/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- Error at 1: Unexpected token '>=', 'benchmark' expected. syntax error rm: cannot remove ‘/tmp/muTerm1804289383846930886.in.dec’: Operation not permitted rm: cannot remove ‘/tmp/muTerm1804289383846930886.in.form’: Operation not permitted Error at 1: Unexpected token '>=', 'benchmark' expected. syntax error rm: cannot remove ‘/tmp/muTerm1957747793424238335.in.dec’: Operation not permitted rm: cannot remove ‘/tmp/muTerm1957747793424238335.in.form’: Operation not permitted YES Problem 1: (VAR I P V V1 V2 X X1 X2 Y Z) (RULES U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(tt,V) -> U12#(isNeList(activate(V))) U11#(tt,V) -> ACTIVATE(V) U11#(tt,V) -> ISNELIST(activate(V)) U21#(tt,V1,V2) -> U22#(isList(activate(V1)),activate(V2)) U21#(tt,V1,V2) -> ACTIVATE(V1) U21#(tt,V1,V2) -> ACTIVATE(V2) U21#(tt,V1,V2) -> ISLIST(activate(V1)) U22#(tt,V2) -> U23#(isList(activate(V2))) U22#(tt,V2) -> ACTIVATE(V2) U22#(tt,V2) -> ISLIST(activate(V2)) U31#(tt,V) -> U32#(isQid(activate(V))) U31#(tt,V) -> ACTIVATE(V) U31#(tt,V) -> ISQID(activate(V)) U41#(tt,V1,V2) -> U42#(isList(activate(V1)),activate(V2)) U41#(tt,V1,V2) -> ACTIVATE(V1) U41#(tt,V1,V2) -> ACTIVATE(V2) U41#(tt,V1,V2) -> ISLIST(activate(V1)) U42#(tt,V2) -> U43#(isNeList(activate(V2))) U42#(tt,V2) -> ACTIVATE(V2) U42#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V1,V2) -> U52#(isNeList(activate(V1)),activate(V2)) U51#(tt,V1,V2) -> ACTIVATE(V1) U51#(tt,V1,V2) -> ACTIVATE(V2) U51#(tt,V1,V2) -> ISNELIST(activate(V1)) U52#(tt,V2) -> U53#(isList(activate(V2))) U52#(tt,V2) -> ACTIVATE(V2) U52#(tt,V2) -> ISLIST(activate(V2)) U61#(tt,V) -> U62#(isQid(activate(V))) U61#(tt,V) -> ACTIVATE(V) U61#(tt,V) -> ISQID(activate(V)) U71#(tt,V) -> U72#(isNePal(activate(V))) U71#(tt,V) -> ACTIVATE(V) U71#(tt,V) -> ISNEPAL(activate(V)) __#(__(X,Y),Z) -> __#(X,__(Y,Z)) __#(__(X,Y),Z) -> __#(Y,Z) ACTIVATE(n____(X1,X2)) -> __#(X1,X2) ACTIVATE(n__a) -> A ACTIVATE(n__and(X1,X2)) -> AND(X1,X2) ACTIVATE(n__e) -> E ACTIVATE(n__i) -> I ACTIVATE(n__isPalListKind(X)) -> ISPALLISTKIND(X) ACTIVATE(n__nil) -> NIL ACTIVATE(n__o) -> O ACTIVATE(n__u) -> U AND(tt,X) -> ACTIVATE(X) ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISLIST(n____(V1,V2)) -> ACTIVATE(V1) ISLIST(n____(V1,V2)) -> ACTIVATE(V2) ISLIST(n____(V1,V2)) -> AND(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) ISLIST(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) ISLIST(V) -> U11#(isPalListKind(activate(V)),activate(V)) ISLIST(V) -> ACTIVATE(V) ISLIST(V) -> ISPALLISTKIND(activate(V)) ISNELIST(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISNELIST(n____(V1,V2)) -> ACTIVATE(V1) ISNELIST(n____(V1,V2)) -> ACTIVATE(V2) ISNELIST(n____(V1,V2)) -> AND(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) ISNELIST(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) ISNELIST(V) -> U31#(isPalListKind(activate(V)),activate(V)) ISNELIST(V) -> ACTIVATE(V) ISNELIST(V) -> ISPALLISTKIND(activate(V)) ISNEPAL(n____(I,__(P,I))) -> ACTIVATE(I) ISNEPAL(n____(I,__(P,I))) -> ACTIVATE(P) ISNEPAL(n____(I,__(P,I))) -> AND(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) ISNEPAL(n____(I,__(P,I))) -> AND(isQid(activate(I)),n__isPalListKind(activate(I))) ISNEPAL(n____(I,__(P,I))) -> ISPAL(activate(P)) ISNEPAL(n____(I,__(P,I))) -> ISQID(activate(I)) ISNEPAL(V) -> U61#(isPalListKind(activate(V)),activate(V)) ISNEPAL(V) -> ACTIVATE(V) ISNEPAL(V) -> ISPALLISTKIND(activate(V)) ISPAL(V) -> U71#(isPalListKind(activate(V)),activate(V)) ISPAL(V) -> ACTIVATE(V) ISPAL(V) -> ISPALLISTKIND(activate(V)) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V1) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V2) ISPALLISTKIND(n____(V1,V2)) -> AND(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) ISPALLISTKIND(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u