MAYBE Problem: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(U101(tt(),M,N)) -> isNatKind#(M) active#(U101(tt(),M,N)) -> U102#(isNatKind(M),M,N) active#(U102(tt(),M,N)) -> isNat#(N) active#(U102(tt(),M,N)) -> U103#(isNat(N),M,N) active#(U103(tt(),M,N)) -> isNatKind#(N) active#(U103(tt(),M,N)) -> U104#(isNatKind(N),M,N) active#(U104(tt(),M,N)) -> x#(N,M) active#(U104(tt(),M,N)) -> plus#(x(N,M),N) active#(U11(tt(),V1,V2)) -> isNatKind#(V1) active#(U11(tt(),V1,V2)) -> U12#(isNatKind(V1),V1,V2) active#(U12(tt(),V1,V2)) -> isNatKind#(V2) active#(U12(tt(),V1,V2)) -> U13#(isNatKind(V2),V1,V2) active#(U13(tt(),V1,V2)) -> isNatKind#(V2) active#(U13(tt(),V1,V2)) -> U14#(isNatKind(V2),V1,V2) active#(U14(tt(),V1,V2)) -> isNat#(V1) active#(U14(tt(),V1,V2)) -> U15#(isNat(V1),V2) active#(U15(tt(),V2)) -> isNat#(V2) active#(U15(tt(),V2)) -> U16#(isNat(V2)) active#(U21(tt(),V1)) -> isNatKind#(V1) active#(U21(tt(),V1)) -> U22#(isNatKind(V1),V1) active#(U22(tt(),V1)) -> isNat#(V1) active#(U22(tt(),V1)) -> U23#(isNat(V1)) active#(U31(tt(),V1,V2)) -> isNatKind#(V1) active#(U31(tt(),V1,V2)) -> U32#(isNatKind(V1),V1,V2) active#(U32(tt(),V1,V2)) -> isNatKind#(V2) active#(U32(tt(),V1,V2)) -> U33#(isNatKind(V2),V1,V2) active#(U33(tt(),V1,V2)) -> isNatKind#(V2) active#(U33(tt(),V1,V2)) -> U34#(isNatKind(V2),V1,V2) active#(U34(tt(),V1,V2)) -> isNat#(V1) active#(U34(tt(),V1,V2)) -> U35#(isNat(V1),V2) active#(U35(tt(),V2)) -> isNat#(V2) active#(U35(tt(),V2)) -> U36#(isNat(V2)) active#(U41(tt(),V2)) -> isNatKind#(V2) active#(U41(tt(),V2)) -> U42#(isNatKind(V2)) active#(U61(tt(),V2)) -> isNatKind#(V2) active#(U61(tt(),V2)) -> U62#(isNatKind(V2)) active#(U71(tt(),N)) -> isNatKind#(N) active#(U71(tt(),N)) -> U72#(isNatKind(N),N) active#(U81(tt(),M,N)) -> isNatKind#(M) active#(U81(tt(),M,N)) -> U82#(isNatKind(M),M,N) active#(U82(tt(),M,N)) -> isNat#(N) active#(U82(tt(),M,N)) -> U83#(isNat(N),M,N) active#(U83(tt(),M,N)) -> isNatKind#(N) active#(U83(tt(),M,N)) -> U84#(isNatKind(N),M,N) active#(U84(tt(),M,N)) -> plus#(N,M) active#(U84(tt(),M,N)) -> s#(plus(N,M)) active#(U91(tt(),N)) -> isNatKind#(N) active#(U91(tt(),N)) -> U92#(isNatKind(N)) active#(isNat(plus(V1,V2))) -> isNatKind#(V1) active#(isNat(plus(V1,V2))) -> U11#(isNatKind(V1),V1,V2) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNat(x(V1,V2))) -> isNatKind#(V1) active#(isNat(x(V1,V2))) -> U31#(isNatKind(V1),V1,V2) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1) active#(isNatKind(plus(V1,V2))) -> U41#(isNatKind(V1),V2) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatKind(s(V1))) -> U51#(isNatKind(V1)) active#(isNatKind(x(V1,V2))) -> isNatKind#(V1) active#(isNatKind(x(V1,V2))) -> U61#(isNatKind(V1),V2) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> U71#(isNat(N),N) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> U81#(isNat(M),M,N) active#(x(N,0())) -> isNat#(N) active#(x(N,0())) -> U91#(isNat(N),N) active#(x(N,s(M))) -> isNat#(M) active#(x(N,s(M))) -> U101#(isNat(M),M,N) active#(U101(X1,X2,X3)) -> active#(X1) active#(U101(X1,X2,X3)) -> U101#(active(X1),X2,X3) active#(U102(X1,X2,X3)) -> active#(X1) active#(U102(X1,X2,X3)) -> U102#(active(X1),X2,X3) active#(U103(X1,X2,X3)) -> active#(X1) active#(U103(X1,X2,X3)) -> U103#(active(X1),X2,X3) active#(U104(X1,X2,X3)) -> active#(X1) active#(U104(X1,X2,X3)) -> U104#(active(X1),X2,X3) active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(x(X1,X2)) -> active#(X1) active#(x(X1,X2)) -> x#(active(X1),X2) active#(x(X1,X2)) -> active#(X2) active#(x(X1,X2)) -> x#(X1,active(X2)) active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(U13(X1,X2,X3)) -> active#(X1) active#(U13(X1,X2,X3)) -> U13#(active(X1),X2,X3) active#(U14(X1,X2,X3)) -> active#(X1) active#(U14(X1,X2,X3)) -> U14#(active(X1),X2,X3) active#(U15(X1,X2)) -> active#(X1) active#(U15(X1,X2)) -> U15#(active(X1),X2) active#(U16(X)) -> active#(X) active#(U16(X)) -> U16#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X1,X2)) -> active#(X1) active#(U22(X1,X2)) -> U22#(active(X1),X2) active#(U23(X)) -> active#(X) active#(U23(X)) -> U23#(active(X)) active#(U31(X1,X2,X3)) -> active#(X1) active#(U31(X1,X2,X3)) -> U31#(active(X1),X2,X3) active#(U32(X1,X2,X3)) -> active#(X1) active#(U32(X1,X2,X3)) -> U32#(active(X1),X2,X3) active#(U33(X1,X2,X3)) -> active#(X1) active#(U33(X1,X2,X3)) -> U33#(active(X1),X2,X3) active#(U34(X1,X2,X3)) -> active#(X1) active#(U34(X1,X2,X3)) -> U34#(active(X1),X2,X3) active#(U35(X1,X2)) -> active#(X1) active#(U35(X1,X2)) -> U35#(active(X1),X2) active#(U36(X)) -> active#(X) active#(U36(X)) -> U36#(active(X)) active#(U41(X1,X2)) -> active#(X1) active#(U41(X1,X2)) -> U41#(active(X1),X2) active#(U42(X)) -> active#(X) active#(U42(X)) -> U42#(active(X)) active#(U51(X)) -> active#(X) active#(U51(X)) -> U51#(active(X)) active#(U61(X1,X2)) -> active#(X1) active#(U61(X1,X2)) -> U61#(active(X1),X2) active#(U62(X)) -> active#(X) active#(U62(X)) -> U62#(active(X)) active#(U71(X1,X2)) -> active#(X1) active#(U71(X1,X2)) -> U71#(active(X1),X2) active#(U72(X1,X2)) -> active#(X1) active#(U72(X1,X2)) -> U72#(active(X1),X2) active#(U81(X1,X2,X3)) -> active#(X1) active#(U81(X1,X2,X3)) -> U81#(active(X1),X2,X3) active#(U82(X1,X2,X3)) -> active#(X1) active#(U82(X1,X2,X3)) -> U82#(active(X1),X2,X3) active#(U83(X1,X2,X3)) -> active#(X1) active#(U83(X1,X2,X3)) -> U83#(active(X1),X2,X3) active#(U84(X1,X2,X3)) -> active#(X1) active#(U84(X1,X2,X3)) -> U84#(active(X1),X2,X3) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(U91(X1,X2)) -> active#(X1) active#(U91(X1,X2)) -> U91#(active(X1),X2) active#(U92(X)) -> active#(X) active#(U92(X)) -> U92#(active(X)) U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3) U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3) U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3) U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) x#(mark(X1),X2) -> x#(X1,X2) x#(X1,mark(X2)) -> x#(X1,X2) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U13#(mark(X1),X2,X3) -> U13#(X1,X2,X3) U14#(mark(X1),X2,X3) -> U14#(X1,X2,X3) U15#(mark(X1),X2) -> U15#(X1,X2) U16#(mark(X)) -> U16#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X1),X2) -> U22#(X1,X2) U23#(mark(X)) -> U23#(X) U31#(mark(X1),X2,X3) -> U31#(X1,X2,X3) U32#(mark(X1),X2,X3) -> U32#(X1,X2,X3) U33#(mark(X1),X2,X3) -> U33#(X1,X2,X3) U34#(mark(X1),X2,X3) -> U34#(X1,X2,X3) U35#(mark(X1),X2) -> U35#(X1,X2) U36#(mark(X)) -> U36#(X) U41#(mark(X1),X2) -> U41#(X1,X2) U42#(mark(X)) -> U42#(X) U51#(mark(X)) -> U51#(X) U61#(mark(X1),X2) -> U61#(X1,X2) U62#(mark(X)) -> U62#(X) U71#(mark(X1),X2) -> U71#(X1,X2) U72#(mark(X1),X2) -> U72#(X1,X2) U81#(mark(X1),X2,X3) -> U81#(X1,X2,X3) U82#(mark(X1),X2,X3) -> U82#(X1,X2,X3) U83#(mark(X1),X2,X3) -> U83#(X1,X2,X3) U84#(mark(X1),X2,X3) -> U84#(X1,X2,X3) s#(mark(X)) -> s#(X) U91#(mark(X1),X2) -> U91#(X1,X2) U92#(mark(X)) -> U92#(X) proper#(U101(X1,X2,X3)) -> proper#(X3) proper#(U101(X1,X2,X3)) -> proper#(X2) proper#(U101(X1,X2,X3)) -> proper#(X1) proper#(U101(X1,X2,X3)) -> U101#(proper(X1),proper(X2),proper(X3)) proper#(U102(X1,X2,X3)) -> proper#(X3) proper#(U102(X1,X2,X3)) -> proper#(X2) proper#(U102(X1,X2,X3)) -> proper#(X1) proper#(U102(X1,X2,X3)) -> U102#(proper(X1),proper(X2),proper(X3)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) proper#(U103(X1,X2,X3)) -> proper#(X3) proper#(U103(X1,X2,X3)) -> proper#(X2) proper#(U103(X1,X2,X3)) -> proper#(X1) proper#(U103(X1,X2,X3)) -> U103#(proper(X1),proper(X2),proper(X3)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U104(X1,X2,X3)) -> proper#(X3) proper#(U104(X1,X2,X3)) -> proper#(X2) proper#(U104(X1,X2,X3)) -> proper#(X1) proper#(U104(X1,X2,X3)) -> U104#(proper(X1),proper(X2),proper(X3)) proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(x(X1,X2)) -> proper#(X2) proper#(x(X1,X2)) -> proper#(X1) proper#(x(X1,X2)) -> x#(proper(X1),proper(X2)) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U13(X1,X2,X3)) -> proper#(X3) proper#(U13(X1,X2,X3)) -> proper#(X2) proper#(U13(X1,X2,X3)) -> proper#(X1) proper#(U13(X1,X2,X3)) -> U13#(proper(X1),proper(X2),proper(X3)) proper#(U14(X1,X2,X3)) -> proper#(X3) proper#(U14(X1,X2,X3)) -> proper#(X2) proper#(U14(X1,X2,X3)) -> proper#(X1) proper#(U14(X1,X2,X3)) -> U14#(proper(X1),proper(X2),proper(X3)) proper#(U15(X1,X2)) -> proper#(X2) proper#(U15(X1,X2)) -> proper#(X1) proper#(U15(X1,X2)) -> U15#(proper(X1),proper(X2)) proper#(U16(X)) -> proper#(X) proper#(U16(X)) -> U16#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X1,X2)) -> proper#(X2) proper#(U22(X1,X2)) -> proper#(X1) proper#(U22(X1,X2)) -> U22#(proper(X1),proper(X2)) proper#(U23(X)) -> proper#(X) proper#(U23(X)) -> U23#(proper(X)) proper#(U31(X1,X2,X3)) -> proper#(X3) proper#(U31(X1,X2,X3)) -> proper#(X2) proper#(U31(X1,X2,X3)) -> proper#(X1) proper#(U31(X1,X2,X3)) -> U31#(proper(X1),proper(X2),proper(X3)) proper#(U32(X1,X2,X3)) -> proper#(X3) proper#(U32(X1,X2,X3)) -> proper#(X2) proper#(U32(X1,X2,X3)) -> proper#(X1) proper#(U32(X1,X2,X3)) -> U32#(proper(X1),proper(X2),proper(X3)) proper#(U33(X1,X2,X3)) -> proper#(X3) proper#(U33(X1,X2,X3)) -> proper#(X2) proper#(U33(X1,X2,X3)) -> proper#(X1) proper#(U33(X1,X2,X3)) -> U33#(proper(X1),proper(X2),proper(X3)) proper#(