/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR I P V V1 V2 X X1 X2 Y Z) (RULES U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: U21#(tt,V2) -> U22#(isList(activate(V2))) U21#(tt,V2) -> ACTIVATE(V2) U21#(tt,V2) -> ISLIST(activate(V2)) U41#(tt,V2) -> U42#(isNeList(activate(V2))) U41#(tt,V2) -> ACTIVATE(V2) U41#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V2) -> U52#(isList(activate(V2))) U51#(tt,V2) -> ACTIVATE(V2) U51#(tt,V2) -> ISLIST(activate(V2)) U71#(tt,P) -> U72#(isPal(activate(P))) U71#(tt,P) -> ACTIVATE(P) U71#(tt,P) -> ISPAL(activate(P)) __#(__(X,Y),Z) -> __#(X,__(Y,Z)) __#(__(X,Y),Z) -> __#(Y,Z) ACTIVATE(n____(X1,X2)) -> __#(activate(X1),activate(X2)) ACTIVATE(n____(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n____(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__a) -> A ACTIVATE(n__e) -> E ACTIVATE(n__i) -> I ACTIVATE(n__nil) -> NIL ACTIVATE(n__o) -> O ACTIVATE(n__u) -> U ISLIST(n____(V1,V2)) -> U21#(isList(activate(V1)),activate(V2)) ISLIST(n____(V1,V2)) -> ACTIVATE(V1) ISLIST(n____(V1,V2)) -> ACTIVATE(V2) ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> U11#(isNeList(activate(V))) ISLIST(V) -> ACTIVATE(V) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ACTIVATE(V1) ISNELIST(n____(V1,V2)) -> ACTIVATE(V2) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) ISNELIST(V) -> U31#(isQid(activate(V))) ISNELIST(V) -> ACTIVATE(V) ISNELIST(V) -> ISQID(activate(V)) ISNEPAL(n____(I,n____(P,I))) -> U71#(isQid(activate(I)),activate(P)) ISNEPAL(n____(I,n____(P,I))) -> ACTIVATE(I) ISNEPAL(n____(I,n____(P,I))) -> ACTIVATE(P) ISNEPAL(n____(I,n____(P,I))) -> ISQID(activate(I)) ISNEPAL(V) -> U61#(isQid(activate(V))) ISNEPAL(V) -> ACTIVATE(V) ISNEPAL(V) -> ISQID(activate(V)) ISPAL(V) -> U81#(isNePal(activate(V))) ISPAL(V) -> ACTIVATE(V) ISPAL(V) -> ISNEPAL(activate(V)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: U21#(tt,V2) -> U22#(isList(activate(V2))) U21#(tt,V2) -> ACTIVATE(V2) U21#(tt,V2) -> ISLIST(activate(V2)) U41#(tt,V2) -> U42#(isNeList(activate(V2))) U41#(tt,V2) -> ACTIVATE(V2) U41#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V2) -> U52#(isList(activate(V2))) U51#(tt,V2) -> ACTIVATE(V2) U51#(tt,V2) -> ISLIST(activate(V2)) U71#(tt,P) -> U72#(isPal(activate(P))) U71#(tt,P) -> ACTIVATE(P) U71#(tt,P) -> ISPAL(activate(P)) __#(__(X,Y),Z) -> __#(X,__(Y,Z)) __#(__(X,Y),Z) -> __#(Y,Z) ACTIVATE(n____(X1,X2)) -> __#(activate(X1),activate(X2)) ACTIVATE(n____(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n____(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__a) -> A ACTIVATE(n__e) -> E ACTIVATE(n__i) -> I ACTIVATE(n__nil) -> NIL ACTIVATE(n__o) -> O ACTIVATE(n__u) -> U ISLIST(n____(V1,V2)) -> U21#(isList(activate(V1)),activate(V2)) ISLIST(n____(V1,V2)) -> ACTIVATE(V1) ISLIST(n____(V1,V2)) -> ACTIVATE(V2) ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> U11#(isNeList(activate(V))) ISLIST(V) -> ACTIVATE(V) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ACTIVATE(V1) ISNELIST(n____(V1,V2)) -> ACTIVATE(V2) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) ISNELIST(V) -> U31#(isQid(activate(V))) ISNELIST(V) -> ACTIVATE(V) ISNELIST(V) -> ISQID(activate(V)) ISNEPAL(n____(I,n____(P,I))) -> U71#(isQid(activate(I)),activate(P)) ISNEPAL(n____(I,n____(P,I))) -> ACTIVATE(I) ISNEPAL(n____(I,n____(P,I))) -> ACTIVATE(P) ISNEPAL(n____(I,n____(P,I))) -> ISQID(activate(I)) ISNEPAL(V) -> U61#(isQid(activate(V))) ISNEPAL(V) -> ACTIVATE(V) ISNEPAL(V) -> ISQID(activate(V)) ISPAL(V) -> U81#(isNePal