Problem 1: SCC Processor: -> Pairs: U11#(tt,V) -> U12#(isNeList(activate(V))) U11#(tt,V) -> ACTIVATE(V) U11#(tt,V) -> ISNELIST(activate(V)) U21#(tt,V1,V2) -> U22#(isList(activate(V1)),activate(V2)) U21#(tt,V1,V2) -> ACTIVATE(V1) U21#(tt,V1,V2) -> ACTIVATE(V2) U21#(tt,V1,V2) -> ISLIST(activate(V1)) U22#(tt,V2) -> U23#(isList(activate(V2))) U22#(tt,V2) -> ACTIVATE(V2) U22#(tt,V2) -> ISLIST(activate(V2)) U31#(tt,V) -> U32#(isQid(activate(V))) U31#(tt,V) -> ACTIVATE(V) U31#(tt,V) -> ISQID(activate(V)) U41#(tt,V1,V2) -> U42#(isList(activate(V1)),activate(V2)) U41#(tt,V1,V2) -> ACTIVATE(V1) U41#(tt,V1,V2) -> ACTIVATE(V2) U41#(tt,V1,V2) -> ISLIST(activate(V1)) U42#(tt,V2) -> U43#(isNeList(activate(V2))) U42#(tt,V2) -> ACTIVATE(V2) U42#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V1,V2) -> U52#(isNeList(activate(V1)),activate(V2)) U51#(tt,V1,V2) -> ACTIVATE(V1) U51#(tt,V1,V2) -> ACTIVATE(V2) U51#(tt,V1,V2) -> ISNELIST(activate(V1)) U52#(tt,V2) -> U53#(isList(activate(V2))) U52#(tt,V2) -> ACTIVATE(V2) U52#(tt,V2) -> ISLIST(activate(V2)) U61#(tt,V) -> U62#(isQid(activate(V))) U61#(tt,V) -> ACTIVATE(V) U61#(tt,V) -> ISQID(activate(V)) U71#(tt,V) -> U72#(isNePal(activate(V))) U71#(tt,V) -> ACTIVATE(V) U71#(tt,V) -> ISNEPAL(activate(V)) __#(__(X,Y),Z) -> __#(X,__(Y,Z)) __#(__(X,Y),Z) -> __#(Y,Z) ACTIVATE(n____(X1,X2)) -> __#(X1,X2) ACTIVATE(n__a) -> A ACTIVATE(n__and(X1,X2)) -> AND(X1,X2) ACTIVATE(n__e) -> E ACTIVATE(n__i) -> I ACTIVATE(n__isPalListKind(X)) -> ISPALLISTKIND(X) ACTIVATE(n__nil) -> NIL ACTIVATE(n__o) -> O ACTIVATE(n__u) -> U AND(tt,X) -> ACTIVATE(X) ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISLIST(n____(V1,V2)) -> ACTIVATE(V1) ISLIST(n____(V1,V2)) -> ACTIVATE(V2) ISLIST(n____(V1,V2)) -> AND(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) ISLIST(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) ISLIST(V) -> U11#(isPalListKind(activate(V)),activate(V)) ISLIST(V) -> ACTIVATE(V) ISLIST(V) -> ISPALLISTKIND(activate(V)) ISNELIST(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISNELIST(n____(V1,V2)) -> ACTIVATE(V1) ISNELIST(n____(V1,V2)) -> ACTIVATE(V2) ISNELIST(n____(V1,V2)) -> AND(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) ISNELIST(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) ISNELIST(V) -> U31#(isPalListKind(activate(V)),activate(V)) ISNELIST(V) -> ACTIVATE(V) ISNELIST(V) -> ISPALLISTKIND(activate(V)) ISNEPAL(n____(I,__(P,I))) -> ACTIVATE(I) ISNEPAL(n____(I,__(P,I))) -> ACTIVATE(P) ISNEPAL(n____(I,__(P,I))) -> AND(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) ISNEPAL(n____(I,__(P,I))) -> AND(isQid(activate(I)),n__isPalListKind(activate(I))) ISNEPAL(n____(I,__(P,I))) -> ISPAL(activate(P)) ISNEPAL(n____(I,__(P,I))) -> ISQID(activate(I)) ISNEPAL(V) -> U61#(isPalListKind(activate(V)),activate(V)) ISNEPAL(V) -> ACTIVATE(V) ISNEPAL(V) -> ISPALLISTKIND(activate(V)) ISPAL(V) -> U71#(isPalListKind(activate(V)),activate(V)) ISPAL(V) -> ACTIVATE(V) ISPAL(V) -> ISPALLISTKIND(activate(V)) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V1) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V2) ISPALLISTKIND(n____(V1,V2)) -> AND(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) ISPALLISTKIND(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: __#(__(X,Y),Z) -> __#(X,__(Y,Z)) __#(__(X,Y),Z) -> __#(Y,Z) ->->-> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->->Cycle: ->->-> Pairs: