/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,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(tt,N) -> ACTIVATE(N) U21#(tt,M,N) -> ACTIVATE(M) U21#(tt,M,N) -> ACTIVATE(N) U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U21#(tt,M,N) -> S(plus(activate(N),activate(M))) U31#(tt) -> 0# U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__0) -> 0# ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U11#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> U31#(isNat(N)) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: SCC Processor: -> Pairs: U11#(tt,N) -> ACTIVATE(N) U21#(tt,M,N) -> ACTIVATE(M) U21#(tt,M,N) -> ACTIVATE(N) U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U21#(tt,M,N) -> S(plus(activate(N),activate(M))) U31#(tt) -> 0# U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__0) -> 0# ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U11#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> U31#(isNat(N)) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(tt,N) -> ACTIVATE(N) U21#(tt,M,N) -> ACTIVATE(M) U21#(tt,M,N) -> ACTIVATE(N) U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U11#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U11#(tt,N) -> ACTIVATE(N) U21#(tt,M,N) -> ACTIVATE(M) U21#(tt,M,N) -> ACTIVATE(N) U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U11#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 + 1 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2.X + 2 [U41](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](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__isNat](X) = X + 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 [U11#](X1,X2) = X2 + 2 [U21#](X1,X2,X3) = X2 + X3 + 2 [U41#](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X [AND](X1,X2) = X2 [ISNAT](X) = X + 2 [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U21#(tt,M,N) -> ACTIVATE(M) U21#(tt,M,N) -> ACTIVATE(N) U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> U11#(isNat(N),N) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(tt,M,N) -> ACTIVATE(M) U21#(tt,M,N) -> ACTIVATE(N) U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U21#(tt,M,N) -> ACTIVATE(M) U21#(tt,M,N) -> ACTIVATE(N) U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 + 1 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2.X + 2 [U41](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](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__isNat](X) = X + 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 [U21#](X1,X2,X3) = X2 + X3 + 2 [U41#](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X [AND](X1,X2) = X2 [ISNAT](X) = X + 2 [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = 2.X1.X2 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U21#(tt,M,N) -> ACTIVATE(N) U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(tt,M,N) -> ACTIVATE(N) U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U21#(tt,M,N) -> ACTIVATE(N) U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2.X + 2 [U41](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 + 1 [isNat](X) = X [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__isNat](X) = X [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 [U21#](X1,X2,X3) = X2 + X3 + 2 [U41#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X + 1 [AND](X1,X2) = X1 + X2 + 1 [ISNAT](X) = X [PLUS](X1,X2) = X1 + X2 + 1 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 Problem 1: SCC Processor: -> Pairs: U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U21#(tt,M,N) -> PLUS(activate(N),activate(M)) U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 + 1 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = X + 2 [U41](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](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__isNat](X) = X + 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 [U21#](X1,X2,X3) = X2 + X3 + 2 [U41#](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X [AND](X1,X2) = X2 [ISNAT](X) = X + 2 [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U41#(tt,M,N) -> ACTIVATE(M) U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 + 1 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2.X [U41](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](X) = X [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__isNat](X) = X [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 2 [U41#](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 [ACTIVATE](X) = X + 2 [AND](X1,X2) = X1 + X2 + 2 [ISNAT](X) = X + 2 [PLUS](X1,X2) = X1 + X2 + 2 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U41#(tt,M,N) -> ACTIVATE(N) U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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) = X2 + 2 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2.X + 2 [U41](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 + 1 [isNat](X) = X + 1 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 1 [n__isNat](X) = X + 1 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 2 [U41#](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X + 2 [AND](X1,X2) = X2 + 2 [ISNAT](X) = X + 2 [PLUS](X1,X2) = X1 + X2 + 2 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U41#(tt,M,N) -> PLUS(x(activate(N),activate(M)),activate(N)) U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2 [U41](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](X) = X + 1 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__isNat](X) = X + 1 [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [tt] = 0 [U41#](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X [AND](X1,X2) = X2 [ISNAT](X) = X + 1 [PLUS](X1,X2) = X1 + X2 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: U41#(tt,M,N) -> X(activate(N),activate(M)) ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2 [U41](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 + 1 [isNat](X) = X [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__isNat](X) = X [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 2 [U41#](X1,X2,X3) = 2.X2.X3 + 2.X2 + X3 + 2 [ACTIVATE](X) = 2.X + 2 [AND](X1,X2) = X1 + 2.X2 + 2 [ISNAT](X) = 2.X + 1 [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> U41#(and(isNat(M),n__isNat(N)),M,N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__isNat(X)) -> ISNAT(X) 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 + 1 