MAYBE Problem: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Proof: DP Processor: DPs: active#(U11(tt(),V1,V2)) -> isNatKind#(V1) active#(U11(tt(),V1,V2)) -> U12#(isNatKind(V1),V1,V2) active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) active#(U12(tt(),V1,V2)) -> isNatKind#(V2) active#(U12(tt(),V1,V2)) -> U13#(isNatKind(V2),V1,V2) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> isNatKind#(V2) active#(U13(tt(),V1,V2)) -> U14#(isNatKind(V2),V1,V2) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> isNat#(V1) active#(U14(tt(),V1,V2)) -> U15#(isNat(V1),V2) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) active#(U15(tt(),V2)) -> isNat#(V2) active#(U15(tt(),V2)) -> U16#(isNat(V2)) active#(U15(tt(),V2)) -> mark#(U16(isNat(V2))) active#(U16(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> isNatKind#(V1) active#(U21(tt(),V1)) -> U22#(isNatKind(V1),V1) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> isNat#(V1) active#(U22(tt(),V1)) -> U23#(isNat(V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) active#(U23(tt())) -> mark#(tt()) active#(U31(tt(),V2)) -> isNatKind#(V2) active#(U31(tt(),V2)) -> U32#(isNatKind(V2)) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) active#(U51(tt(),N)) -> isNatKind#(N) active#(U51(tt(),N)) -> U52#(isNatKind(N),N) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) active#(U52(tt(),N)) -> mark#(N) active#(U61(tt(),M,N)) -> isNatKind#(M) active#(U61(tt(),M,N)) -> U62#(isNatKind(M),M,N) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) active#(U62(tt(),M,N)) -> isNat#(N) active#(U62(tt(),M,N)) -> U63#(isNat(N),M,N) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) -> isNatKind#(N) active#(U63(tt(),M,N)) -> U64#(isNatKind(N),M,N) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> plus#(N,M) active#(U64(tt(),M,N)) -> s#(plus(N,M)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) active#(isNat(0())) -> mark#(tt()) active#(isNat(plus(V1,V2))) -> isNatKind#(V1) active#(isNat(plus(V1,V2))) -> U11#(isNatKind(V1),V1,V2) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1) active#(isNatKind(plus(V1,V2))) -> U31#(isNatKind(V1),V2) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatKind(s(V1))) -> U41#(isNatKind(V1)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> U51#(isNat(N),N) active#(plus(N,0())) -> mark#(U51(isNat(N),N)) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> U61#(isNat(M),M,N) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) mark#(tt()) -> active#(tt()) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> U12#(mark(X1),X2,X3) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> U13#(mark(X1),X2,X3) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> U14#(mark(X1),X2,X3) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U15(X1,X2)) -> mark#(X1) mark#(U15(X1,X2)) -> U15#(mark(X1),X2) mark#(U15(X1,X2)) -> active#(U15(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> U16#(mark(X)) mark#(U16(X)) -> active#(U16(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> U21#(mark(X1),X2) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> U22#(mark(X1),X2) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> U23#(mark(X)) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> U31#(mark(X1),X2) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> U32#(mark(X)) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> U41#(mark(X)) mark#(U41(X)) -> active#(U41(mark(X))) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> U51#(mark(X1),X2) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> U52#(mark(X1),X2) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) mark#(U62(X1,X2,X3)) -> mark#(X1) mark#(U62(X1,X2,X3)) -> U62#(mark(X1),X2,X3) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) mark#(U63(X1,X2,X3)) -> mark#(X1) mark#(U63(X1,X2,X3)) -> U63#(mark(X1),X2,X3) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) mark#(U64(X1,X2,X3)) -> mark#(X1) mark#(U64(X1,X2,X3)) -> U64#(mark(X1),X2,X3) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(0()) -> active#(0()) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) isNatKind#(mark(X)) -> isNatKind#(X) isNatKind#(active(X)) -> isNatKind#(X) U13#(mark(X1),X2,X3) -> U13#(X1,X2,X3) U13#(X1,mark(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) U13#(active(X1),X2,X3) -> U13#(X1,X2,X3) U13#(X1,active(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U14#(mark(X1),X2,X3) -> U14#(X1,X2,X3) U14#(X1,mark(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) U14#(active(X1),X2,X3) -> U14#(X1,X2,X3) U14#(X1,active(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U15#(mark(X1),X2) -> U15#(X1,X2) U15#(X1,mark(X2)) -> U15#(X1,X2) U15#(active(X1),X2) -> U15#(X1,X2) U15#(X1,active(X2)) -> U15#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) U16#(mark(X)) -> U16#(X) U16#(active(X)) -> U16#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U22#(mark(X1),X2) -> U22#(X1,X2) U22#(X1,mark(X2)) -> U22#(X1,X2) U22#(active(X1),X2) -> U22#(X1,X2) U22#(X1,active(X2)) -> U22#(X1,X2) U23#(mark(X)) -> U23#(X) U23#(active(X)) -> U23#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U32#(mark(X)) -> U32#(X) U32#(active(X)) -> U32#(X) U41#(mark(X)) -> U41#(X) U41#(active(X)) -> U41#(X) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) U51#(active(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U52#(mark(X1),X2) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) U52#(active(X1),X2) -> U52#(X1,X2) U52#(X1,active(X2)) -> U52#(X1,X2) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U62#(mark(X1),X2,X3) -> U62#(X1,X2,X3) U62#(X1,mark(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) U62#(active(X1),X2,X3) -> U62#(X1,X2,X3) U62#(X1,active(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U63#(mark(X1),X2,X3) -> U63#(X1,X2,X3) U63#(X1,mark(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) U63#(active(X1),X2,X3) -> U63#(X1,X2,X3) U63#(X1,active(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U64#(mark(X1),X2,X3) -> U64#(X1,X2,X3) U64#(X1,mark(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) U64#(active(X1),X2,X3) -> U64#(X1,X2,X3) U64#(X1,active(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) TDG