/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR M N V1 V2 X X1 X2) (RULES 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(tt,V2) -> U12#(isNat(activate(V2))) U11#(tt,V2) -> ACTIVATE(V2) U11#(tt,V2) -> ISNAT(activate(V2)) U31#(tt,V2) -> U32#(isNat(activate(V2))) U31#(tt,V2) -> ACTIVATE(V2) U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U52#(tt,M,N) -> S(plus(activate(N),activate(M))) U61#(tt) -> 0# U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__0) -> 0# ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> S(activate(X)) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> U21#(isNat(activate(V1))) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> U61#(isNat(N)) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: SCC Processor: -> Pairs: U11#(tt,V2) -> U12#(isNat(activate(V2))) U11#(tt,V2) -> ACTIVATE(V2) U11#(tt,V2) -> ISNAT(activate(V2)) U31#(tt,V2) -> U32#(isNat(activate(V2))) U31#(tt,V2) -> ACTIVATE(V2) U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U52#(tt,M,N) -> S(plus(activate(N),activate(M))) U61#(tt) -> 0# U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__0) -> 0# ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> S(activate(X)) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> U21#(isNat(activate(V1))) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> U61#(isNat(N)) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(tt,V2) -> ACTIVATE(V2) U11#(tt,V2) -> ISNAT(activate(V2)) U31#(tt,V2) -> ACTIVATE(V2) U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U11#(tt,V2) -> ACTIVATE(V2) U11#(tt,V2) -> ISNAT(activate(V2)) U31#(tt,V2) -> ACTIVATE(V2) U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1.X2 + 2.X1 [U12](X) = X.X + 2.X [U21](X) = 2.X [U31](X1,X2) = X1.X2 + 2.X1 [U32](X) = X.X [U41](X1,X2) = X2 [U51](X1,X2,X3) = 2.X1.X3 + X2 + X3 + 2 [U52](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X2 + X3 + 2 [U61](X) = X.X + 2.X + 2 [U71](X1,X2,X3) = 2.X2.X3 + 2.X1 + X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X1.X2 + 2.X2.X3 + 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 + 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] = 0 [U11#](X1,X2) = X2 + 2 [U31#](X1,X2) = 2.X1 + X2 + 2 [U41#](X1,X2) = 2.X1 + X2 + 2 [U51#](X1,X2,X3) = X1.X2 + X2 + X3 + 1 [U52#](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + X2 + X3 + 1 [U71#](X1,X2,X3) = 2.X2.X3 + X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X1.X2 + 2.X2.X3 + 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 + X2 + 2 Problem 1: SCC Processor: -> Pairs: U11#(tt,V2) -> ISNAT(activate(V2)) U31#(tt,V2) -> ACTIVATE(V2) U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(tt,V2) -> ISNAT(activate(V2)) U31#(tt,V2) -> ACTIVATE(V2) U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U11#(tt,V2) -> ISNAT(activate(V2)) U31#(tt,V2) -> ACTIVATE(V2) U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + 2.X2 + 1 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = X1 + 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 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 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) = 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) = X1.X2 + X1 + 2.X2 + 1 [tt] = 2 [U11#](X1,X2) = X2 + 2 [U31#](X1,X2) = 2.X2 + 2 [U41#](X1,X2) = X2 + 2 [U51#](X1,X2,X3) = X2 + X3 + 2 [U52#](X1,X2,X3) = X2 + X3 + 2 [U71#](X1,X2,X3) = X2.X3 + 2.X2 + 2.X3 + 2 [U72#](X1,X2,X3) = X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X + 2 [ISNAT](X) = X + 1 [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U31#(tt,V2) -> ACTIVATE(V2) U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U31#(tt,V2) -> ACTIVATE(V2) U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U31#(tt,V2) -> ACTIVATE(V2) U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->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 + 1 [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 [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 [U31#](X1,X2) = 2.X2 + 1 [U41#](X1,X2) = X2 + 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 [ISNAT](X) = X [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + 2.X2 Problem 1: SCC Processor: -> Pairs: U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U31#(tt,V2) -> ISNAT(activate(V2)) U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2.X2 + 1 [U32](X) = 0 [U41](X1,X2) = X2 + 1 [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 [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 [U31#](X1,X2) = X2 + 1 [U41#](X1,X2) = X2 + 2 [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 [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U41#(tt,N) -> ACTIVATE(N) U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 0 [U11](X1,X2) = X2 + 2 [U12](X) = 2 [U21](X) = X [U31](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [U32](X) = 2 [U41](X1,X2) = X2 + 1 