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2 [U41](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 [activate](X) = X [and](X1,X2) = X2 + 1 [isNat](X) = X [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [n__0] = 2 [n__isNat](X) = X [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 [AND](X1,X2) = 2.X1 + 2.X2 + 2 [ISNAT](X) = 2.X + 1 [PLUS](X1,X2) = 2.X1 + 2.X2 + 1 [X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 [U21](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U31](X) = X.X + 2 [U41](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + X3 + 1 [activate](X) = X [and](X1,X2) = X2 [isNat](X) = 2 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [n__0] = 2 [n__isNat](X) = 2 [n__plus](X1,X2) = X1 + 2.X2 + 2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [tt] = 2 [ACTIVATE](X) = 2.X + 2 [AND](X1,X2) = X1 + 2.X2 [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [X](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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 + 1 [U21](X1,X2,X3) = X1 + X2 + X3 + 2 [U31](X) = X.X + 1 [U41](X1,X2,X3) = X1.X2 + 2.X2.X3 + X1 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](X) = 2 [plus](X1,X2) = X1 + X2 + 2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__isNat](X) = 2 [n__plus](X1,X2) = X1 + X2 + 2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 2 [ACTIVATE](X) = 2.X + 2 [AND](X1,X2) = X1.X2 + 2 [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [X](X1,X2) = 2.X1.X2 + 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2 [U41](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](X) = X [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__isNat](X) = X [n__plus](X1,X2) = X1 + X2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 0 [ACTIVATE](X) = 2.X + 2 [AND](X1,X2) = 2.X2 + 2 [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 + 1 [X](X1,X2) = 2.X1.X2 + 2.X2 + 2 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) PLUS(N,0) -> ISNAT(N) PLUS(N,s(M)) -> AND(isNat(M),n__isNat(N)) PLUS(N,s(M)) -> ISNAT(M) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2.X + 2 [U41](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](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__isNat](X) = 2.X [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) = X [AND](X1,X2) = X1 + X2 + 2 [ISNAT](X) = 2.X + 2 [X](X1,X2) = 2.X1.X2 + X1 + 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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 + 1 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2 [U41](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 + 2 [isNat](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__isNat](X) = 2.X + 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) = X + 2 [AND](X1,X2) = X2 + 2 [ISNAT](X) = 2.X + 2 [X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__x(X1,X2)) -> ACTIVATE(X2) ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 + X2 [U21](X1,X2,X3) = 2.X1 + X2 + X3 [U31](X) = 2.X + 1 [U41](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](X) = 2 [plus](X1,X2) = X1 + X2 + 2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [n__0] = 2 [n__isNat](X) = 2 [n__plus](X1,X2) = X1 + X2 + 2 [n__s](X) = X + 2 [n__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [tt] = 2 [ACTIVATE](X) = 2.X + 2 [AND](X1,X2) = 2.X2 + 2 [ISNAT](X) = 2.X + 2 [X](X1,X2) = 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__x(X1,X2)) -> X(activate(X1),activate(X2)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) ->->-> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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)) AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2.X + 2 [U41](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 [activate](X) = X [and](X1,X2) = X2 [isNat](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__isNat](X) = 2.X [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) = X + 1 [AND](X1,X2) = X1 + X2 + 1 [ISNAT](X) = 2.X + 2 [X](X1,X2) = X1.X2 + X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: AND(tt,X) -> ACTIVATE(X) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V1) ISNAT(n__plus(V1,V2)) -> ACTIVATE(V2) ISNAT(n__plus(V1,V2)) -> AND(isNat(activate(V1)),n__isNat(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)) -> AND(isNat(activate(V1)),n__isNat(activate(V2))) ISNAT(n__x(V1,V2)) -> ISNAT(activate(V1)) X(N,0) -> ISNAT(N) X(N,s(M)) -> AND(isNat(M),n__isNat(N)) X(N,s(M)) -> ISNAT(M) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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) = X2 [U21](X1,X2,X3) = X3 [U31](X) = X + 2 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](X) = 2 [plus](X1,X2) = X1 [s](X) = X [x](X1,X2) = 2.X1 + 2.X2 + 2 [n__0] = 2 [n__isNat](X) = 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,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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) = X2 + 2 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2 [U41](X1,X2,X3) = 2.X2.X3 + X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X1 + X2 + 2 [isNat](X) = 2.X.X + 2.X [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + X1 + X2 + 1 [n__0] = 1 [n__isNat](X) = 2.X.X + 2.X [n__plus](X1,X2) = X1 + X2 + 1 [n__s](X) = X + 1 [n__x](X1,X2) = 2.X1.X2 + X1 + X2 + 1 [tt] = 2 [ISNAT](X) = 2.X.X + 2.X Problem 1: SCC Processor: -> Pairs: ISNAT(n__s(V1)) -> ISNAT(activate(V1)) -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) -> Usable rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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 + 1 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2 [U41](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](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__isNat](X) = 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 [ISNAT](X) = 2.X.X + 2.X Problem 1: SCC Processor: -> Pairs: Empty -> Rules: 0 -> n__0 U11(tt,N) -> activate(N) U21(tt,M,N) -> s(plus(activate(N),activate(M))) U31(tt) -> 0 U41(tt,M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(n__0) -> 0 activate(n__isNat(X)) -> isNat(X) 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 and(tt,X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(X) -> n__isNat(X) plus(N,0) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) ->Strongly Connected Components: There is no strongly connected component The problem is finite.