YES Problem: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Proof: DP Processor: DPs: U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U15#(tt(),V2) -> U16#(isNat(activate(V2))) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U22#(tt(),V1) -> U23#(isNat(activate(V1))) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) U31#(tt(),V2) -> U32#(isNatKind(activate(V2))) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U64#(tt(),M,N) -> s#(plus(activate(N),activate(M))) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> U51#(isNat(N),N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U61#(isNat(M),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> plus#(X1,X2) activate#(n__s(X)) -> s#(X) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X TDG Processor: DPs: U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U15#(tt(),V2) -> U16#(isNat(activate(V2))) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U22#(tt(),V1) -> U23#(isNat(activate(V1))) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) U31#(tt(),V2) -> U32#(isNatKind(activate(V2))) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U64#(tt(),M,N) -> s#(plus(activate(N),activate(M))) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> U51#(isNat(N),N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U61#(isNat(M),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> plus#(X1,X2) activate#(n__s(X)) -> s#(X) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X graph: plus#(N,0()) -> U51#(isNat(N),N) -> U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) plus#(N,0()) -> U51#(isNat(N),N) -> U51#(tt(),N) -> isNatKind#(activate(N)) plus#(N,0()) -> U51#(isNat(N),N) -> U51#(tt(),N) -> activate#(N) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V1) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V2) plus#(N,s(M)) -> U61#(isNat(M),M,N) -> U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) plus#(N,s(M)) -> U61#(isNat(M),M,N) -> U61#(tt(),M,N) -> isNatKind#(activate(M)) plus#(N,s(M)) -> U61#(isNat(M),M,N) -> U61#(tt(),M,N) -> activate#(M) plus#(N,s(M)) -> U61#(isNat(M),M,N) -> U61#(tt(),M,N) -> activate#(N) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V1) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> U61#(isNat(M),M,N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> isNat#(M) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> U51#(isNat(N),N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> isNat#(N) U64#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X) U64#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U64#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U64#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U64#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U64#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -> U64#(tt(),M,N) -> s#(plus(activate(N),activate(M))) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -> U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -> U64#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -> U64#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> activate#(V1) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U63#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X) U63#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U63#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U63#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U63#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U63#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -> U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -> U63#(tt(),M,N) -> isNatKind#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -> U63#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -> U63#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__s(V1)) -> activate#(V1) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U62#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X) U62#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U62#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U62#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U62#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U62#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -> U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -> U62#(tt(),M,N) -> isNat#(activate(N)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -> U62#