U34(X1,X2,X3)) -> proper#(X3) proper#(U34(X1,X2,X3)) -> proper#(X2) proper#(U34(X1,X2,X3)) -> proper#(X1) proper#(U34(X1,X2,X3)) -> U34#(proper(X1),proper(X2),proper(X3)) proper#(U35(X1,X2)) -> proper#(X2) proper#(U35(X1,X2)) -> proper#(X1) proper#(U35(X1,X2)) -> U35#(proper(X1),proper(X2)) proper#(U36(X)) -> proper#(X) proper#(U36(X)) -> U36#(proper(X)) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2)) proper#(U42(X)) -> proper#(X) proper#(U42(X)) -> U42#(proper(X)) proper#(U51(X)) -> proper#(X) proper#(U51(X)) -> U51#(proper(X)) proper#(U61(X1,X2)) -> proper#(X2) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2)) proper#(U62(X)) -> proper#(X) proper#(U62(X)) -> U62#(proper(X)) proper#(U71(X1,X2)) -> proper#(X2) proper#(U71(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2)) proper#(U72(X1,X2)) -> proper#(X2) proper#(U72(X1,X2)) -> proper#(X1) proper#(U72(X1,X2)) -> U72#(proper(X1),proper(X2)) proper#(U81(X1,X2,X3)) -> proper#(X3) proper#(U81(X1,X2,X3)) -> proper#(X2) proper#(U81(X1,X2,X3)) -> proper#(X1) proper#(U81(X1,X2,X3)) -> U81#(proper(X1),proper(X2),proper(X3)) proper#(U82(X1,X2,X3)) -> proper#(X3) proper#(U82(X1,X2,X3)) -> proper#(X2) proper#(U82(X1,X2,X3)) -> proper#(X1) proper#(U82(X1,X2,X3)) -> U82#(proper(X1),proper(X2),proper(X3)) proper#(U83(X1,X2,X3)) -> proper#(X3) proper#(U83(X1,X2,X3)) -> proper#(X2) proper#(U83(X1,X2,X3)) -> proper#(X1) proper#(U83(X1,X2,X3)) -> U83#(proper(X1),proper(X2),proper(X3)) proper#(U84(X1,X2,X3)) -> proper#(X3) proper#(U84(X1,X2,X3)) -> proper#(X2) proper#(U84(X1,X2,X3)) -> proper#(X1) proper#(U84(X1,X2,X3)) -> U84#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(U91(X1,X2)) -> proper#(X2) proper#(U91(X1,X2)) -> proper#(X1) proper#(U91(X1,X2)) -> U91#(proper(X1),proper(X2)) proper#(U92(X)) -> proper#(X) proper#(U92(X)) -> U92#(proper(X)) U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3) U102#(ok(X1),ok(X2),ok(X3)) -> U102#(X1,X2,X3) isNatKind#(ok(X)) -> isNatKind#(X) U103#(ok(X1),ok(X2),ok(X3)) -> U103#(X1,X2,X3) isNat#(ok(X)) -> isNat#(X) U104#(ok(X1),ok(X2),ok(X3)) -> U104#(X1,X2,X3) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) x#(ok(X1),ok(X2)) -> x#(X1,X2) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) U13#(ok(X1),ok(X2),ok(X3)) -> U13#(X1,X2,X3) U14#(ok(X1),ok(X2),ok(X3)) -> U14#(X1,X2,X3) U15#(ok(X1),ok(X2)) -> U15#(X1,X2) U16#(ok(X)) -> U16#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X1),ok(X2)) -> U22#(X1,X2) U23#(ok(X)) -> U23#(X) U31#(ok(X1),ok(X2),ok(X3)) -> U31#(X1,X2,X3) U32#(ok(X1),ok(X2),ok(X3)) -> U32#(X1,X2,X3) U33#(ok(X1),ok(X2),ok(X3)) -> U33#(X1,X2,X3) U34#(ok(X1),ok(X2),ok(X3)) -> U34#(X1,X2,X3) U35#(ok(X1),ok(X2)) -> U35#(X1,X2) U36#(ok(X)) -> U36#(X) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) U42#(ok(X)) -> U42#(X) U51#(ok(X)) -> U51#(X) U61#(ok(X1),ok(X2)) -> U61#(X1,X2) U62#(ok(X)) -> U62#(X) U71#(ok(X1),ok(X2)) -> U71#(X1,X2) U72#(ok(X1),ok(X2)) -> U72#(X1,X2) U81#(ok(X1),ok(X2),ok(X3)) -> U81#(X1,X2,X3) U82#(ok(X1),ok(X2),ok(X3)) -> U82#(X1,X2,X3) U83#(ok(X1),ok(X2),ok(X3)) -> U83#(X1,X2,X3) U84#(ok(X1),ok(X2),ok(X3)) -> U84#(X1,X2,X3) s#(ok(X)) -> s#(X) U91#(ok(X1),ok(X2)) -> U91#(X1,X2) U92#(ok(X)) -> U92#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) EDG Processor: DPs: active#(U101(tt(),M,N)) -> isNatKind#(M) active#(U101(tt(),M,N)) -> U102#(isNatKind(M),M,N) active#(U102(tt(),M,N)) -> isNat#(N) active#(U102(tt(),M,N)) -> U103#(isNat(N),M,N) active#(U103(tt(),M,N)) -> isNatKind#(N) active#(U103(tt(),M,N)) -> U104#(isNatKind(N),M,N) active#(U104(tt(),M,N)) -> x#(N,M) active#(U104(tt(),M,N)) -> plus#(x(N,M),N) active#(U11(tt(),V1,V2)) -> isNatKind#(V1) active#(U11(tt(),V1,V2)) -> U12#(isNatKind(V1),V1,V2) active#(U12(tt(),V1,V2)) -> isNatKind#(V2) active#(U12(tt(),V1,V2)) -> U13#(isNatKind(V2),V1,V2) active#(U13(tt(),V1,V2)) -> isNatKind#(V2) active#(U13(tt(),V1,V2)) -> U14#(isNatKind(V2),V1,V2) active#(U14(tt(),V1,V2)) -> isNat#(V1) active#(U14(tt(),V1,V2)) -> U15#(isNat(V1),V2) active#(U15(tt(),V2)) -> isNat#(V2) active#(U15(tt(),V2)) -> U16#(isNat(V2)) active#(U21(tt(),V1)) -> isNatKind#(V1) active#(U21(tt(),V1)) -> U22#(isNatKind(V1),V1) active#(U22(tt(),V1)) -> isNat#(V1) active#(U22(tt(),V1)) -> U23#(isNat(V1)) active#(U31(tt(),V1,V2)) -> isNatKind#(V1) active#(U31(tt(),V1,V2)) -> U32#(isNatKind(V1),V1,V2) active#(U32(tt(),V1,V2)) -> isNatKind#(V2) active#(U32(tt(),V1,V2)) -> U33#(isNatKind(V2),V1,V2) active#(U33(tt(),V1,V2)) -> isNatKind#(V2) active#(U33(tt(),V1,V2)) -> U34#(isNatKind(V2),V1,V2) active#(U34(tt(),V1,V2)) -> isNat#(V1) active#(U34(tt(),V1