ACTIVATE(n__and(X1,X2)) -> AND(X1,X2) ACTIVATE(n__isPalListKind(X)) -> ISPALLISTKIND(X) AND(tt,X) -> ACTIVATE(X) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V1) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V2) ISPALLISTKIND(n____(V1,V2)) -> AND(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) ISPALLISTKIND(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) ->->-> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->->Cycle: ->->-> Pairs: U71#(tt,V) -> ISNEPAL(activate(V)) ISNEPAL(n____(I,__(P,I))) -> ISPAL(activate(P)) ISPAL(V) -> U71#(isPalListKind(activate(V)),activate(V)) ->->-> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->->Cycle: ->->-> Pairs: U11#(tt,V) -> ISNELIST(activate(V)) U21#(tt,V1,V2) -> U22#(isList(activate(V1)),activate(V2)) U21#(tt,V1,V2) -> ISLIST(activate(V1)) U22#(tt,V2) -> ISLIST(activate(V2)) U41#(tt,V1,V2) -> U42#(isList(activate(V1)),activate(V2)) U41#(tt,V1,V2) -> ISLIST(activate(V1)) U42#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V1,V2) -> U52#(isNeList(activate(V1)),activate(V2)) U51#(tt,V1,V2) -> ISNELIST(activate(V1)) U52#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISLIST(V) -> U11#(isPalListKind(activate(V)),activate(V)) ISNELIST(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ->->-> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u The problem is decomposed in 4 subproblems. Problem 1.1: Subterm Processor: -> Pairs: __#(__(X,Y),Z) -> __#(X,__(Y,Z)) __#(__(X,Y),Z) -> __#(Y,Z) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Projection: pi(__#) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pair Processor: -> Pairs: ACTIVATE(n__and(X1,X2)) -> AND(X1,X2) ACTIVATE(n__isPalListKind(X)) -> ISPALLISTKIND(X) AND(tt,X) -> ACTIVATE(X) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V1) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V2) ISPALLISTKIND(n____(V1,V2)) -> AND(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) ISPALLISTKIND(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u -> Usable rules: __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) nil -> n__nil o -> n__o u -> n__u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [a] = 1 [activate](X) = X [and](X1,X2) = X1 + X2 [e] = 2 [i] = 2 [isPalListKind](X) = X + 2 [nil] = 2 [o] = 2 [u] = 2 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 1 [n__and](X1,X2) = X1 + X2 [n__e] = 2 [n__i] = 2 [n__isPalListKind](X) = X + 2 [n__nil] = 2 [n__o] = 2 [n__u] = 2 [tt] = 2 [ACTIVATE](X) = 2.X [AND](X1,X2) = X1 + 2.X2 [ISPALLISTKIND](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: ACTIVATE(n__and(X1,X2)) -> AND(X1,X2) ACTIVATE(n__isPalListKind(X)) -> ISPALLISTKIND(X) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V1) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V2) ISPALLISTKIND(n____(V1,V2)) -> AND(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) ISPALLISTKIND(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__isPalListKind(X)) -> ISPALLISTKIND(X) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V1) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V2) ISPALLISTKIND(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) ->->-> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u Problem 1.2: Reduction Pair Processor: -> Pairs: ACTIVATE(n__isPalListKind(X)) -> ISPALLISTKIND(X) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V1) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V2) ISPALLISTKIND(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u -> Usable rules: __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) nil -> n__nil o -> n__o u -> n__u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [a] = 