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 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) = 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] = 0 [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 + 1 [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 [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: U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U41#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U51#(tt,M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2 [U12](X) = 2 [U21](X) = X + 1 [U31](X1,X2) = 2.X1.X2 + 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] = 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 [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 [ISNAT](X) = X + 2 [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U52#(tt,M,N) -> ACTIVATE(M) U52#(tt,M,N) -> ACTIVATE(N) U52#(tt,M,N) -> PLUS(activate(N),activate(M)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U51#(tt,M,N) -> ACTIVATE(M) U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 0 [U12](X) = 0 [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 + 1 [U52](X1,X2,X3) = X2 + X3 + 1 [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 + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 0 [U51#](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 + 1 [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = 2.X1.X2 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U51#(tt,M,N) -> ACTIVATE(N) U51#(tt,M,N) -> ISNAT(activate(N)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->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 [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 + 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) = 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 + 1 [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: U51#(tt,M,N) -> ISNAT(activate(N)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(tt,M,N) -> ISNAT(activate(N)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U51#(tt,M,N) -> ISNAT(activate(N)) U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->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 + 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 [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 + 2 [ISNAT](X) = X + 1 [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U51#(isNat(M),M,N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U71#(tt,M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X1 + X2 [U12](X) = X + 2 [U21](X) = 2 [U31](X1,X2) = X1 + 2.X2 [U32](X) = 2.X [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = X [U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 1 [U72](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 [activate](X) = X [isNat](X) = X [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 [tt] = 2 [U71#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [U72#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 1 [ACTIVATE](X) = X + 1 [ISNAT](X) = X + 2 [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) U72#(tt,M,N) -> ACTIVATE(M) U72#(tt,M,N) -> ACTIVATE(N) U72#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U72#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U71#(tt,M,N) -> ACTIVATE(M) U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 1 [U11](X1,X2) = 2.X2 + 2 [U12](X) = 0 [U21](X) = X + 2 [U31](X1,X2) = 2.X1.X2 + X1 + 2 [U32](X) = 1 [U41](X1,X2) = X2 + 1 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = X2 + X3 + 2 [U61](X) = 2 [U71](X1,X2,X3) = X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 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) = 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) = X1.X2 + X1 + 2.X2 + 1 [tt] = 0 [U71#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 1 [ACTIVATE](X) = 2.X [ISNAT](X) = 2.X + 1 [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U71#(tt,M,N) -> ACTIVATE(N) U71#(tt,M,N) -> ISNAT(activate(N)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 1 [U11](X1,X2) = X1 + 2.X2 + 2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2.X1.X2 + 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) = 1 [U71](X1,X2,X3) = X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = 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) = 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) = X1.X2 + X1 + 2.X2 + 1 [tt] = 2 [U71#](X1,X2,X3) = 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 + 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U71#(tt,M,N) -> ISNAT(activate(N)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U71#(tt,M,N) -> ISNAT(activate(N)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U71#(tt,M,N) -> ISNAT(activate(N)) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->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) = 1 [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) = 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.