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -> U62#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__s(V1)) -> activate#(V1) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U61#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X) U61#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U61#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U61#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U61#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U61#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U52#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U52#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U52#(tt(),N) -> activate#(N) -> activate#(n__0()) -> 0#() U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) -> U52#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> activate#(V1) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U51#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U51#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U51#(tt(),N) -> activate#(N) -> activate#(n__0()) -> 0#() U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> activate#(V1) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U31#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) U31#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U31#(tt(),V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U22#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U22#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U22#(tt(),V1) -> activate#(V1) -> activate#(n__0()) -> 0#() U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) -> U22#(tt(),V1) -> U23#(isNat(activate(V1))) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) -> U22#(tt(),V1) -> isNat#(activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) -> U22#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U21#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__0()) -> 0#() U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> activate#(V1) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U15#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) U15#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U15#(tt(),V2) -> activate#(V2) -> activate#(n__0()) -> 0#() isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> activate#(V1) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) -> U15#(tt(),V2) -> U16#(isNat(activate(V2))) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) -> U15#(tt(),V2) -> isNat#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) -> U15#(tt(),V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U14#(tt(),V1,V2) -> isNat#(activate(V1)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U14#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U14#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> activate#(V1) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U13#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U13#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> activate#(V1) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) -> U31#(tt(),V2) -> U32#(isNatKind(activate(V2))) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) -> U31#(tt(),V2) -> isNatKind#(activate(V2)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) -> U31#(tt(),V2) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> U61#(isNat(M),M,N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> isNat#(M) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> U51#(isNat(N),N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> isNat#(N) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U12#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U12#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() SCC Processor: #sccs: 1 #rules: 62 #arcs: 266/4761 DPs: plus#(N,0()) -> U51#(isNat(N),N) U51#(tt(),N) -> activate#(N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,0()) -> isNat#(N) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U61#(isNat(M),M,N) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U51#(tt(),N) -> isNatKind#(activate(N)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) -> activate#(N) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Matrix Interpretation Processor: dim=3 usable rules: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X interpretation: [0 1 0] [1 1 1] [0] [U61](x0, x1, x2) = [0 0 0]x1 + [0 0 1]x2 + [0] [0 