2 [activate](X) = X + 2 [and](X1,X2) = X2 + 2 [e] = 2 [i] = 2 [isPalListKind](X) = 2.X + 2 [nil] = 0 [o] = 2 [u] = 2 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 2 [n__and](X1,X2) = X2 + 2 [n__e] = 2 [n__i] = 2 [n__isPalListKind](X) = 2.X [n__nil] = 0 [n__o] = 2 [n__u] = 1 [tt] = 0 [ACTIVATE](X) = 2.X + 2 [ISPALLISTKIND](X) = 2.X + 1 Problem 1.2: SCC Processor: -> Pairs: ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V1) ISPALLISTKIND(n____(V1,V2)) -> ACTIVATE(V2) ISPALLISTKIND(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISPALLISTKIND(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) ->->-> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u Problem 1.2: Reduction Pair Processor: -> Pairs: ISPALLISTKIND(n____(V1,V2)) -> ISPALLISTKIND(activate(V1)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u -> Usable rules: __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) nil -> n__nil o -> n__o u -> n__u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [a] = 1 [activate](X) = X + 1 [and](X1,X2) = X1 + X2 [e] = 2 [i] = 2 [isPalListKind](X) = 2.X + 1 [nil] = 0 [o] = 2 [u] = 2 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 0 [n__and](X1,X2) = X1 + X2 [n__e] = 2 [n__i] = 2 [n__isPalListKind](X) = 2.X [n__nil] = 0 [n__o] = 2 [n__u] = 2 [tt] = 1 [ISPALLISTKIND](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Reduction Pair Processor: -> Pairs: U71#(tt,V) -> ISNEPAL(activate(V)) ISNEPAL(n____(I,__(P,I))) -> ISPAL(activate(P)) ISPAL(V) -> U71#(isPalListKind(activate(V)),activate(V)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u -> Usable rules: __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) nil -> n__nil o -> n__o u -> n__u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [a] = 2 [activate](X) = X [and](X1,X2) = 2.X1 + X2 [e] = 1 [i] = 0 [isPalListKind](X) = 2.X + 2 [nil] = 2 [o] = 2 [u] = 2 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 2 [n__and](X1,X2) = 2.X1 + X2 [n__e] = 1 [n__i] = 0 [n__isPalListKind](X) = 2.X + 2 [n__nil] = 2 [n__o] = 2 [n__u] = 2 [tt] = 2 [U71#](X1,X2) = X2 + 2 [ISNEPAL](X) = X + 1 [ISPAL](X) = 2.X + 2 Problem 1.3: SCC Processor: -> Pairs: ISNEPAL(n____(I,__(P,I))) -> ISPAL(activate(P)) ISPAL(V) -> U71#(isPalListKind(activate(V)),activate(V)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Reduction Pair Processor: -> Pairs: U11#(tt,V) -> ISNELIST(activate(V)) U21#(tt,V1,V2) -> U22#(isList(activate(V1)),activate(V2)) U21#(tt,V1,V2) -> ISLIST(activate(V1)) U22#(tt,V2) -> ISLIST(activate(V2)) U41#(tt,V1,V2) -> U42#(isList(activate(V1)),activate(V2)) U41#(tt,V1,V2) -> ISLIST(activate(V1)) U42#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V1,V2) -> U52#(isNeList(activate(V1)),activate(V2)) U51#(tt,V1,V2) -> ISNELIST(activate(V1)) U52#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISLIST(V) -> U11#(isPalListKind(activate(V)),activate(V)) ISNELIST(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u -> Usable rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 2.X2 + 2 [U12](X) = 2 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2 [U22](X1,X2) = X1 + 1 [U23](X) = 2 [U31](X1,X2) = 2.X2 + 2 [U32](X) = 2 [U41](X1,X2,X3) = X1 + 2.X2 + X3 + 1 [U42](X1,X2) = X1 [U43](X) = 2 [U51](X1,X2,X3) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = X1 [U53](X) = 2 [__](X1,X2) = 2.X1 + X2 + 2 [a] = 2 [activate](X) = X [and](X1,X2) = X2 [e] = 2 [i] = 2 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 2 [isPalListKind](X) = X + 2 [isQid](X) = 2.X + 2 [nil] = 2 [o] = 1 [u] = 2 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 2 [n__and](X1,X2) = X2 [n__e] = 2 [n__i] = 