Processor: DPs: active#(U11(tt(),V1,V2)) -> isNatKind#(V1) active#(U11(tt(),V1,V2)) -> U12#(isNatKind(V1),V1,V2) active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) active#(U12(tt(),V1,V2)) -> isNatKind#(V2) active#(U12(tt(),V1,V2)) -> U13#(isNatKind(V2),V1,V2) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> isNatKind#(V2) active#(U13(tt(),V1,V2)) -> U14#(isNatKind(V2),V1,V2) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> isNat#(V1) active#(U14(tt(),V1,V2)) -> U15#(isNat(V1),V2) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) active#(U15(tt(),V2)) -> isNat#(V2) active#(U15(tt(),V2)) -> U16#(isNat(V2)) active#(U15(tt(),V2)) -> mark#(U16(isNat(V2))) active#(U16(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> isNatKind#(V1) active#(U21(tt(),V1)) -> U22#(isNatKind(V1),V1) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> isNat#(V1) active#(U22(tt(),V1)) -> U23#(isNat(V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) active#(U23(tt())) -> mark#(tt()) active#(U31(tt(),V2)) -> isNatKind#(V2) active#(U31(tt(),V2)) -> U32#(isNatKind(V2)) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) active#(U51(tt(),N)) -> isNatKind#(N) active#(U51(tt(),N)) -> U52#(isNatKind(N),N) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) active#(U52(tt(),N)) -> mark#(N) active#(U61(tt(),M,N)) -> isNatKind#(M) active#(U61(tt(),M,N)) -> U62#(isNatKind(M),M,N) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) active#(U62(tt(),M,N)) -> isNat#(N) active#(U62(tt(),M,N)) -> U63#(isNat(N),M,N) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) -> isNatKind#(N) active#(U63(tt(),M,N)) -> U64#(isNatKind(N),M,N) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> plus#(N,M) active#(U64(tt(),M,N)) -> s#(plus(N,M)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) active#(isNat(0())) -> mark#(tt()) active#(isNat(plus(V1,V2))) -> isNatKind#(V1) active#(isNat(plus(V1,V2))) -> U11#(isNatKind(V1),V1,V2) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1) active#(isNatKind(plus(V1,V2))) -> U31#(isNatKind(V1),V2) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatKind(s(V1))) -> U41#(isNatKind(V1)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> U51#(isNat(N),N) active#(plus(N,0())) -> mark#(U51(isNat(N),N)) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> U61#(isNat(M),M,N) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) mark#(tt()) -> active#(tt()) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> U12#(mark(X1),X2,X3) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> U13#(mark(X1),X2,X3) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> U14#(mark(X1),X2,X3) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U15(X1,X2)) -> mark#(X1) mark#(U15(X1,X2)) -> U15#(mark(X1),X2) mark#(U15(X1,X2)) -> active#(U15(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> U16#(mark(X)) mark#(U16(X)) -> active#(U16(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> U21#(mark(X1),X2) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> U22#(mark(X1),X2) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> U23#(mark(X)) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> U31#(mark(X1),X2) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> U32#(mark(X)) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> U41#(mark(X)) mark#(U41(X)) -> active#(U41(mark(X))) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> U51#(mark(X1),X2) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> U52#(mark(X1),X2) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) mark#(U62(X1,X2,X3)) -> mark#(X1) mark#(U62(X1,X2,X3)) -> U62#(mark(X1),X2,X3) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) mark#(U63(X1,X2,X3)) -> mark#(X1) mark#(U63(X1,X2,X3)) -> U63#(mark(X1),X2,X3) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) mark#(U64(X1,X2,X3)) -> mark#(X1) mark#(U64(X1,X2,X3)) -> U64#(mark(X1),X2,X3) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(0()) -> active#(0()) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) isNatKind#(mark(X)) -> isNatKind#(X) isNatKind#(active(X)) -> isNatKind#(X) U13#(mark(X1),X2,X3) -> U13#(X1,X2,X3) U13#(X1,mark(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) U13#(active(X1),X2,X3) -> U13#(X1,X2,X3) U13#(X1,active(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U14#(mark(X1),X2,X3) -> U14#(X1,X2,X3) U14#(X1,mark(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) U14#(active(X1),X2,X3) -> U14#(X1,X2,X3) U14#(X1,active(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U15#(mark(X1),X2) -> U15#(X1,X2) U15#(X1,mark(X2)) -> U15#(X1,X2) U15#(active(X1),X2) -> U15#(X1,X2) U15#(X1,active(X2)) -> U15#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) U16#(mark(X)) -> U16#(X) U16#(active(X)) -> U16#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U22#(mark(X1),X2) -> U22#(X1,X2) U22#(X1,mark(X2)) -> U22#(X1,X2) U22#(active(X1),X2) -> U22#(X1,X2) U22#(X1,active(X2)) -> U22#(X1,X2) U23#(mark(X)) -> U23#(X) U23#(active(X)) -> U23#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U32#(mark(X)) -> U32#(X) U32#(active(X)) -> U32#(X) U41#(mark(X)) -> U41#(X) U41#(active(X)) -> U41#(X) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) U51#(active(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