,V2)) -> U35#(isNat(V1),V2) active#(U35(tt(),V2)) -> isNat#(V2) active#(U35(tt(),V2)) -> U36#(isNat(V2)) active#(U41(tt(),V2)) -> isNatKind#(V2) active#(U41(tt(),V2)) -> U42#(isNatKind(V2)) active#(U61(tt(),V2)) -> isNatKind#(V2) active#(U61(tt(),V2)) -> U62#(isNatKind(V2)) active#(U71(tt(),N)) -> isNatKind#(N) active#(U71(tt(),N)) -> U72#(isNatKind(N),N) active#(U81(tt(),M,N)) -> isNatKind#(M) active#(U81(tt(),M,N)) -> U82#(isNatKind(M),M,N) active#(U82(tt(),M,N)) -> isNat#(N) active#(U82(tt(),M,N)) -> U83#(isNat(N),M,N) active#(U83(tt(),M,N)) -> isNatKind#(N) active#(U83(tt(),M,N)) -> U84#(isNatKind(N),M,N) active#(U84(tt(),M,N)) -> plus#(N,M) active#(U84(tt(),M,N)) -> s#(plus(N,M)) active#(U91(tt(),N)) -> isNatKind#(N) active#(U91(tt(),N)) -> U92#(isNatKind(N)) active#(isNat(plus(V1,V2))) -> isNatKind#(V1) active#(isNat(plus(V1,V2))) -> U11#(isNatKind(V1),V1,V2) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNat(x(V1,V2))) -> isNatKind#(V1) active#(isNat(x(V1,V2))) -> U31#(isNatKind(V1),V1,V2) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1) active#(isNatKind(plus(V1,V2))) -> U41#(isNatKind(V1),V2) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatKind(s(V1))) -> U51#(isNatKind(V1)) active#(isNatKind(x(V1,V2))) -> isNatKind#(V1) active#(isNatKind(x(V1,V2))) -> U61#(isNatKind(V1),V2) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> U71#(isNat(N),N) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> U81#(isNat(M),M,N) active#(x(N,0())) -> isNat#(N) active#(x(N,0())) -> U91#(isNat(N),N) active#(x(N,s(M))) -> isNat#(M) active#(x(N,s(M))) -> U101#(isNat(M),M,N) active#(U101(X1,X2,X3)) -> active#(X1) active#(U101(X1,X2,X3)) -> U101#(active(X1),X2,X3) active#(U102(X1,X2,X3)) -> active#(X1) active#(U102(X1,X2,X3)) -> U102#(active(X1),X2,X3) active#(U103(X1,X2,X3)) -> active#(X1) active#(U103(X1,X2,X3)) -> U103#(active(X1),X2,X3) active#(U104(X1,X2,X3)) -> active#(X1) active#(U104(X1,X2,X3)) -> U104#(active(X1),X2,X3) active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(x(X1,X2)) -> active#(X1) active#(x(X1,X2)) -> x#(active(X1),X2) active#(x(X1,X2)) -> active#(X2) active#(x(X1,X2)) -> x#(X1,active(X2)) active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(U13(X1,X2,X3)) -> active#(X1) active#(U13(X1,X2,X3)) -> U13#(active(X1),X2,X3) active#(U14(X1,X2,X3)) -> active#(X1) active#(U14(X1,X2,X3)) -> U14#(active(X1),X2,X3) active#(U15(X1,X2)) -> active#(X1) active#(U15(X1,X2)) -> U15#(active(X1),X2) active#(U16(X)) -> active#(X) active#(U16(X)) -> U16#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X1,X2)) -> active#(X1) active#(U22(X1,X2)) -> U22#(active(X1),X2) active#(U23(X)) -> active#(X) active#(U23(X)) -> U23#(active(X)) active#(U31(X1,X2,X3)) -> active#(X1) active#(U31(X1,X2,X3)) -> U31#(active(X1),X2,X3) active#(U32(X1,X2,X3)) -> active#(X1) active#(U32(X1,X2,X3)) -> U32#(active(X1),X2,X3) active#(U33(X1,X2,X3)) -> active#(X1) active#(U33(X1,X2,X3)) -> U33#(active(X1),X2,X3) active#(U34(X1,X2,X3)) -> active#(X1) active#(U34(X1,X2,X3)) -> U34#(active(X1),X2,X3) active#(U35(X1,X2)) -> active#(X1) active#(U35(X1,X2)) -> U35#(active(X1),X2) active#(U36(X)) -> active#(X) active#(U36(X)) -> U36#(active(X)) active#(U41(X1,X2)) -> active#(X1) active#(U41(X1,X2)) -> U41#(active(X1),X2) active#(U42(X)) -> active#(X) active#(U42(X)) -> U42#(active(X)) active#(U51(X)) -> active#(X) active#(U51(X)) -> U51#(active(X)) active#(U61(X1,X2)) -> active#(X1) active#(U61(X1,X2)) -> U61#(active(X1),X2) active#(U62(X)) -> active#(X) active#(U62(X)) -> U62#(active(X)) active#(U71(X1,X2)) -> active#(X1) active#(U71(X1,X2)) -> U71#(active(X1),X2) active#(U72(X1,X2)) -> active#(X1) active#(U72(X1,X2)) -> U72#(active(X1),X2) active#(U81(X1,X2,X3)) -> active#(X1) active#(U81(X1,X2,X3)) -> U81#(active(X1),X2,X3) active#(U82(X1,X2,X3)) -> active#(X1) active#(U82(X1,X2,X3)) -> U82#(active(X1),X2,X3) active#(U83(X1,X2,X3)) -> active#(X1) active#(U83(X1,X2,X3)) -> U83#(active(X1),X2,X3) active#(U84(X1,X2,X3)) -> active#(X1) active#(U84(X1,X2,X3)) -> U84#(active(X1),X2,X3) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(U91(X1,X2)) -> active#(X1) active#(U91(X1,X2)) -> U91#(active(X1),X2) active#(U92(X)) -> active#(X) active#(U92(X)) -> U92#(active(X)) U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3) U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3) U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3) U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) x#(mark(X1),X2) -> x#(X1,X2) x#(X1,mark(X2)) -> x#(X1,X2) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U13#(mark(X1),