 27.20/28.28 isBin(0(V1)) -> U51(isBinKind(V1),V1) 27.20/28.28 isBin(mult(V1,V2)) -> U71(isBinKind(V1),V1,V2) 27.20/28.28 isBin(plus(V1,V2)) -> U81(isBinKind(V1),V1,V2) 27.20/28.28 isBin(prod(V1)) -> U91(isBagKind(V1),V1) 27.20/28.28 isBin(sum(V1)) -> U101(isBagKind(V1),V1) 27.20/28.28 isBin(1(V1)) -> U61(isBinKind(V1),V1) 27.20/28.28 isBin(z) -> tt 27.20/28.28 isBinKind(0(V1)) -> U111(isBinKind(V1)) 27.20/28.28 isBinKind(mult(V1,V2)) -> U131(isBinKind(V1),V2) 27.20/28.28 isBinKind(plus(V1,V2)) -> U141(isBinKind(V1),V2) 27.20/28.28 isBinKind(prod(V1)) -> U151(isBagKind(V1)) 27.20/28.28 isBinKind(sum(V1)) -> U161(isBagKind(V1)) 27.20/28.28 isBinKind(1(V1)) -> U121(isBinKind(V1)) 27.20/28.28 isBinKind(z) -> tt 27.20/28.28 mult(0(X),Y) -> U181(isBin(X),X,Y) 27.20/28.28 mult(1(X),Y) -> U191(isBin(X),X,Y) 27.20/28.28 mult(z,X) -> U171(isBin(X),X) 27.20/28.28 plus(0(X),0(Y)) -> U211(isBin(X),X,Y) 27.20/28.28 plus(0(X),1(Y)) -> U221(isBin(X),X,Y) 27.20/28.28 plus(1(X),1(Y)) -> U231(isBin(X),X,Y) 27.20/28.28 plus(z,X) -> U201(isBin(X),X) 27.20/28.28 prod(union(A,B)) -> U251(isBag(A),A,B) 27.20/28.28 prod(empty) -> 1(z) 27.20/28.28 prod(singl(X)) -> U241(isBin(X),X) 27.20/28.28 sum(union(A,B)) -> U271(isBag(A),A,B) 27.20/28.28 sum(empty) -> 0(z) 27.20/28.28 sum(singl(X)) -> U261(isBin(X),X) 27.20/28.28 union(empty,X) -> X 27.20/28.28 union(X,empty) -> X 27.20/28.28 -> SRules: 27.20/28.28 Empty 27.20/28.28 ->Strongly Connected Components: 27.20/28.28 There is no strongly connected component 27.20/28.28 27.20/28.28 The problem is finite. 27.20/28.28 EOF