U52#(mark(X1),X2) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) U52#(active(X1),X2) -> U52#(X1,X2) U52#(X1,active(X2)) -> U52#(X1,X2) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U62#(mark(X1),X2,X3) -> U62#(X1,X2,X3) U62#(X1,mark(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) U62#(active(X1),X2,X3) -> U62#(X1,X2,X3) U62#(X1,active(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U63#(mark(X1),X2,X3) -> U63#(X1,X2,X3) U63#(X1,mark(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) U63#(active(X1),X2,X3) -> U63#(X1,X2,X3) U63#(X1,active(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U64#(mark(X1),X2,X3) -> U64#(X1,X2,X3) U64#(X1,mark(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) U64#(active(X1),X2,X3) -> U64#(X1,X2,X3) U64#(X1,active(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) graph: ... SCC Processor: #sccs: 23 #rules: 161 #arcs: 5227/47961 DPs: mark#(0()) -> active#(0()) active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) -> active#(tt()) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U15(tt(),V2)) -> mark#(U16(isNat(V2))) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) mark#(U15(X1,X2)) -> mark#(X1) mark#(U15(X1,X2)) -> active#(U15(mark(X1),X2)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),N)) -> mark#(N) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> mark#(X1) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> mark#(X1) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> mark#(X1) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,0())) -> mark#(U51(isNat(N),N)) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U64](x0, x1, x2) = x0 + x1 + x2 + 1, [U63](x0, x1, x2) = x0 + x1 + x2 + 1, [U62](x0, x1, x2) = x0 + x1 + x2 + 1, [U61](x0, x1, x2) = x0 + x1 + x2 + 1, [U52](x0, x1) = x0 + x1, [U51](x0, x1) = x0 + x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = x0, [isNat](x0) = 0, [U14](x0, x1, x2) = x0, [U13](x0, x1, x2) = x0, [mark](x0) = x0, [U12](x0, x1, x2) = x0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: mark#(0()) = 0 >= 0 = active#(0()) active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 >= X1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U12(X1,X2,X3)) = X1 >= X1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U15(tt(),V2)) = 0 >= 0 = mark#(U16(isNat(V2))) mark#(U13(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U13(X1,X2,X3)) = X1 >= X1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U14(X1,X2,X3)) = X1 >= X1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) mark#(U15(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U15(X1,X2)) = X1 >= X1 = active#(U15(mark(X1),X2)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= X1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) active#(U51(tt(),N)) = N >= N = mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) active#(U52(tt(),N)) = N >= N = mark#(N) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U51(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U64(isNatKind(N),M,N)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) active#(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U62(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U63(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U64(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(s(X)) = X >= X = mark#(X) mark#(s(X)) = X >= X = active#(s(mark(X))) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active#(plus(N,0())) = N + 1 >= N = mark#(U51(isNat(N),N)) active#(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X1 >= X1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X1 >= X1 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X1 >= X1 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X1 >= X1 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X1 >= X1 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X1 + X2 >= X1 + X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 >= X1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 >= X1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X1 >= X1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X1 >= X1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X1 >= X1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X1 >= X1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X1 >= X1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X1 >= X1 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = X1 >= X1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X1 >= X1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X1 >= X1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X1 >= X1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X1 >= X1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X1 >= X1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X1 >= X1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X1 >= X1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X1 >= X1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X1 >= X1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X1 >= X1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X1 >= X1 = U14(X1,X2,X3) U15(mark(X1),X2) = X1 >= X1 = U15(X1,X2) U15(X1,mark(X2)) = X1 >= X1 = U15(X1,X2) U15(active(X1),X2) = X1 >= X1 = U15(X1,X2) U15(X1,active(X2)) = X1 >= X1 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X1 + X2 >= X1 + X2 = U51(X1,X2) U51(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U51(X1,X2) U51(active(X1),X2) = X1 + X2 >= X1 + X2 = U51(X1,X2) U51(X1,active(X2)) = X1 + X2 >= X1 + X2 = U51(X1,X2) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: mark#(0()) -> active#(0()) active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) -> active#(tt()) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U15(tt(),V2)) -> mark#(U16(isNat(V2))) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) mark#(U15(X1,X2)) -> mark#(X1) mark#(U15(X1,X2)) -> active#(U15(mark(X1),X2)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),N)) -> mark#(N) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 1, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x0 + x1 + 1, [U51](x0, x1) = x0 + x1 + 1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = x0, [isNat](x0) = 0, [U14](x0, x1, x2) = x0, [U13](x0, x1, x2) = x0, [mark](x0) = x0, [U12](x0, x1, x2) = x0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: mark#(0()) = 1 >= 1 = active#(0()) active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 >= X1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U12(X1,X2,X3)) = X1 >= X1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U15(tt(),V2)) = 0 >= 0 = mark#(U16(isNat(V2))) mark#(U13(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U13(X1,X2,X3)) = X1 >= X1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U14(X1,X2,X3)) = X1 >= X1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) mark#(U15(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U15(X1,X2)) = X1 >= X1 = active#(U15(mark(X1),X2)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= X1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) active#(U51(tt(),N)) = N + 1 >= N + 1 = mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) active#(U52(tt(),N)) = N + 1 >= N = mark#(N) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U51(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U64(isNatKind(N),M,N)) mark#(U52(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U52(mark(X1),X2)) active#(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(s(X)) = X >= X = mark#(X) mark#(s(X)) = X >= X = active#(s(mark(X))) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N + 1 >= N + 1 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N + 1 >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 2 >= N + 1 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X1 >= X1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X1 >= X1 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X1 >= X1 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X1 >= X1 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X1 >= X1 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) U11(mark(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 >= X1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 >= X1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X1 >= X1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X1 >= X1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X1 >= X1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X1 >= X1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X1 >= X1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X1 >= X1 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = X1 >= X1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X1 >= X1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X1 >= X1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X1 >= X1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X1 >= X1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X1 >= X1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X1 >= X1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X1 >= X1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X1 >= X1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X1 >= X1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X1 >= X1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X1 >= X1 = U14(X1,X2,X3) U15(mark(X1),X2) = X1 >= X1 = U15(X1,X2) U15(X1,mark(X2)) = X1 >= X1 = U15(X1,X2) U15(active(X1),X2) = X1 >= X1 = U15(X1,X2) U15(X1,active(X2)) = X1 >= X1 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(X1,X2) U51(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(X1,X2) U51(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(X1,X2) U51(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(X1,X2) U52(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: mark#(0()) -> active#(0()) active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) -> active#(tt()) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U15(tt(),V2)) -> mark#(U16(isNat(V2))) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) mark#(U15(X1,X2)) -> mark#(X1) mark#(U15(X1,X2)) -> active#(U15(mark(X1),X2)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0 + x1, [U21](x0, x1) = x0 + x1, [U16](x0) = x0, [U15](x0, x1) = x0 + x1 + 1, [isNat](x0) = x0, [U14](x0, x1, x2) = x0 + x1 + x2 + 1, [U13](x0, x1, x2) = x0 + x1 + x2 + 1, [mark](x0) = x0, [U12](x0, x1, x2) = x0 + x1 + x2 + 1, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x0 + x1 + x2 + 1, [tt] = 0 orientation: mark#(0()) = 0 >= 0 = active#(0()) active#(U11(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U13(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U15(tt(),V2)) = V2 + 1 >= V2 = mark#(U16(isNat(V2))) mark#(U13(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U13(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U14(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNatKind(V1),V1)) mark#(U15(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U15(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U15(mark(X1),X2)) active#(U22(tt(),V1)) = V1 >= V1 = mark#(U23(isNat(V1))) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) active#(U51(tt(),N)) = N >= N = mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) = X2 >= X2 = active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U64(isNatKind(N),M,N)) mark#(U52(X1,X2)) = X2 >= X2 = active#(U52(mark(X1),X2)) active#(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(s(X)) = X >= X = mark#(X) mark#(s(X)) = X >= X = active#(s(mark(X))) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = V2 + 1 >= V2 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = V1 >= V1 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U15(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 + X2 >= X1 + X2 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U14(X1,X2,X3) U15(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U15(X1,X2) U15(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U15(X1,X2) U15(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U15(X1,X2) U15(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U15(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X1),X2) = X1 + X2 >= X1 + X2 = U22(X1,X2) U22(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U22(X1,X2) U22(active(X1),X2) = X1 + X2 >= X1 + X2 = U22(X1,X2) U22(X1,active(X2)) = X1 + X2 >= X1 + X2 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: mark#(0()) -> active#(0()) active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) -> active#(tt()) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) mark#(U15(X1,X2)) -> active#(U15(mark(X1),X2)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 1, [s](x0) = 0, [plus](x0, x1) = 1, [U64](x0, x1, x2) = 1, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 1, [U52](x0, x1) = 1, [U51](x0, x1) = 1, [U41](x0) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U23](x0) = 1, [U22](x0, x1) = 1, [U21](x0, x1) = 1, [U16](x0) = 1, [U15](x0, x1) = 1, [isNat](x0) = 1, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 1 orientation: mark#(0()) = 1 >= 1 = active#(0()) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 1 >= 1 = mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) = 1 >= 1 = active#(tt()) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNatKind(V1),V1)) mark#(U15(X1,X2)) = 1 >= 1 = active#(U15(mark(X1),X2)) active#(U22(tt(),V1)) = 1 >= 1 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U23(tt())) = 1 >= 1 = mark#(tt()) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 1 >= 1 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(U22(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U22(X1,X2)) = 1 >= 1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 1 >= 1 = mark#(tt()) mark#(U23(X)) = 1 >= 1 = mark#(X) mark#(U23(X)) = 1 >= 1 = active#(U23(mark(X))) active#(U51(tt(),N)) = 1 >= 1 = mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 1 >= 1 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = 1 >= 1 = mark#(X) mark#(U41(X)) = 1 >= 1 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 1 >= 1 = mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) = 1 >= 1 = active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) = 1 >= 1 = mark#(U64(isNatKind(N),M,N)) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) active#(U64(tt(),M,N)) = 1 >= 1 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2,X3)) = 1 >= 1 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 1 >= 1 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 1 >= 1 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(s(X)) = 1 >= 0 = active#(s(mark(X))) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = 1 >= 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = 1 >= 1 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,mark(X2)) = 1 >= 1 = U22(X1,X2) U22(active(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,active(X2)) = 1 >= 1 = U22(X1,X2) U23(mark(X)) = 1 >= 1 = U23(X) U23(active(X)) = 1 >= 1 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 1 >= 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 1 >= 1 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) problem: DPs: mark#(0()) -> active#(0()) active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) -> active#(tt()) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) mark#(U15(X1,X2)) -> active#(U15(mark(X1),X2)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 1, [s](x0) = 0, [plus](x0, x1) = 1, [U64](x0, x1, x2) = 1, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 1, [U52](x0, x1) = 0, [U51](x0, x1) = 1, [U41](x0) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U23](x0) = 1, [U22](x0, x1) = 1, [U21](x0, x1) = 1, [U16](x0) = 1, [U15](x0, x1) = 0, [isNat](x0) = 1, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 1 orientation: mark#(0()) = 1 >= 1 = active#(0()) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 1 >= 1 = mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) = 1 >= 1 = active#(tt()) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNatKind(V1),V1)) mark#(U15(X1,X2)) = 1 >= 0 = active#(U15(mark(X1),X2)) active#(U22(tt(),V1)) = 1 >= 1 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U23(tt())) = 1 >= 1 = mark#(tt()) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 1 >= 1 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(U22(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U22(X1,X2)) = 1 >= 1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 1 >= 1 = mark#(tt()) mark#(U23(X)) = 1 >= 1 = mark#(X) mark#(U23(X)) = 1 >= 1 = active#(U23(mark(X))) active#(U51(tt(),N)) = 1 >= 1 = mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 1 >= 1 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = 1 >= 1 = mark#(X) mark#(U41(X)) = 1 >= 1 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 1 >= 1 = mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) = 1 >= 1 = active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) = 1 >= 1 = mark#(U64(isNatKind(N),M,N)) mark#(U52(X1,X2)) = 1 >= 0 = active#(U52(mark(X1),X2)) active#(U64(tt(),M,N)) = 1 >= 1 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2,X3)) = 