X2,X3) -> U13#(X1,X2,X3) U14#(mark(X1),X2,X3) -> U14#(X1,X2,X3) U15#(mark(X1),X2) -> U15#(X1,X2) U16#(mark(X)) -> U16#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X1),X2) -> U22#(X1,X2) U23#(mark(X)) -> U23#(X) U31#(mark(X1),X2,X3) -> U31#(X1,X2,X3) U32#(mark(X1),X2,X3) -> U32#(X1,X2,X3) U33#(mark(X1),X2,X3) -> U33#(X1,X2,X3) U34#(mark(X1),X2,X3) -> U34#(X1,X2,X3) U35#(mark(X1),X2) -> U35#(X1,X2) U36#(mark(X)) -> U36#(X) U41#(mark(X1),X2) -> U41#(X1,X2) U42#(mark(X)) -> U42#(X) U51#(mark(X)) -> U51#(X) U61#(mark(X1),X2) -> U61#(X1,X2) U62#(mark(X)) -> U62#(X) U71#(mark(X1),X2) -> U71#(X1,X2) U72#(mark(X1),X2) -> U72#(X1,X2) U81#(mark(X1),X2,X3) -> U81#(X1,X2,X3) U82#(mark(X1),X2,X3) -> U82#(X1,X2,X3) U83#(mark(X1),X2,X3) -> U83#(X1,X2,X3) U84#(mark(X1),X2,X3) -> U84#(X1,X2,X3) s#(mark(X)) -> s#(X) U91#(mark(X1),X2) -> U91#(X1,X2) U92#(mark(X)) -> U92#(X) proper#(U101(X1,X2,X3)) -> proper#(X3) proper#(U101(X1,X2,X3)) -> proper#(X2) proper#(U101(X1,X2,X3)) -> proper#(X1) proper#(U101(X1,X2,X3)) -> U101#(proper(X1),proper(X2),proper(X3)) proper#(U102(X1,X2,X3)) -> proper#(X3) proper#(U102(X1,X2,X3)) -> proper#(X2) proper#(U102(X1,X2,X3)) -> proper#(X1) proper#(U102(X1,X2,X3)) -> U102#(proper(X1),proper(X2),proper(X3)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) proper#(U103(X1,X2,X3)) -> proper#(X3) proper#(U103(X1,X2,X3)) -> proper#(X2) proper#(U103(X1,X2,X3)) -> proper#(X1) proper#(U103(X1,X2,X3)) -> U103#(proper(X1),proper(X2),proper(X3)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U104(X1,X2,X3)) -> proper#(X3) proper#(U104(X1,X2,X3)) -> proper#(X2) proper#(U104(X1,X2,X3)) -> proper#(X1) proper#(U104(X1,X2,X3)) -> U104#(proper(X1),proper(X2),proper(X3)) proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(x(X1,X2)) -> proper#(X2) proper#(x(X1,X2)) -> proper#(X1) proper#(x(X1,X2)) -> x#(proper(X1),proper(X2)) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U13(X1,X2,X3)) -> proper#(X3) proper#(U13(X1,X2,X3)) -> proper#(X2) proper#(U13(X1,X2,X3)) -> proper#(X1) proper#(U13(X1,X2,X3)) -> U13#(proper(X1),proper(X2),proper(X3)) proper#(U14(X1,X2,X3)) -> proper#(X3) proper#(U14(X1,X2,X3)) -> proper#(X2) proper#(U14(X1,X2,X3)) -> proper#(X1) proper#(U14(X1,X2,X3)) -> U14#(proper(X1),proper(X2),proper(X3)) proper#(U15(X1,X2)) -> proper#(X2) proper#(U15(X1,X2)) -> proper#(X1) proper#(U15(X1,X2)) -> U15#(proper(X1),proper(X2)) proper#(U16(X)) -> proper#(X) proper#(U16(X)) -> U16#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X1,X2)) -> proper#(X2) proper#(U22(X1,X2)) -> proper#(X1) proper#(U22(X1,X2)) -> U22#(proper(X1),proper(X2)) proper#(U23(X)) -> proper#(X) proper#(U23(X)) -> U23#(proper(X)) proper#(U31(X1,X2,X3)) -> proper#(X3) proper#(U31(X1,X2,X3)) -> proper#(X2) proper#(U31(X1,X2,X3)) -> proper#(X1) proper#(U31(X1,X2,X3)) -> U31#(proper(X1),proper(X2),proper(X3)) proper#(U32(X1,X2,X3)) -> proper#(X3) proper#(U32(X1,X2,X3)) -> proper#(X2) proper#(U32(X1,X2,X3)) -> proper#(X1) proper#(U32(X1,X2,X3)) -> U32#(proper(X1),proper(X2),proper(X3)) proper#(U33(X1,X2,X3)) -> proper#(X3) proper#(U33(X1,X2,X3)) -> proper#(X2) proper#(U33(X1,X2,X3)) -> proper#(X1) proper#(U33(X1,X2,X3)) -> U33#(proper(X1),proper(X2),proper(X3)) proper#(U34(X1,X2,X3)) -> proper#(X3) proper#(U34(X1,X2,X3)) -> proper#(X2) proper#(U34(X1,X2,X3)) -> proper#(X1) proper#(U34(X1,X2,X3)) -> U34#(proper(X1),proper(X2),proper(X3)) proper#(U35(X1,X2)) -> proper#(X2) proper#(U35(X1,X2)) -> proper#(X1) proper#(U35(X1,X2)) -> U35#(proper(X1),proper(X2)) proper#(U36(X)) -> proper#(X) proper#(U36(X)) -> U36#(proper(X)) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2)) proper#(U42(X)) -> proper#(X) proper#(U42(X)) -> U42#(proper(X)) proper#(U51(X)) -> proper#(X) proper#(U51(X)) -> U51#(proper(X)) proper#(U61(X1,X2)) -> proper#(X2) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2)) proper#(U62(X)) -> proper#(X) proper#(U62(X)) -> U62#(proper(X)) proper#(U71(X1,X2)) -> proper#(X2) proper#(U71(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2)) proper#(U72(X1,X2)) -> proper#(X2) proper#(U72(X1,X2)) -> proper#(X1) proper#(U72(X1,X2)) -> U72#(proper(X1),proper(X2)) proper#(U81(X1,X2,X3)) -> proper#(X3) proper#(U81(X1,X2,X3)) -> proper#(X2) proper#(U81(X1,X2,X3)) -> proper#(X1) proper#(U81(X1,X2,X3)) -> U81#(proper(X1),proper(X2),proper(X3)) proper#(U82(X1,X2,X3)) -> proper#(X3) proper#(U82(X1,X2,X3)) -> proper#(X2) proper#(U82(X1,X2,X3)) -> proper#(X1) proper#(U82(X1,X2,X3)) -> U82#(proper(X1),proper(X2),proper(X3)) proper#(U83(X1,X2,X3)) -> proper#(X3) proper#(U83(X1,X2,X3)) -> proper#(X2) proper#(U83(X1,X