(activate(V))) ISPAL(V) -> ACTIVATE(V) ISPAL(V) -> ISNEPAL(activate(V)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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____(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n____(X1,X2)) -> ACTIVATE(X2) ->->-> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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,P) -> ISPAL(activate(P)) ISNEPAL(n____(I,n____(P,I))) -> U71#(isQid(activate(I)),activate(P)) ISPAL(V) -> ISNEPAL(activate(V)) ->->-> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: U21#(tt,V2) -> ISLIST(activate(V2)) U41#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> U21#(isList(activate(V1)),activate(V2)) ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) ->->-> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: Subterm Processor: -> Pairs: ACTIVATE(n____(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n____(X1,X2)) -> ACTIVATE(X2) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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(ACTIVATE) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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,P) -> ISPAL(activate(P)) ISNEPAL(n____(I,n____(P,I))) -> U71#(isQid(activate(I)),activate(P)) ISPAL(V) -> ISNEPAL(activate(V)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i 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: [__](X1,X2) = X1 + X2 + 2 [a] = 2 [activate](X) = X [e] = 0 [i] = 2 [isQid](X) = 2.X + 2 [nil] = 2 [o] = 2 [u] = 2 [n____](X1,X2) = X1 + X2 + 2 [n__a] = 2 [n__e] = 0 [n__i] = 2 [n__nil] = 2 [n__o] = 2 [n__u] = 2 [tt] = 2 [U71#](X1,X2) = 2.X1 + 2.X2 [ISNEPAL](X) = 2.X [ISPAL](X) = 2.X Problem 1.3: SCC Processor: -> Pairs: ISNEPAL(n____(I,n____(P,I))) -> U71#(isQid(activate(I)),activate(P)) ISPAL(V) -> ISNEPAL(activate(V)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: U21#(tt,V2) -> ISLIST(activate(V2)) U41#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> U21#(isList(activate(V1)),activate(V2)) ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) 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](X) = 0 [U21](X1,X2) = 2.X1 [U22](X) = 2.X [U31](X) = 2 [U41](X1,X2) = 2.X1 + 2.X2 + 2 [U42](X) = X [U51](X1,X2) = 2.X1 + X2 + 1 [U52](X) = 2.X + 1 [__](X1,X2) = 2.X1 + X2 + 2 [a] = 1 [activate](X) = X [e] = 1 [i] = 0 [isList](X) = 0 [isNeList](X) = 2.X + 2 [isQid](X) = 2 [nil] = 1 [o] = 2 [u] = 2 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 1 [n__e] = 1 [n__i] = 0 [n__nil] = 1 [n__o] = 2 [n__u] = 2 [tt] = 0 [U21#](X1,X2) = 2.X2 + 2 [U41#](X1,X2) = 2.X1 + 2.X2 + 2 [U51#](X1,X2) = X1 + 2.X2 + 2 [ISLIST](X) = 2.X + 1 [ISNELIST](X) = 2.X + 1 Problem 1.4: SCC Processor: -> Pairs: U41#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> U21#(isList(activate(V1)),activate(V2)) ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: U41#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) ->->-> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: Reduction Pair Processor: -> Pairs: U41#(tt,V2) -> ISNELIST(activate(V2)) U51#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) 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](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 1 [U22](X) = X + 2 [U31](X) = X [U41](X1,X2) = 2.X1 + 2.X2 + 2 [U42](X) = X [U51](X1,X2) = X1 + 2 [U52](X) = 2 [__](X1,X2) = 2.X1 + X2 + 2 [a] = 2 [activate](X) = X [e] = 2 [i] = 2 [isList](X) = 2.X + 1 [isNeList](X) = 2.X + 1 [isQid](X) = 2.X + 1 [nil] = 2 [o] = 2 [u] = 2 