0 1] [1 0 1] [1], [0] [U12](x0, x1, x2) = [0] [0], [0] [U41](x0) = [1] [0], [0 0 0] [U15](x0, x1) = [0 0 0]x0 [0 1 0] , [0 0 1] [U11](x0, x1, x2) = [0 0 0]x2 [0 0 0] , [0] [isNatKind](x0) = [0] [0], [1 1 1] [0] [U51](x0, x1) = [0 1 0]x1 + [0] [1 0 1] [1], [0 1 0] [0 1 1] [0] [U64](x0, x1, x2) = [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [1 0 1] [1], [0 1 0] [0] [n__s](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0 1 0] [0 1 1] [0] [U62](x0, x1, x2) = [0 0 0]x1 + [0 0 1]x2 + [0] [0 0 1] [1 0 1] [1], [U13#](x0, x1, x2) = [0 0 1]x1 + [0 0 1]x2, [1 1 0] [U52](x0, x1) = [0 1 0]x1 [0 0 1] , [0] [0] = [0] [1], [isNat#](x0) = [0 0 1]x0, [1 1 1] [1 0 0] [plus](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [1 0 1] [0 0 1] , [U11#](x0, x1, x2) = [1 0 1]x1 + [0 0 1]x2, [U52#](x0, x1) = [0 0 1]x1, [0] [U13](x0, x1, x2) = [0] [0], [U14#](x0, x1, x2) = [0 0 1]x1 + [0 0 1]x2, [0] [U21](x0, x1) = [0] [0], [0] [tt] = [0] [0], [0] [isNat](x0) = [1] [0], [0 1 0] [0 1 1] [0] [U63](x0, x1, x2) = [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [1 0 1] [1], [1 1 1] [1 0 0] [n__plus](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [1 0 1] [0 0 1] , [plus#](x0, x1) = [0 0 1]x0 + [0 0 1]x1, [isNatKind#](x0) = [0 0 1]x0, [0] [U32](x0) = [0] [0], [U12#](x0, x1, x2) = [1 0 1]x1 + [0 0 1]x2, [0] [U23](x0) = [0] [0], [U61#](x0, x1, x2) = [0 0 1]x1 + [0 0 1]x2, [0 1 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0] [n__0] = [0] [1], [0] [U22](x0, x1) = [0] [0], [0] [U16](x0) = [0] [0], [0 0 0] [U31](x0, x1) = [0 0 1]x1 [0 0 0] , [U31#](x0, x1) = [0 0 1]x1, [0 1 0] [U14](x0, x1, x2) = [0 0 0]x1 [0 0 0] , [U22#](x0, x1) = [0 0 1]x1, [U64#](x0, x1, x2) = [0 0 1]x1 + [0 0 1]x2, [U63#](x0, x1, x2) = [0 0 1]x1 + [0 0 1]x2, [U21#](x0, x1) = [0 0 1]x1 + [1], [U62#](x0, x1, x2) = [0 0 1]x1 + [0 0 1]x2, [U15#](x0, x1) = [0 0 1]x1, [activate#](x0) = [0 0 1]x0, [U51#](x0, x1) = [0 0 1]x1 + [1], [activate](x0) = x0 orientation: plus#(N,0()) = [0 0 1]N + [1] >= [0 0 1]N + [1] = U51#(isNat(N),N) U51#(tt(),N) = [0 0 1]N + [1] >= [0 0 1]N = activate#(N) activate#(n__plus(X1,X2)) = [1 0 1]X1 + [0 0 1]X2 >= [0 0 1]X1 + [0 0 1]X2 = plus#(X1,X2) plus#(N,0()) = [0 0 1]N + [1] >= [0 0 1]N = isNat#(N) isNat#(n__plus(V1,V2)) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V2 = activate#(V2) isNat#(n__plus(V1,V2)) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 = activate#(V1) isNat#(n__plus(V1,V2)) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V2 = activate#(V2) isNatKind#(n__plus(V1,V2)) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 = activate#(V1) isNatKind#(n__plus(V1,V2)) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V2 = U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) = [0 0 1]V2 >= [0 0 1]V2 = activate#(V2) U31#(tt(),V2) = [0 0 1]V2 >= [0 0 1]V2 = isNatKind#(activate(V2)) isNatKind#(n__s(V1)) = [0 0 1]V1 + [1] >= [0 0 1]V1 = activate#(V1) isNatKind#(n__s(V1)) = [0 0 1]V1 + [1] >= [0 0 1]V1 = isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) = [1 0 1]V1 + [0 0 1]V2 >= [1 0 1]V1 + [0 0 1]V2 = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V2 = activate#(V2) U11#(tt(),V1,V2) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 = activate#(V1) U11#(tt(),V1,V2) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 = isNatKind#(activate(V1)) U11#(tt(),V1,V2) = [1 0 1]V1 + [0 0 1]V2 >= [1 0 1]V1 + [0 0 1]V2 = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 = activate#(V1) U12#(tt(),V1,V2) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V2 = activate#(V2) U12#(tt(),V1,V2) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V2 = isNatKind#(activate(V2)) U12#(tt(),V1,V2) = [1 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 + [0 0 1]V2 = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = [0 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 = activate#(V1) U13#(tt(),V1,V2) = [0 0 1]V1 + [0 0 1]V2 >= [0 0 1]V2 = activate#(V2) U13#(tt(),V1,V2) = [0 0 1]V1 + [0 0 1]V2 >= [0 0 1]V2 = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = [0 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 + [0 0 1]V2 = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) = [0 0 1]V1 + [0 0 1]V2 >= [0 0 1]V2 = activate#(V2) U14#(tt(),V1,V2) = [0 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 = activate#(V1) U14#(tt(),V1,V2) = [0 0 1]V1 + [0 0 1]V2 >= [0 0 1]V1 = isNat#(activate(V1)) isNat#(n__s(V1)) = [0 0 1]V1 + [1] >= [0 0 1]V1 = activate#(V1) isNat#(n__s(V1)) = [0 0 1]V1 + [1] >= [0 0 1]V1 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [0 0 1]V1 + [1] >= [0 0 1]V1 + [1] = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [0 0 1]V1 + [1] >= [0 0 1]V1 = activate#(V1) U21#(tt(),V1) = [0 0 1]V1 + [1] >= [0 0 1]V1 = isNatKind#(activate(V1)) U21#(tt(),V1) = [0 0 1]V1 + [1] >= [0 0 1]V1 = U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) = [0 0 1]V1 >= [0 0 1]V1 = activate#(V1) U22#(tt(),V1) = [0 0 1]V1 >= [0 0 1]V1 = isNat#(activate(V1)) U14#(tt(),V1,V2) = [0 0 1]V1 + [0 0 1]V2 >= [0 0 1]V2 = U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) = [0 0 1]V2 >= [0 0 1]V2 = activate#(V2) U15#(tt(),V2) = [0 0 1]V2 >= [0 0 1]V2 = isNat#(activate(V2)) plus#(N,s(M)) = [0 0 1]M + [0 0 1]N + [1] >= [0 0 1]M = isNat#(M) plus#(N,s(M)) = [0 0 1]M + [0 0 1]N + [1] >= [0 0 1]M + [0 0 1]N = U61#(isNat(M),M,N) U61#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]N = activate#(N) U61#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]M = activate#(M) U61#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]M = isNatKind#(activate(M)) U61#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]M + [0 0 1]N = U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]M = activate#(M) U62#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]N = activate#(N) U62#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]N = isNat#(activate(N)) U62#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]M + [0 0 1]N = U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]M = activate#(M) U63#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]N = activate#(N) U63#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]N = isNatKind#(activate(N)) U63#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]M + [0 0 1]N = U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]M = activate#(M) U64#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]N = activate#(N) U64#(tt(),M,N) = [0 0 1]M + [0 0 1]N >= [0 0 1]M + [0 0 1]N = plus#(activate(N),activate(M)) U51#(tt(),N) = [0 0 1]N + [1] >= [0 0 1]N = isNatKind#(activate(N)) U51#(tt(),N) = [0 0 1]N + [1] >= [0 0 1]N = U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) = [0 0 1]N >= [0 0 1]N = activate#(N) [0 0 1] [0] U11(tt(),V1,V2) = [0 0 0]V2 >= [0] = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) [0 0 0] [0] [0] [0] U12(tt(),V1,V2) = [0] >= [0] = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) [0] [0] [0] [0 1 0] U13(tt(),V1,V2) = [0] >= [0 0 0]V1 = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) [0] [0 0 0] [0 1 0] [0] U14(tt(),V1,V2) = [0 0 0]V1 >= [0] = U15(isNat(activate(V1)),activate(V2)) [0 0 0] [1] [0] [0] U15(tt(),V2) = [0] >= [0] = U16(isNat(activate(V2))) [0] [0] [0] [0] U16(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] U21(tt(),V1) = [0] >= [0] = U22(isNatKind(activate(V1)),activate(V1)) [0] [0] [0] [0] U22(tt(),V1) = [0] >= [0] = U23(isNat(activate(V1))) [0] [0] [0] [0] U23(tt()) = [0] >= [0] = tt() [0] [0] [0 0 0] [0] U31(tt(),V2) = [0 0 1]V2 >= [0] = U32(isNatKind(activate(V2))) [0 0 0] [0] [0] [0] U32(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] U41(tt()) = [1] >= [0] = tt() [0] [0] [1 1 1] [0] [1 1 0] U51(tt(),N) = [0 1 0]N + [0] >= [0 1 0]N = U52(isNatKind(activate(N)),activate(N)) [1 0 1] [1] [0 0 1] [1 1 0] U52(tt(),N) = [0 1 0]N >= N = activate(N) [0 0 1] [0 1 0] [1 1 1] [0] [0 1 0] [0 1 1] [0] U61(tt(),M,N) = [0 0 0]M + [0 0 1]N + [0] >= [0 0 0]M + [0 0 1]N + [0] = U62(isNatKind(activate(M)),activate(M),activate(N)) [0 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1] [0 1 0] [0 1 1] [0] [0 1 0] [0 1 1] [0] U62(tt(),M,N) = [0 0 0]M + [0 0 1]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = U63(isNat(activate(N)),activate(M),activate(N)) [0 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1] [0 1 0] [0 1 1] [0] [0 1 0] [0 1 1] [0] U63(tt(),M,N) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = U64(isNatKind(activate(N)),activate(M),activate(N)) [0 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1] [0 1 0] [0 1 1] [0] [0 0 0] [0 1 1] [0] U64(tt(),M,N) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = s(plus(activate(N),activate(M))) [0 