2 [n__isPalListKind](X) = X + 2 [n__nil] = 2 [n__o] = 1 [n__u] = 2 [tt] = 2 [U11#](X1,X2) = 2.X2 + 2 [U21#](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U22#](X1,X2) = 2.X2 + 2 [U41#](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U42#](X1,X2) = 2.X2 + 2 [U51#](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U52#](X1,X2) = X1 + 2.X2 [ISLIST](X) = 2.X + 2 [ISNELIST](X) = 2.X + 1 Problem 1.4: SCC Processor: -> Pairs: U21#(tt,V1,V2) -> U22#(isList(activate(V1)),activate(V2)) U21#(tt,V1,V2) -> ISLIST(activate(V1)) U22#(tt,V2) -> ISLIST(activate(V2)) U41#(tt,V1,V2) -> U42#(isList(activate(V1)),activate(V2)) U41#(tt,V1,V2) -> ISLIST(activate(V1)) U42#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V1,V2) -> U52#(isNeList(activate(V1)),activate(V2)) U51#(tt,V1,V2) -> ISNELIST(activate(V1)) U52#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISLIST(V) -> U11#(isPalListKind(activate(V)),activate(V)) ISNELIST(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(tt,V1,V2) -> U22#(isList(activate(V1)),activate(V2)) U21#(tt,V1,V2) -> ISLIST(activate(V1)) U22#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ->->-> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->->Cycle: ->->-> Pairs: U41#(tt,V1,V2) -> U42#(isList(activate(V1)),activate(V2)) U42#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V1,V2) -> ISNELIST(activate(V1)) ISNELIST(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ->->-> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u The problem is decomposed in 2 subproblems. Problem 1.4.1: Reduction Pair Processor: -> Pairs: U21#(tt,V1,V2) -> U22#(isList(activate(V1)),activate(V2)) U21#(tt,V1,V2) -> ISLIST(activate(V1)) U22#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u -> Usable rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 2 [U12](X) = 2 [U21](X1,X2,X3) = 2 [U22](X1,X2) = 2 [U23](X) = 2 [U31](X1,X2) = 2.X1 + 2 [U32](X) = 2.X + 2 [U41](X1,X2,X3) = 2.X1 + 2 [U42](X1,X2) = X1 + 2 [U43](X) = 2 [U51](X1,X2,X3) = 2.X1 + 2 [U52](X1,X2) = 2 [U53](X) = 2 [__](X1,X2) = 2.X1 + X2 + 2 [a] = 2 [activate](X) = X [and](X1,X2) = X2 + 2 [e] = 2 [i] = 2 [isList](X) = 2 [isNeList](X) = 2.X + 2 [isPalListKind](X) = X [isQid](X) = 2 [nil] = 2 [o] = 2 [u] = 2 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 2 [n__and](X1,X2) = X2 + 2 [n__e] = 2 [n__i] = 2 [n__isPalListKind](X) = X [n__nil] = 2 [n__o] = 2 [n__u] = 2 [tt] = 2 [U21#](X1,X2,X3) = 2.X2 + 2.X3 + 1 [U22#](X1,X2) = 2.X2 [ISLIST](X) = 2.X Problem 1.4.1: SCC Processor: -> Pairs: U21#(tt,V1,V2) -> ISLIST(activate(V1)) U22#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(tt,V1,V2) -> ISLIST(activate(V1)) ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ->->-> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u Problem 1.4.1: Reduction Pair Processor: -> Pairs: U21#(tt,V1,V2) -> ISLIST(activate(V1)) ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u -> Usable rules: __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) nil -> n__nil o -> n__o u -> n__u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [a] = 2 [activate](X) = X [and](X1,X2) = X2 + 1 [e] = 2 [i] = 2 [isPalListKind](X) = X [nil] = 0 [o] = 2 [u] = 1 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 2 [n__and](X1,X2) = X2 + 1 [n__e] = 2 [n__i] = 2 [n__isPalListKind](X) = X [n__nil] = 0 [n__o] = 2 [n__u] = 1 [tt] = 0 [U21#](X1,X2,X3) = 2.X1 + 2.X2 + 1 [ISLIST](X) = 2.X Problem 1.4.1: SCC Processor: -> Pairs: ISLIST(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4.2: Reduction Pair Processor: -> Pairs: U41#(tt,V1,V2) -> U42#(isList(activate(V1)),activate(V2)) U42#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V1,V2) -> ISNELIST(activate(V1)) ISNELIST(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u -> Usable rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 0 [U12](X) = 0 [U21](X1,X2,X3) = 0 [U22](X1,X2) = 2.X1 [U23](X) = 2.X [U31](X1,X2) = 2.X2 + 2 [U32](X) = 2.X [U41](X1,X2,X3) = 2.X3 + 2 [U42](X1,X2) = 2.X1 + 2.X2 + 2 [U43](X) = 2 [U51](X1,X2,X3) = X2 + 2.X3 + 2 [U52](X1,X2) = 2.X2 + 1 [U53](X) = 0 [__](X1,X2) = 2.X1 + X2 + 2 [a] = 2 [activate](X) = X [and](X1,X2) = X2 [e] = 2 [i] = 0 [isList](X) = 0 [isNeList](X) = 2.X + 2 [isPalListKind](X) = 1 [isQid](X) = X + 1 [nil] = 0 [o] = 0 [u] = 2 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 2 [n__and](X1,X2) = X2 [n__e] = 2 [n__i] = 0 [n__isPalListKind](X) = 1 [n__nil] = 0 [n__o] = 0 [n__u] = 2 [tt] = 0 [U41#](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [U42#](X1,X2) = 2.X1 + 2.X2 + 1 [U51#](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [ISNELIST](X) = 2.X + 1 Problem 1.4.2: SCC Processor: -> Pairs: U42#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V1,V2) -> ISNELIST(activate(V1)) ISNELIST(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(tt,V1,V2) -> ISNELIST(activate(V1)) ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) ->->-> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u Problem 1.4.2: Reduction Pair Processor: -> Pairs: U51#(tt,V1,V2) -> ISNELIST(activate(V1)) ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u -> Usable rules: __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) nil -> n__nil o -> n__o u -> n__u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [a] = 2 [activate](X) = X [and](X1,X2) = X2 + 1 [e] = 2 [i] = 2 [isPalListKind](X) = X [nil] = 0 [o] = 2 [u] = 1 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 2 [n__and](X1,X2) = X2 + 1 [n__e] = 2 [n__i] = 2 [n__isPalListKind](X) = X [n__nil] = 0 [n__o] = 2 [n__u] = 1 [tt] = 0 [U51#](X1,X2,X3) = 2.X1 + 2.X2 + 1 [ISNELIST](X) = 2.X Problem 1.4.2: SCC Processor: -> Pairs: ISNELIST(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) -> Rules: U11(tt,V) -> U12(isNeList(activate(V))) U12(tt) -> tt U21(tt,V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt,V2) -> U23(isList(activate(V2))) U23(tt) -> tt U31(tt,V) -> U32(isQid(activate(V))) U32(tt) -> tt U41(tt,V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt,V2) -> U43(isNeList(activate(V2))) U43(tt) -> tt U51(tt,V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt,V2) -> U53(isList(activate(V2))) U53(tt) -> tt U61(tt,V) -> U62(isQid(activate(V))) U62(tt) -> tt U71(tt,V) -> U72(isNePal(activate(V))) U72(tt) -> tt __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil,X) -> X __(X,nil) -> X __(X1,X2) -> n____(X1,X2) a -> n__a activate(n____(X1,X2)) -> __(X1,X2) activate(n__a) -> a activate(n__and(X1,X2)) -> and(X1,X2) activate(n__e) -> e activate(n__i) -> i activate(n__isPalListKind(X)) -> isPalListKind(X) activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X and(tt,X) -> activate(X) and(X1,X2) -> n__and(X1,X2) e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P)))) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isPal(n__nil) -> tt isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) isPalListKind(n__a) -> tt isPalListKind(n__e) -> tt isPalListKind(n__i) -> tt isPalListKind(n__nil) -> tt isPalListKind(n__o) -> tt isPalListKind(n__u) -> tt isPalListKind(X) -> n__isPalListKind(X) isQid(n__a) -> tt isQid(n__e) -> tt isQid(n__i) -> tt isQid(n__o) -> tt isQid(n__u) -> tt nil -> n__nil o -> n__o u -> n__u ->Strongly Connected Components: There is no strongly connected component The problem is finite.