2,X3)) -> proper#(X1) proper#(U83(X1,X2,X3)) -> U83#(proper(X1),proper(X2),proper(X3)) proper#(U84(X1,X2,X3)) -> proper#(X3) proper#(U84(X1,X2,X3)) -> proper#(X2) proper#(U84(X1,X2,X3)) -> proper#(X1) proper#(U84(X1,X2,X3)) -> U84#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(U91(X1,X2)) -> proper#(X2) proper#(U91(X1,X2)) -> proper#(X1) proper#(U91(X1,X2)) -> U91#(proper(X1),proper(X2)) proper#(U92(X)) -> proper#(X) proper#(U92(X)) -> U92#(proper(X)) U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3) U102#(ok(X1),ok(X2),ok(X3)) -> U102#(X1,X2,X3) isNatKind#(ok(X)) -> isNatKind#(X) U103#(ok(X1),ok(X2),ok(X3)) -> U103#(X1,X2,X3) isNat#(ok(X)) -> isNat#(X) U104#(ok(X1),ok(X2),ok(X3)) -> U104#(X1,X2,X3) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) x#(ok(X1),ok(X2)) -> x#(X1,X2) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) U13#(ok(X1),ok(X2),ok(X3)) -> U13#(X1,X2,X3) U14#(ok(X1),ok(X2),ok(X3)) -> U14#(X1,X2,X3) U15#(ok(X1),ok(X2)) -> U15#(X1,X2) U16#(ok(X)) -> U16#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X1),ok(X2)) -> U22#(X1,X2) U23#(ok(X)) -> U23#(X) U31#(ok(X1),ok(X2),ok(X3)) -> U31#(X1,X2,X3) U32#(ok(X1),ok(X2),ok(X3)) -> U32#(X1,X2,X3) U33#(ok(X1),ok(X2),ok(X3)) -> U33#(X1,X2,X3) U34#(ok(X1),ok(X2),ok(X3)) -> U34#(X1,X2,X3) U35#(ok(X1),ok(X2)) -> U35#(X1,X2) U36#(ok(X)) -> U36#(X) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) U42#(ok(X)) -> U42#(X) U51#(ok(X)) -> U51#(X) U61#(ok(X1),ok(X2)) -> U61#(X1,X2) U62#(ok(X)) -> U62#(X) U71#(ok(X1),ok(X2)) -> U71#(X1,X2) U72#(ok(X1),ok(X2)) -> U72#(X1,X2) U81#(ok(X1),ok(X2),ok(X3)) -> U81#(X1,X2,X3) U82#(ok(X1),ok(X2),ok(X3)) -> U82#(X1,X2,X3) U83#(ok(X1),ok(X2),ok(X3)) -> U83#(X1,X2,X3) U84#(ok(X1),ok(X2),ok(X3)) -> U84#(X1,X2,X3) s#(ok(X)) -> s#(X) U91#(ok(X1),ok(X2)) -> U91#(X1,X2) U92#(ok(X)) -> U92#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... SCC Processor: #sccs: 40 #rules: 193 #arcs: 15292/113569 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(U92(X)) -> proper#(X) proper#(U91(X1,X2)) -> proper#(X1) proper#(U91(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) proper#(U84(X1,X2,X3)) -> proper#(X1) proper#(U84(X1,X2,X3)) -> proper#(X2) proper#(U84(X1,X2,X3)) -> proper#(X3) proper#(U83(X1,X2,X3)) -> proper#(X1) proper#(U83(X1,X2,X3)) -> proper#(X2) proper#(U83(X1,X2,X3)) -> proper#(X3) proper#(U82(X1,X2,X3)) -> proper#(X1) proper#(U82(X1,X2,X3)) -> proper#(X2) proper#(U82(X1,X2,X3)) -> proper#(X3) proper#(U81(X1,X2,X3)) -> proper#(X1) proper#(U81(X1,X2,X3)) -> proper#(X2) proper#(U81(X1,X2,X3)) -> proper#(X3) proper#(U72(X1,X2)) -> proper#(X1) proper#(U72(X1,X2)) -> proper#(X2) proper#(U71(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> proper#(X2) proper#(U62(X)) -> proper#(X) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> proper#(X2) proper#(U51(X)) -> proper#(X) proper#(U42(X)) -> proper#(X) proper#(U41(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> proper#(X2) proper#(U36(X)) -> proper#(X) proper#(U35(X1,X2)) -> proper#(X1) proper#(U35(X1,X2)) -> proper#(X2) proper#(U34(X1,X2,X3)) -> proper#(X1) proper#(U34(X1,X2,X3)) -> proper#(X2) proper#(U34(X1,X2,X3)) -> proper#(X3) proper#(U33(X1,X2,X3)) -> proper#(X1) proper#(U33(X1,X2,X3)) -> proper#(X2) proper#(U33(X1,X2,X3)) -> proper#(X3) proper#(U32(X1,X2,X3)) -> proper#(X1) proper#(U32(X1,X2,X3)) -> proper#(X2) proper#(U32(X1,X2,X3)) -> proper#(X3) proper#(U31(X1,X2,X3)) -> proper#(X1) proper#(U31(X1,X2,X3)) -> proper#(X2) proper#(U31(X1,X2,X3)) -> proper#(X3) proper#(U23(X)) -> proper#(X) proper#(U22(X1,X2)) -> proper#(X1) proper#(U22(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(U16(X)) -> proper#(X) proper#(U15(X1,X2)) -> proper#(X1) proper#(U15(X1,X2)) -> proper#(X2) proper#(U14(X1,X2,X3)) -> proper#(X1) proper#(U14(X1,X2,X3)) -> proper#(X2) proper#(U14(X1,X2,X3)) -> proper#(X3) proper#(U13(X1,X2,X3)) -> proper#(X1) proper#(U13(X1,X2,X3)) -> proper#(X2) proper#(U13(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(x(X1,X2)) -> proper#(X1) proper#(x(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X2) proper#(U104(X1,X2,X3)) -> proper#(X1) proper#(U104(X1,X2,X3)) -> proper#(X2) proper#(U104(X1,X2,X3)) -> proper#(X3) proper#(isNat(X)) -> proper#(X) proper#(U103(X1,X2,X3)) -> proper#(X1) proper#(U103(X1,X2,X3)) -> proper#(X2) proper#(U103(X1,X2,X3)) -> proper#(X3) proper#(isNatKind(X)) -> proper#(X) proper#(U102(X1,X2,X3)) -> proper#(X1) proper#(U102(X1,X2,X3)) -> proper#(X2) proper#(U102(X1,X2,X3)) -> proper#(X3) proper#(U101(X1,X2,X3)) -> proper#(X1) proper#(U101(X1,X2,X3)) -> proper#(X2) proper#