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 [U71#](X1,X2,X3) = X2.X3 + 2.X3 + 2 [ACTIVATE](X) = 2.X + 2 [ISNAT](X) = 2.X + 1 [PLUS](X1,X2) = 2.X1 + 2.X2 + 1 [X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U71#(isNat(M),M,N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->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) = X1 + X2 + 2 [U32](X) = 1 [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] = 0 [ACTIVATE](X) = 2.X + 2 [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 [X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__plus(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->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 + 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 + 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 [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,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__plus(X1,X2)) -> PLUS(activate(X1),activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2.X1 [U12](X) = 2.X [U21](X) = 2.X.X + 2.X [U31](X1,X2) = 0 [U32](X) = 2.X.X [U41](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1 + X2 + X3 + 2 [U61](X) = 2.X.X + X + 2 [U71](X1,X2,X3) = 2.X2.X3 + X2 + 2.X3 + 2 [U72](X1,X2,X3) = 2.X1.X2 + 2.X2.X3 + 2.X1 + 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 + X2 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + X2 + 2 [tt] = 0 [ACTIVATE](X) = 2.X + 2 [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 + 1 [X](X1,X2) = 2.X1.X2 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2.X1.X2 + X1 + X2 + 2 [U32](X) = X + 1 [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 [ACTIVATE](X) = 2.X + 2 [ISNAT](X) = 2.X + 1 [X](X1,X2) = 2.X1.X2 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X1) ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->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) = X1 + 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 [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 + 1 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 0 [U11](X1,X2) = 2.X1.X2 [U12](X) = 2.X.X [U21](X) = 2.X.X [U31](X1,X2) = 2.X1.X2 + 2.X1 [U32](X) = 2.X.X + X [U41](X1,X2) = 2.X1.X2 + 2.X1 + X2 [U51](X1,X2,X3) = X2 + X3 + 2 [U52](X1,X2,X3) = 2.X1.X2.X3 + X2 + X3 + 2 [U61](X) = 2.X.X + 1 [U71](X1,X2,X3) = X2.X3 + 2.X2 + 2.X3 + 2 [U72](X1,X2,X3) = X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = X1.X2 + X1 + 2.X2 + 2 [n__0] = 0 [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = X1.X2 + X1 + 2.X2 + 2 [tt] = 0 [ACTIVATE](X) = 2.X [ISNAT](X) = 2.X + 1 [X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->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) = X1 + 2.X2 [U32](X) = 2 [U41](X1,X2) = X2 + 1 [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 [ACTIVATE](X) = 2.X [ISNAT](X) = 2.X [X](X1,X2) = 2.X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ACTIVATE(V1) ISNAT(n__x(V1,V2)) -> ACTIVATE(V2) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2 [U12](X) = 2 [U21](X) = 2 [U31](X1,X2) = 2.X1 + 2.X2 + 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,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ISNAT(n__plus(V1,V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 0 [U11](X1,X2) = X1 + 2 [U12](X) = 2 [U21](X) = X + 2 [U31](X1,X2) = 2.X2 + 2 [U32](X) = 0 [U41](X1,X2) = X2 + 1 [U51](X1,X2,X3) = 2.X2 + X3 + 2 [U52](X1,X2,X3) = 2.X2 + X3 + 2 [U61](X) = 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) = 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] = 0 [n__plus](X1,X2) = X1 + 2.X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + 2.X2 + 1 [tt] = 0 [ISNAT](X) = 2.X.X Problem 1: SCC Processor: -> Pairs: ISNAT(n__s(V1)) -> ISNAT(activate(V1)) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ->->-> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ISNAT(n__s(V1)) -> ISNAT(activate(V1)) -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2.X2 [U12](X) = 0 [U21](X) = X + 2 [U31](X1,X2) = 2.X2 + 2 [U32](X) = 0 [U41](X1,X2) = X2 + 2 [U51](X1,X2,X3) = 2.X2 + X3 + 2 [U52](X1,X2,X3) = 2.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 + 2.X2 [s](X) = X + 2 [x](X1,X2) = X1.X2 + X2 + 2 [n__0] = 2 [n__plus](X1,X2) = X1 + 2.X2 [n__s](X) = X + 2 [n__x](X1,X2) = X1.X2 + X2 + 2 [tt] = 0 [ISNAT](X) = X Problem 1: SCC Processor: -> Pairs: Empty -> Rules: 0 -> n__0 U11(tt,V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt,V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt,N) -> activate(N) U51(tt,M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt,M,N) -> s(plus(activate(N),activate(M))) U61(tt) -> 0 U71(tt,M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1,X2)) -> x(activate(X1),activate(X2)) activate(X) -> X isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: There is no strongly connected component The problem is finite.