YES Problem 1: (VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S) (RULES 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(tt,V2:S) -> U12#(isNat(activate(V2:S))) U11#(tt,V2:S) -> ACTIVATE(V2:S) U11#(tt,V2:S) -> ISNAT(activate(V2:S)) U31#(tt,V2:S) -> U32#(isNat(activate(V2:S))) U31#(tt,V2:S) -> ACTIVATE(V2:S) U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U52#(tt,M:S,N:S) -> S(plus(activate(N:S),activate(M:S))) U61#(tt) -> 0# U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__0) -> 0# ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__s(X:S)) -> S(activate(X:S)) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> U21#(isNat(activate(V1:S))) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> U61#(isNat(N:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: SCC Processor: -> Pairs: U11#(tt,V2:S) -> U12#(isNat(activate(V2:S))) U11#(tt,V2:S) -> ACTIVATE(V2:S) U11#(tt,V2:S) -> ISNAT(activate(V2:S)) U31#(tt,V2:S) -> U32#(isNat(activate(V2:S))) U31#(tt,V2:S) -> ACTIVATE(V2:S) U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U52#(tt,M:S,N:S) -> S(plus(activate(N:S),activate(M:S))) U61#(tt) -> 0# U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__0) -> 0# ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__s(X:S)) -> S(activate(X:S)) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> U21#(isNat(activate(V1:S))) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> U61#(isNat(N:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(tt,V2:S) -> ACTIVATE(V2:S) U11#(tt,V2:S) -> ISNAT(activate(V2:S)) U31#(tt,V2:S) -> ACTIVATE(V2:S) U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U11#(tt,V2:S) -> ACTIVATE(V2:S) U11#(tt,V2:S) -> ISNAT(activate(V2:S)) U31#(tt,V2:S) -> ACTIVATE(V2:S) U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X2 + 2 [U12](X) = 2 [U21](X) = X [U31](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [U32](X) = 2 [U41](X1,X2) = X2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [tt] = 0 [U11#](X1,X2) = X2 + 2 [U31#](X1,X2) = X2 + 2 [U41#](X1,X2) = X2 [U51#](X1,X2,X3) = X2 + X3 + 2 [U52#](X1,X2,X3) = X2 + X3 + 2 [U71#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X [ISNAT](X) = X + 1 [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: U11#(tt,V2:S) -> ISNAT(activate(V2:S)) U31#(tt,V2:S) -> ACTIVATE(V2:S) U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(tt,V2:S) -> ISNAT(activate(V2:S)) U31#(tt,V2:S) -> ACTIVATE(V2:S) U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U11#(tt,V2:S) -> ISNAT(activate(V2:S)) U31#(tt,V2:S) -> ACTIVATE(V2:S) U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X2 + 2 [U12](X) = 2 [U21](X) = X + 2 [U31](X1,X2) = 2.X1.X2 + 2.X2 + 2 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [tt] = 2 [U11#](X1,X2) = X2 + 2 [U31#](X1,X2) = X2 + 2 [U41#](X1,X2) = X2 + 1 [U51#](X1,X2,X3) = X2 + X3 + 1 [U52#](X1,X2,X3) = X2 + X3 + 1 [U71#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X + 1 [ISNAT](X) = X + 1 [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U31#(tt,V2:S) -> ACTIVATE(V2:S) U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U31#(tt,V2:S) -> ACTIVATE(V2:S) U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U31#(tt,V2:S) -> ACTIVATE(V2:S) U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 0 [U11](X1,X2) = X1 + 2.X2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2.X1.X2 + 2.X2 + 2 [U32](X) = 2 [U41](X1,X2) = X2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 0 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X + 2 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 0 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 2 [U31#](X1,X2) = 2.X2 + 2 [U41#](X1,X2) = X2 [U51#](X1,X2,X3) = X2 + X3 [U52#](X1,X2,X3) = X2 + X3 [U71#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X [ISNAT](X) = X [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U31#(tt,V2:S) -> ISNAT(activate(V2:S)) U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + X2 [U12](X) = 2 [U21](X) = X + 1 [U31](X1,X2) = X1 + 2.X2 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X + 2 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 2 [U31#](X1,X2) = X2 + 2 [U41#](X1,X2) = X2 + 2 [U51#](X1,X2,X3) = X2 + X3 + 1 [U52#](X1,X2,X3) = X2 + X3 + 1 [U71#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X [ISNAT](X) = X [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = 2.X1.X2 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> U31#(isNat(activate(V1:S)),activate(V2:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U41#(tt,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2.X2 + 2 [U12](X) = X [U21](X) = 2 [U31](X1,X2) = 2.X1.X2 + X1 + X2 + 1 [U32](X) = 2.X + 2 [U41](X1,X2) = X2 + 1 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2.X + 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [tt] = 2 [U41#](X1,X2) = X2 + 2 [U51#](X1,X2,X3) = X2 + X3 + 1 [U52#](X1,X2,X3) = X2 + X3 + 1 [U71#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X + 1 [ISNAT](X) = X + 