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1] [0] [0] isNat(n__0()) = [1] >= [0] = tt() [0] [0] [0] [0 0 1] isNat(n__plus(V1,V2)) = [1] >= [0 0 0]V2 = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) [0] [0 0 0] [0] [0] isNat(n__s(V1)) = [1] >= [0] = U21(isNatKind(activate(V1)),activate(V1)) [0] [0] [0] [0] isNatKind(n__0()) = [0] >= [0] = tt() [0] [0] [0] [0 0 0] isNatKind(n__plus(V1,V2)) = [0] >= [0 0 1]V2 = U31(isNatKind(activate(V1)),activate(V2)) [0] [0 0 0] [0] [0] isNatKind(n__s(V1)) = [0] >= [1] = U41(isNatKind(activate(V1))) [0] [0] [1 1 1] [0] [1 1 1] [0] plus(N,0()) = [0 1 1]N + [0] >= [0 1 0]N + [0] = U51(isNat(N),N) [1 0 1] [1] [1 0 1] [1] [0 1 0] [1 1 1] [0] [0 1 0] [1 1 1] [0] plus(N,s(M)) = [0 0 0]M + [0 1 1]N + [0] >= [0 0 0]M + [0 0 1]N + [0] = U61(isNat(M),M,N) [0 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1] [0] [0] 0() = [0] >= [0] = n__0() [1] [1] [1 1 1] [1 0 0] [1 1 1] [1 0 0] plus(X1,X2) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = n__plus(X1,X2) [1 0 1] [0 0 1] [1 0 1] [0 0 1] [0 1 0] [0] [0 1 0] [0] s(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = n__s(X) [0 0 1] [1] [0 0 1] [1] [0] [0] activate(n__0()) = [0] >= [0] = 0() [1] [1] [1 1 1] [1 0 0] [1 1 1] [1 0 0] activate(n__plus(X1,X2)) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = plus(X1,X2) [1 0 1] [0 0 1] [1 0 1] [0 0 1] [0 1 0] [0] [0 1 0] [0] activate(n__s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 1] [1] [0 0 1] [1] activate(X) = X >= X = X problem: DPs: plus#(N,0()) -> U51#(isNat(N),N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U52#(tt(),N) -> activate#(N) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Restore Modifier: DPs: plus#(N,0()) -> U51#(isNat(N),N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U52#(tt(),N) -> activate#(N) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X SCC Processor: #sccs: 2 #rules: 10 #arcs: 194/2401 DPs: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> isNat#(activate(V2)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X interpretation: [U61](x0, x1, x2) = 3/2x2 + 2, [U12](x0, x1, x2) = 2x0 + x1 + 3x2, [U41](x0) = 1, [U15](x0, x1) = 1/2x0, [U11](x0, x1, x2) = 2x0 + 2x2, [isNatKind](x0) = 1, [U51](x0, x1) = x1 + 3/2, [U64](x0, x1, x2) = 1/2x2 + 1/2, [n__s](x0) = 1/2, [U62](x0, x1, x2) = 3/2x0 + 3/2x2 + 1/2, [U13#](x0, x1, x2) = x0 + 2x1 + 2x2 + 3/2, [U52](x0, x1) = x1 + 3/2, [0] = 0, [isNat#](x0) = 2x0 + 1/2, [plus](x0, x1) = 3/2x0 + 3x1 + 2, [U11#](x0, x1, x2) = 2x0 + 2x1 + 2x2 + 5/2, [U13](x0, x1, x2) = 7/2x0 + 2x1 + 1/2, [U14#](x0, x1, x2) = 1/2x0 + 2x1 + 2x2 + 1/2, [U21](x0, x1) = 1/2x0, [tt] = 0, [isNat](x0) = 2x0 + 3, [U63](x0, x1, x2) = x2 + 1/2, [n__plus](x0, x1) = 3/2x0 + 3x1 + 2, [U32](x0) = 1/2, [U12#](x0, x1, x2) = 2x1 + 2x2 + 5/2, [U23](x0) = 3/2, [s](x0) = 1/2, [n__0] = 0, [U22](x0, x1) = 1/2, [U16](x0) = 3/2x0 + 5/2, [U31](x0, x1) = 1, [U14](x0, x1, x2) = 1/2x0 + 1, [U15#](x0, x1) = 2x1 + 1/2, [activate](x0) = x0 orientation: isNat#(n__plus(V1,V2)) = 3V1 + 6V2 + 9/2 >= 2V1 + 2V2 + 9/2 = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = 2V1 + 2V2 + 5/2 >= 2V1 + 2V2 + 5/2 = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = 2V1 + 2V2 + 5/2 >= 2V1 + 2V2 + 5/2 = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = 2V1 + 2V2 + 3/2 >= 2V1 + 2V2 + 1 = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) = 2V1 + 2V2 + 1/2 >= 2V1 + 1/2 = isNat#(activate(V1)) U14#(tt(),V1,V2) = 2V1 + 2V2 + 1/2 >= 2V2 + 1/2 = U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) = 2V2 + 1/2 >= 2V2 + 1/2 = isNat#(activate(V2)) U11(tt(),V1,V2) = 2V2 >= V1 + 3V2 + 2 = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) = V1 + 3V2 >= 2V1 + 4 = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) = 2V1 + 1/2 >= 3/2 = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) = 1 >= V1 + 3/2 = U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) = 0 >= 3V2 + 7 = U16(isNat(activate(V2))) U16(tt()) = 5/2 >= 0 = tt() U21(tt(),V1) = 0 >= 1/2 = U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) = 1/2 >= 3/2 = U23(isNat(activate(V1))) U23(tt()) = 3/2 >= 0 = tt() U31(tt(),V2) = 1 >= 1/2 = U32(isNatKind(activate(V2))) U32(tt()) = 1/2 >= 0 = tt() U41(tt()) = 1 >= 0 = tt() U51(tt(),N) = N + 3/2 >= N + 3/2 = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = N + 3/2 >= N = activate(N) U61(tt(),M,N) = 3/2N + 2 >= 3/2N + 2 = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = 3/2N + 1/2 >= N + 1/2 = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = N + 1/2 >= 1/2N + 1/2 = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = 1/2N + 1/2 >= 1/2 = s(plus(activate(N),activate(M))) isNat(n__0()) = 3 >= 0 = tt() isNat(n__plus(V1,V2)) = 3V1 + 6V2 + 7 >= 2V2 + 2 = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) = 4 >= 1/2 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 1 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 1 >= 1 = U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = 1 >= 1 = U41(isNatKind(activate(V1))) plus(N,0()) = 3/2N + 2 >= N + 3/2 = U51(isNat(N),N) plus(N,s(M)) = 3/2N + 7/2 >= 3/2N + 2 = U61(isNat(M),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = 3/2X1 + 3X2 + 2 >= 3/2X1 + 3X2 + 2 = n__plus(X1,X2) s(X) = 1/2 >= 1/2 = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 3/2X1 + 3X2 + 2 >= 3/2X1 + 3X2 + 2 = plus(X1,X2) activate(n__s(X)) = 1/2 >= 1/2 = s(X) activate(X) = X >= X = X problem: DPs: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> isNat#(activate(V2)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Restore Modifier: DPs: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> isNat#(activate(V2)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 8/36 DPs: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) -> isNatKind#(activate(V2)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X interpretation: [U61](x0, x1, x2) = 3x2 + 7, [U12](x0, x1, x2) = x0 + 2x1 + 3, [U41](x0) = 2, [U15](x0, x1) = 6x0 + 4x1, [U11](x0, x1, x2) = 4x0 + 4x1 + 4, [isNatKind](x0) = 4x0 + 1, [U51](x0, x1) = x1, [U64](x0, x1, x2) = 0, [n__s](x0) = 0, [U62](x0, x1, x2) = 7, [U52](x0, x1) = x1, [0] = 3, [plus](x0, x1) = 3x0 + 4x1 + 7, [U13](x0, x1, x2) = 2, [U21](x0, x1) = 2x0 + 2x1, [tt] = 2, [isNat](x0) = 4x0 + 1, [U63](x0, x1, x2) = 7, [n__plus](x0, x1) = 3x0 + 4x1 + 7, [isNatKind#](x0) = 2x0, [U32](x0) = 2x0, [U23](x0) = 4x0 + 4, [s](x0) = 0, [n__0] = 3, [U22](x0, x1) = x0 + 6, [U16](x0) = 6x0 + 4, [U31](x0, x1) = 4, [U31#](x0, x1) = 4x1, [U14](x0, x1, x2) = 4x0 + 4x2, [activate](x0) = x0 orientation: isNatKind#(n__plus(V1,V2)) = 6V1 + 8V2 + 14 >= 2V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 6V1 + 8V2 + 14 >= 4V2 = U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) = 4V2 >= 2V2 = isNatKind#(activate(V2)) U11(tt(),V1,V2) = 4V1 + 12 >= 6V1 + 4 = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) = 2V1 + 5 >= 2 = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) = 2 >= 20V2 + 4 = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) = 4V2 + 8 >= 24V1 + 4V2 + 6 = U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) = 4V2 + 12 >= 24V2 + 10 = U16(isNat(activate(V2))) U16(tt()) = 16 >= 2 = tt() U21(tt(),V1) = 2V1 + 4 >= 4V1 + 7 = U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) = 8 >= 16V1 + 8 = U23(isNat(activate(V1))) U23(tt()) = 12 >= 2 = tt() U31(tt(),V2) = 4 >= 8V2 + 2 = U32(isNatKind(activate(V2))) U32(tt()) = 4 >= 2 = tt() U41(tt()) = 2 >= 2 = tt() U51(tt(),N) = N >= N = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = N >= N = activate(N) U61(tt(),M,N) = 3N + 7 >= 7 = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = 7 >= 7 = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = 7 >= 0 = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = 0 >= 0 = s(plus(activate(N),activate(M))) isNat(n__0()) = 13 >= 2 = tt() isNat(n__plus(V1,V2)) = 12V1 + 16V2 + 29 >= 20V1 + 8 = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) = 1 >= 10V1 + 2 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 13 >= 2 = tt() isNatKind(n__plus(V1,V2)) = 12V1 + 16V2 + 29 >= 4 = U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = 1 >= 2 = U41(isNatKind(activate(V1))) plus(N,0()) = 3N + 19 >= N = U51(isNat(N),N) plus(N,s(M)) = 3N + 7 >= 3N + 7 = U61(isNat(M),M,N) 0() = 3 >= 3 = n__0() plus(X1,X2) = 3X1 + 4X2 + 7 >= 3X1 + 4X2 + 7 = n__plus(X1,X2) s(X) = 0 >= 0 = n__s(X) activate(n__0()) = 3 >= 3 = 0() activate(n__plus(X1,X2)) = 3X1 + 4X2 + 7 >= 3X1 + 4X2 + 7 = plus(X1,X2) activate(n__s(X)) = 0 >= 0 = s(X) activate(X) = X >= X = X problem: DPs: U31#(tt(),V2) -> isNatKind#(activate(V2)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Restore Modifier: DPs: U31#(tt(),V2) -> isNatKind#(activate(V2)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 5/1