1 >= 1 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 1 >= 1 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 1 >= 1 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(s(X)) = 1 >= 1 = mark#(X) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = 1 >= 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = 1 >= 1 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,mark(X2)) = 1 >= 1 = U22(X1,X2) U22(active(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,active(X2)) = 1 >= 1 = U22(X1,X2) U23(mark(X)) = 1 >= 1 = U23(X) U23(active(X)) = 1 >= 1 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 1 >= 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 1 >= 1 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) problem: DPs: mark#(0()) -> active#(0()) active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) -> active#(tt()) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 1, [U64](x0, x1, x2) = 1, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 1, [U52](x0, x1) = 1, [U51](x0, x1) = 1, [U41](x0) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U23](x0) = 1, [U22](x0, x1) = 1, [U21](x0, x1) = 1, [U16](x0) = 1, [U15](x0, x1) = 1, [isNat](x0) = 1, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 1 orientation: mark#(0()) = 1 >= 0 = active#(0()) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 1 >= 1 = mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) = 1 >= 1 = active#(tt()) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 1 >= 1 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U23(tt())) = 1 >= 1 = mark#(tt()) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 1 >= 1 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(U22(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U22(X1,X2)) = 1 >= 1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 1 >= 1 = mark#(tt()) mark#(U23(X)) = 1 >= 1 = mark#(X) mark#(U23(X)) = 1 >= 1 = active#(U23(mark(X))) active#(U51(tt(),N)) = 1 >= 1 = mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 1 >= 1 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = 1 >= 1 = mark#(X) mark#(U41(X)) = 1 >= 1 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 1 >= 1 = mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) = 1 >= 1 = active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) = 1 >= 1 = mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) = 1 >= 1 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2,X3)) = 1 >= 1 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 1 >= 1 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 1 >= 1 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(s(X)) = 1 >= 1 = mark#(X) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = 1 >= 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = 1 >= 1 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,mark(X2)) = 1 >= 1 = U22(X1,X2) U22(active(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,active(X2)) = 1 >= 1 = U22(X1,X2) U23(mark(X)) = 1 >= 1 = U23(X) U23(active(X)) = 1 >= 1 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 1 >= 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 1 >= 1 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) -> active#(tt()) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= X1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) active#(U51(tt(),N)) = N >= N = mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) = X2 >= X2 = active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(s(X)) = X + 1 >= X = mark#(X) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) -> active#(tt()) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 1, [U64](x0, x1, x2) = 1, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 1, [U52](x0, x1) = 1, [U51](x0, x1) = 1, [U41](x0) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U23](x0) = 1, [U22](x0, x1) = 1, [U21](x0, x1) = 1, [U16](x0) = 1, [U15](x0, x1) = 1, [isNat](x0) = 1, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 1, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 1, [active](x0) = x0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 1 >= 1 = mark#(U13(isNatKind(V2),V1,V2)) mark#(tt()) = 1 >= 0 = active#(tt()) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 1 >= 1 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U23(tt())) = 1 >= 1 = mark#(tt()) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 1 >= 1 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(U22(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U22(X1,X2)) = 1 >= 1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 1 >= 1 = mark#(tt()) mark#(U23(X)) = 1 >= 1 = mark#(X) mark#(U23(X)) = 1 >= 1 = active#(U23(mark(X))) active#(U51(tt(),N)) = 1 >= 1 = mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 1 >= 1 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = 1 >= 1 = mark#(X) mark#(U41(X)) = 1 >= 1 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 1 >= 1 = mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) = 1 >= 1 = active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) = 1 >= 1 = mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) = 1 >= 1 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2,X3)) = 1 >= 1 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 1 >= 1 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 1 >= 1 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = 1 >= 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = 1 >= 1 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 1 >= 1 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 1 >= 1 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 1 >= 1 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 1 >= 1 = mark(U16(isNat(V2))) active(U16(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 1 >= 1 = mark(U23(isNat(V1))) active(U23(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V2)) = 1 >= 1 = mark(U32(isNatKind(V2))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),N)) = 1 >= 1 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 1 >= 1 = mark(N) active(U61(tt(),M,N)) = 1 >= 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 1 >= 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 1 >= 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 1 >= 1 = mark(s(plus(N,M))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 1 >= 1 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 1 >= 1 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 1 >= 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 1 >= 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 1 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 1 >= 1 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 1 >= 1 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 1 >= 1 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 1 >= 1 = active(U15(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U16(X)) = 1 >= 1 = active(U16(mark(X))) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 1 >= 1 = active(U22(mark(X1),X2)) mark(U23(X)) = 1 >= 1 = active(U23(mark(X))) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X)) = 1 >= 1 = active(U41(mark(X))) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 1 >= 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 1 >= 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 1 >= 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 1 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 1 >= 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,mark(X2)) = 1 >= 1 = U22(X1,X2) U22(active(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,active(X2)) = 1 >= 1 = U22(X1,X2) U23(mark(X)) = 1 >= 1 = U23(X) U23(active(X)) = 1 >= 1 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 1 >= 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 1 >= 1 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) active#(U51(tt(),N)) -> mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 1, [s](x0) = 0, [plus](x0, x1) = x0 + x1, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = x1, [U51](x0, x1) = x1 + 1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= X1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) active#(U51(tt(),N)) = N + 1 >= N = mark#(U52(isNatKind(N),N)) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 0 >= 0 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 0 >= 0 = mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) = 0 >= 0 = mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) = 0 >= 0 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = 0 >= 0 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = 0 >= 0 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 0 >= 0 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 0 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = N >= 0 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N + 1 >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N + 1 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 + 1 >= X2 + 1 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(active(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,active(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [0] = 1, [s](x0) = 0, [plus](x0, x1) = x0 + x1, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = x1, [U51](x0, x1) = x1 + 1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= 0 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= 0 = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 0 >= 0 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= 0 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 0 >= 0 = mark#(U63(isNat(N),M,N)) mark#(U51(X1,X2)) = X2 + 1 >= 0 = active#(U51(mark(X1),X2)) active#(U63(tt(),M,N)) = 0 >= 0 = mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) = 0 >= 0 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = 0 >= 0 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = 0 >= 0 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 0 >= 0 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 0 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = X1 + X2 >= 0 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = 0 >= 0 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N + 1 >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N + 1 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 + 1 >= X2 + 1 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(active(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,active(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = x1, [U51](x0, x1) = x1 + 1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= 0 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= 0 = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 0 >= 0 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= 0 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 0 >= 0 = mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) = 0 >= 0 = mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) = 0 >= 0 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = 0 >= 0 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = 0 >= 0 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 0 >= 0 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 0 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) mark#(plus(X1,X2)) = X1 + 1 >= 0 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = 0 >= 0 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N + 1 >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N + 1 