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 2 [n__e] = 2 [n__i] = 2 [n__nil] = 2 [n__o] = 2 [n__u] = 2 [tt] = 2 [U41#](X1,X2) = 2.X1 + 2.X2 + 2 [U51#](X1,X2) = 2.X1 + 2.X2 + 2 [ISLIST](X) = 2.X + 2 [ISNELIST](X) = 2.X Problem 1.4: SCC Processor: -> Pairs: U51#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) ->->-> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: Reduction Pair Processor: -> Pairs: U51#(tt,V2) -> ISLIST(activate(V2)) ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) 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](X) = 2 [U21](X1,X2) = 2 [U22](X) = 2 [U31](X) = X [U41](X1,X2) = 2.X2 [U42](X) = X [U51](X1,X2) = 2.X1 [U52](X) = 2 [__](X1,X2) = 2.X1 + X2 [a] = 2 [activate](X) = X [e] = 2 [i] = 2 [isList](X) = 2 [isNeList](X) = 2.X [isQid](X) = 2.X [nil] = 1 [o] = 2 [u] = 2 [n____](X1,X2) = 2.X1 + X2 [n__a] = 2 [n__e] = 2 [n__i] = 2 [n__nil] = 1 [n__o] = 2 [n__u] = 2 [tt] = 2 [U51#](X1,X2) = 2.X1 + 2.X2 + 2 [ISLIST](X) = 2.X + 2 [ISNELIST](X) = 2.X + 2 Problem 1.4: SCC Processor: -> Pairs: ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) ->->-> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: Reduction Pair Processor: -> Pairs: ISLIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i 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 [e] = 1 [i] = 0 [nil] = 1 [o] = 2 [u] = 1 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 2 [n__e] = 1 [n__i] = 0 [n__nil] = 1 [n__o] = 2 [n__u] = 1 [ISLIST](X) = 2.X + 1 [ISNELIST](X) = 2.X Problem 1.4: SCC Processor: -> Pairs: ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) ->->-> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: Reduction Pair Processor: -> Pairs: ISLIST(V) -> ISNELIST(activate(V)) ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i 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 [e] = 1 [i] = 0 [nil] = 2 [o] = 1 [u] = 0 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 1 [n__e] = 1 [n__i] = 0 [n__nil] = 2 [n__o] = 1 [n__u] = 0 [ISLIST](X) = 2.X + 1 [ISNELIST](X) = 2.X Problem 1.4: SCC Processor: -> Pairs: ISNELIST(n____(V1,V2)) -> ISLIST(activate(V1)) ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) ->->-> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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: Reduction Pair Processor: -> Pairs: ISNELIST(n____(V1,V2)) -> ISNELIST(activate(V1)) -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i 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) = 2.X + 1 [e] = 2 [i] = 2 [nil] = 2 [o] = 2 [u] = 2 [n____](X1,X2) = 2.X1 + X2 + 2 [n__a] = 2 [n__e] = 2 [n__i] = 2 [n__nil] = 2 [n__o] = 1 [n__u] = 2 [ISNELIST](X) = 2.X Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: U11(tt) -> tt U21(tt,V2) -> U22(isList(activate(V2))) U22(tt) -> tt U31(tt) -> tt U41(tt,V2) -> U42(isNeList(activate(V2))) U42(tt) -> tt U51(tt,V2) -> U52(isList(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt,P) -> U72(isPal(activate(P))) U72(tt) -> tt U81(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)) -> __(activate(X1),activate(X2)) activate(n__a) -> a activate(n__e) -> e activate(n__i) -> i activate(n__nil) -> nil activate(n__o) -> o activate(n__u) -> u activate(X) -> X e -> n__e i -> n__i isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil) -> tt isList(V) -> U11(isNeList(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isNePal(V) -> U61(isQid(activate(V))) isPal(n__nil) -> tt isPal(V) -> U81(isNePal(activate(V))) 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.