(U101(X1,X2,X3)) -> proper#(X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(U92(X)) -> active#(X) active#(U91(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(U84(X1,X2,X3)) -> active#(X1) active#(U83(X1,X2,X3)) -> active#(X1) active#(U82(X1,X2,X3)) -> active#(X1) active#(U81(X1,X2,X3)) -> active#(X1) active#(U72(X1,X2)) -> active#(X1) active#(U71(X1,X2)) -> active#(X1) active#(U62(X)) -> active#(X) active#(U61(X1,X2)) -> active#(X1) active#(U51(X)) -> active#(X) active#(U42(X)) -> active#(X) active#(U41(X1,X2)) -> active#(X1) active#(U36(X)) -> active#(X) active#(U35(X1,X2)) -> active#(X1) active#(U34(X1,X2,X3)) -> active#(X1) active#(U33(X1,X2,X3)) -> active#(X1) active#(U32(X1,X2,X3)) -> active#(X1) active#(U31(X1,X2,X3)) -> active#(X1) active#(U23(X)) -> active#(X) active#(U22(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> active#(X1) active#(U16(X)) -> active#(X) active#(U15(X1,X2)) -> active#(X1) active#(U14(X1,X2,X3)) -> active#(X1) active#(U13(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> active#(X1) active#(x(X1,X2)) -> active#(X2) active#(x(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> active#(X1) active#(U104(X1,X2,X3)) -> active#(X1) active#(U103(X1,X2,X3)) -> active#(X1) active#(U102(X1,X2,X3)) -> active#(X1) active#(U101(X1,X2,X3)) -> active#(X1) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: isNatKind#(ok(X)) -> isNatKind#(X) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: isNat#(ok(X)) -> isNat#(X) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3) U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U102#(ok(X1),ok(X2),ok(X3)) -> U102#(X1,X2,X3) U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U103#(ok(X1),ok(X2),ok(X3)) -> U103#(X1,X2,X3) U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U104#(ok(X1),ok(X2),ok(X3)) -> U104#(X1,X2,X3) U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: plus#(ok(X1),ok(X2)) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(mark(X1),X2) -> plus#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: x#(ok(X1),ok(X2)) -> x#(X1,X2) x#(X1,mark(X2)) -> x#(X1,X2) x#(mark(X1),X2) -> x#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U13#(ok(X1),ok(X2),ok(X3)) -> U13#(X1,X2,X3) U13#(mark(X1),X2,X3) -> U13#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U14#(ok(X1),ok(X2),ok(X3)) -> U14#(X1,X2,X3) U14#(mark(X1),X2,X3) -> U14#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U15#(ok(X1),ok(X2)) -> U15#(X1,X2) U15#(mark(X1),X2) -> U15#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U16#(ok(X)) -> U16#(X) U16#(mark(X)) -> U16#(X) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U22#(ok(X1),ok(X2)) -> U22#(X1,X2) U22#(mark(X1),X2) -> U22#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U23#(ok(X)) -> U23#(X) U23#(mark(X)) -> U23#(X) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U31#(ok(X1),ok(X2),ok(X3)) -> U31#(X1,X2,X3) U31#(mark(X1),X2,X3) -> U31#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U32#(ok(X1),ok(X2),ok(X3)) -> U32#(X1,X2,X3) U32#(mark(X1),X2,X3) -> U32#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U33#(ok(X1),ok(X2),ok(X3)) -> U33#(X1,X2,X3) U33#(mark(X1),X2,X3) -> U33#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U34#(ok(X1),ok(X2),ok(X3)) -> U34#(X1,X2,X3) U34#(mark(X1),X2,X3) -> U34#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U35#(ok(X1),ok(X2)) -> U35#(X1,X2) U35#(mark(X1),X2) -> U35#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U36#(ok(X)) -> U36#(X) U36#(mark(X)) -> U36#(X) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U41#(ok(X1),ok(X2)) -> U41#(X1,X2) U41#(mark(X1),X2) -> U41#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U42#(ok(X)) -> U42#(X) U42#(mark(X)) -> U42#(X) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U51#(ok(X)) -> U51#(X) U51#(mark(X)) -> U51#(X) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U61#(ok(X1),ok(X2)) -> U61#(X1,X2) U61#(mark(X1),X2) -> U61#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U62#(ok(X)) -> U62#(X) U62#(mark(X)) -> U62#(X) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U71#(ok(X1),ok(X2)) -> U71#(X1,X2) U71#(mark(X1),X2) -> U71#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U72#(ok(X1),ok(X2)) -> U72#(X1,X2) U72#(mark(X1),X2) -> U72#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U81#(ok(X1),ok(X2),ok(X3)) -> U81#(X1,X2,X3) U81#(mark(X1),X2,X3) -> U81#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U82#(ok(X1),ok(X2),ok(X3)) -> U82#(X1,X2,X3) U82#(mark(X1),X2,X3) -> U82#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U83#(ok(X1),ok(X2),ok(X3)) -> U83#(X1,X2,X3) U83#(mark(X1),X2,X3) -> U83#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U84#(ok(X1),ok(X2),ok(X3)) -> U84#(X1,X2,X3) U84#(mark(X1),X2,X3) -> U84#(X1,X2,X3) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U91#(ok(X1),ok(X2)) -> U91#(X1,X2) U91#(mark(X1),X2) -> U91#(X1,X2) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U92#(ok(X)) -> U92#(X) U92#(mark(X)) -> U92#(X) TRS: active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N)) active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N)) active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N)) active(U104(tt(),M,N)) -> mark(plus(x(N,M),N)) active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) -> mark(U16(isNat(V2))) active(U16(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2)) active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2)) active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2)) active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2)) active(U35(tt(),V2)) -> mark(U36(isNat(V2))) active(U36(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatKind(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt(),N)) -> mark(U72(isNatKind(N),N)) active(U72(tt(),N)) -> mark(N) active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N)) active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N)) active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N)) active(U84(tt(),M,N)) -> mark(s(plus(N,M))) active(U91(tt(),N)) -> mark(U92(isNatKind(N))) active(U92(tt())) -> mark(0()) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2)) active(plus(N,0())) -> mark(U71(isNat(N),N)) active(plus(N,s(M))) -> mark(U81(isNat(M),M,N)) active(x(N,0())) -> mark(U91(isNat(N),N)) active(x(N,s(M))) -> mark(U101(isNat(M),M,N)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) active(U15(X1,X2)) -> U15(active(X1),X2) active(U16(X)) -> U16(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3) active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3) active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3) active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3) active(U35(X1,X2)) -> U35(active(X1),X2) active(U36(X)) -> U36(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(U72(X1,X2)) -> U72(active(X1),X2) active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3) active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3) active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3) active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(U91(X1,X2)) -> U91(active(X1),X2) active(U92(X)) -> U92(active(X)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) U15(mark(X1),X2) -> mark(U15(X1,X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X1),X2) -> mark(U22(X1,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3)) U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3)) U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3)) U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3)) U35(mark(X1),X2) -> mark(U35(X1,X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U72(mark(X1),X2) -> mark(U72(X1,X2)) U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3)) U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3)) U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3)) U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1),X2) -> mark(U91(X1,X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3)) proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3)) proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3)) proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3)) proper(U35(X1,X2)) -> U35(proper(X1),proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U72(X1,X2)) -> U72(proper(X1),proper(X2)) proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3)) proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3)) proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3)) proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0()) -> ok(0()) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3)) U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3)) U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3)) U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3)) U35(ok(X1),ok(X2)) -> ok(U35(X1,X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U72(ok(X1),ok(X2)) -> ok(U72(X1,X2)) U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3)) U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3)) U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3)) U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open