1 [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U41#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U51#(tt,M:S,N:S) -> U52#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + 2.X2 + 2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X + 2 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [tt] = 2 [U51#](X1,X2,X3) = X2 + X3 + 2 [U52#](X1,X2,X3) = X2 + X3 + 1 [U71#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X + 1 [ISNAT](X) = X + 2 [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U52#(tt,M:S,N:S) -> ACTIVATE(M:S) U52#(tt,M:S,N:S) -> ACTIVATE(N:S) U52#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U51#(tt,M:S,N:S) -> ACTIVATE(M:S) U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X2 + 2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + X2 + 1 [tt] = 2 [U51#](X1,X2,X3) = X2 + X3 + 2 [U71#](X1,X2,X3) = 2.X2.X3 + X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X2.X3 + X2 + 2.X3 + 2 [ACTIVATE](X) = X + 1 [ISNAT](X) = X + 2 [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + X1 + X2 + 2 Problem 1: SCC Processor: -> Pairs: U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U51#(tt,M:S,N:S) -> ACTIVATE(N:S) U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + 2.X2 [U12](X) = 0 [U21](X) = X [U31](X1,X2) = 2.X2 + 2 [U32](X) = 0 [U41](X1,X2) = X2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X + 2 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 0 [U51#](X1,X2,X3) = X2 + X3 + 2 [U71#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X + 1 [ISNAT](X) = X + 1 [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = 2.X1.X2 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U51#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 [U12](X) = 2 [U21](X) = X [U31](X1,X2) = 2.X1.X2 + 2.X2 + 2 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = X2.X3 + X2 + 2.X3 + 2 [U72](X1,X2,X3) = X2.X3 + X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = X1.X2 + X1 + X2 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = X1.X2 + X1 + X2 + 2 [tt] = 2 [U51#](X1,X2,X3) = X2 + X3 + 2 [U71#](X1,X2,X3) = X2.X3 + X2 + 2.X3 + 2 [U72#](X1,X2,X3) = X2.X3 + X2 + 2.X3 + 2 [ACTIVATE](X) = X + 1 [ISNAT](X) = X + 1 [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = X1.X2 + X2 + 1 Problem 1: SCC Processor: -> Pairs: U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U51#(isNat(M:S),M:S,N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U71#(tt,M:S,N:S) -> U72#(isNat(activate(N:S)),activate(M:S),activate(N:S)) U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2.X1 [U12](X) = 2.X.X [U21](X) = 2.X [U31](X1,X2) = 2.X1.X2 + 2.X1 [U32](X) = 2.X.X + 2.X [U41](X1,X2) = 2.X1 + X2 + 1 [U51](X1,X2,X3) = 2.X1.X2 + X1.X3 + X2 + X3 + 2 [U52](X1,X2,X3) = 2.X1.X2.X3 + X1 + X2 + X3 + 2 [U61](X) = X.X + 2.X + 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [tt] = 0 [U71#](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 1 [ACTIVATE](X) = X + 1 [ISNAT](X) = X + 2 [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) U72#(tt,M:S,N:S) -> ACTIVATE(M:S) U72#(tt,M:S,N:S) -> ACTIVATE(N:S) U72#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U72#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U71#(tt,M:S,N:S) -> ACTIVATE(M:S) U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + 2.X2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2.X1.X2 + X1 + 2 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 2 [U71#](X1,X2,X3) = X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = 2.X [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 [X](X1,X2) = X1.X2 + 2.X2 Problem 1: SCC Processor: -> Pairs: U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U71#(tt,M:S,N:S) -> ACTIVATE(N:S) U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + 2.X2 [U12](X) = 0 [U21](X) = 0 [U31](X1,X2) = 2.X1.X2 + X1 + X2 + 2 [U32](X) = 0 [U41](X1,X2) = X2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = X2.X3 + X2 + 2.X3 + 2 [U72](X1,X2,X3) = X2.X3 + X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = X1.X2 + X1 + X2 + 1 [tt] = 0 [U71#](X1,X2,X3) = 2.X2.X3 + X2 + 2.X3 + 2 [ACTIVATE](X) = 2.X + 1 [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 + 1 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: U71#(tt,M:S,N:S) -> ISNAT(activate(N:S)) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + 2.X2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2.X2 + 2 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 2 [U71#](X1,X2,X3) = 2.X2 + 2.X3 + 2 [ACTIVATE](X) = 2.X + 1 [ISNAT](X) = 2.X + 1 [PLUS](X1,X2) = 2.X1 + 2.X2 [X](X1,X2) = 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> U71#(isNat(M:S),M:S,N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + 2.X2 + 2 [U12](X) = 2 [U21](X) = X + 1 [U31](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X + 2 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + X2 + 1 [tt] = 2 [ACTIVATE](X) = 2.X [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + X2 + 2 [U12](X) = 2 [U21](X) = X + 2 [U31](X1,X2) = 2.X1.X2 + X1 + 2.X2 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [tt] = 2 [ACTIVATE](X) = 2.X + 1 [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [X](X1,X2) = 2.X1 + 2.X2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X2 + 1 [U12](X) = 0 [U21](X) = X + 1 [U31](X1,X2) = 2.X1.X2 + X2 + 2 [U32](X) = 0 [U41](X1,X2) = X2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [tt] = 0 [ACTIVATE](X) = 2.X + 2 [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [X](X1,X2) = 2.X1.X2 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> ISNAT(M:S) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 1 [U11](X1,X2) = 2.X1.X2 + 2.X1 [U12](X) = 2.X [U21](X) = 0 [U31](X1,X2) = 2.X1.X2 + 2.X1 [U32](X) = X [U41](X1,X2) = 2.X1 + X2 + 2 [U51](X1,X2,X3) = 2.X1.X2 + 2.X1 + X2 + X3 + 2 [U52](X1,X2,X3) = X1 + X2 + X3 + 2 [U61](X) = 2.X.X + X + 1 [U71](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 0 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 [n__0] = 1 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 [tt] = 0 [ACTIVATE](X) = 2.X + 2 [ISNAT](X) = 2.X + 2 [X](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + 2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = X1 + 2 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + X + 2 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [tt] = 2 [ACTIVATE](X) = 2.X + 2 [ISNAT](X) = 2.X [X](X1,X2) = 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__x(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 1 [U11](X1,X2) = X1 + X2 + 2 [U12](X) = 2 [U21](X) = X + 2 [U31](X1,X2) = 2.X1.X2 + 2 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 1 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [tt] = 2 [ACTIVATE](X) = 2.X [ISNAT](X) = 2.X [X](X1,X2) = 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__x(X1:S,X2:S)) -> X(activate(X1:S),activate(X2:S)) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2 [U32](X) = 2 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 2 [ACTIVATE](X) = 2.X [ISNAT](X) = 2.X [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 Problem 1: SCC Processor: -> Pairs: ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__x(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) X(N:S,0) -> ISNAT(N:S) X(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ISNAT(n__x(V1:S,V2:S)) -> ISNAT(activate(V1:S)) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 [U12](X) = 2 [U21](X) = X [U31](X1,X2) = 2.X1 + 2 [U32](X) = 2 [U41](X1,X2) = X2 [U51](X1,X2,X3) = X3 [U52](X1,X2,X3) = X3 [U61](X) = X + 2 [U71](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X + 2 [plus](X1,X2) = X1 [s](X) = X [x](X1,X2) = 2.X1 + 2.X2 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 [n__s](X) = X [n__x](X1,X2) = 2.X1 + 2.X2 + 2 [tt] = 2 [ISNAT](X) = 2.X Problem 1: SCC Processor: -> Pairs: ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + X2 + 2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2.X2 + 2 [U32](X) = 2 [U41](X1,X2) = X2 + 1 [U51](X1,X2,X3) = 2.X2 + X3 + 2 [U52](X1,X2,X3) = 2.X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X + 2.X [plus](X1,X2) = X1 + 2.X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + 2.X2 + 1 [n__0] = 2 [n__plus](X1,X2) = X1 + 2.X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + 2.X2 + 1 [tt] = 2 [ISNAT](X) = X.X + 2.X Problem 1: SCC Processor: -> Pairs: ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ->->-> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) Problem 1: Reduction Pair Processor: -> Pairs: ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) -> Usable rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2 [U32](X) = 2 [U41](X1,X2) = X2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X2.X3 + 2.X3 + 2 [activate](X) = X [isNat](X) = 2.X.X [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2 [tt] = 2 [ISNAT](X) = X Problem 1: SCC Processor: -> Pairs: Empty -> Rules: 0 -> n__0 U11(tt,V2:S) -> U12(isNat(activate(V2:S))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2:S) -> U32(isNat(activate(V2:S))) U32(tt) -> tt U41(tt,N:S) -> activate(N:S) U51(tt,M:S,N:S) -> U52(isNat(activate(N:S)),activate(M:S),activate(N:S)) U52(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U61(tt) -> 0 U71(tt,M:S,N:S) -> U72(isNat(activate(N:S)),activate(M:S),activate(N:S)) U72(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(n__0) -> 0 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(activate(X:S)) activate(n__x(X1:S,X2:S)) -> x(activate(X1:S),activate(X2:S)) activate(X:S) -> X:S isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> U11(isNat(activate(V1:S)),activate(V2:S)) isNat(n__s(V1:S)) -> U21(isNat(activate(V1:S))) isNat(n__x(V1:S,V2:S)) -> U31(isNat(activate(V1:S)),activate(V2:S)) plus(N:S,0) -> U41(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) x(N:S,0) -> U61(isNat(N:S)) x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) x(X1:S,X2:S) -> n__x(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.