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N + 1 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 + 1 >= X2 + 1 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + 1 >= X1 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(active(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,active(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) -> mark#(U61(isNat(M),M,N)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 0, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 1, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = 0 >= 0 = mark#(X) mark#(U16(X)) = 0 >= 0 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U21(X1,X2)) = 0 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U22(X1,X2)) = 0 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = 0 >= 0 = mark#(X) mark#(U23(X)) = 0 >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U31(X1,X2)) = 0 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 0 >= 0 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = 0 >= 0 = mark#(X) mark#(U41(X)) = 0 >= 0 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 0 >= 0 = mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) = 0 >= 0 = mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) = 0 >= 0 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = 0 >= 0 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = 0 >= 0 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 0 >= 0 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 0 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active#(plus(N,s(M))) = 1 >= 0 = mark#(U61(isNat(M),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 0 >= 0 = U16(X) U16(active(X)) = 0 >= 0 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 0 >= 0 = U41(X) U41(active(X)) = 0 >= 0 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [U64](x0, x1, x2) = x0, [U63](x0, x1, x2) = x0, [U62](x0, x1, x2) = x0, [U61](x0, x1, x2) = x0 + 1, [U52](x0, x1) = x1 + 1, [U51](x0, x1) = x1 + 1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= 0 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= 0 = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 0 >= 0 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= 0 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 0 >= 0 = mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) = 0 >= 0 = mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) = 0 >= 0 = mark#(s(plus(N,M))) mark#(U61(X1,X2,X3)) = X1 + 1 >= 0 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = X1 >= 0 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = X1 >= 0 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = X1 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N + 1 >= N + 1 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N + 1 >= N = mark(N) active(U61(tt(),M,N)) = 1 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N + 1 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N + 1 >= 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 + 1 >= X2 + 1 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 + 1 >= X2 + 1 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X1 + 1 >= X1 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X1 >= X1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X1 >= X1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X1 >= X1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + 1 >= X1 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(active(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,active(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U52(mark(X1),X2) = X2 + 1 >= X2 + 1 = U52(X1,X2) U52(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U52(X1,X2) U52(active(X1),X2) = X2 + 1 >= X2 + 1 = U52(X1,X2) U52(X1,active(X2)) = X2 + 1 >= X2 + 1 = U52(X1,X2) U61(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 + 1 >= X1 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 + 1 >= X1 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 + 1 >= X1 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 + 1 >= X1 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 + 1 >= X1 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X1 >= X1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X1 >= X1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X1 >= X1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X1 >= X1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X1 >= X1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X1 >= X1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X1 >= X1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X1 >= X1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X1 >= X1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X1 >= X1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X1 >= X1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X1 >= X1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X1 >= X1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X1 >= X1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X1 >= X1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X1 >= X1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X1 >= X1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X1 >= X1 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) -> mark#(s(plus(N,M))) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [U64](x0, x1, x2) = 1, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= X1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 1 >= 1 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 1 >= 1 = mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) = 1 >= 1 = mark#(U64(isNatKind(N),M,N)) active#(U64(tt(),M,N)) = 1 >= 0 = mark#(s(plus(N,M))) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = 1 >= 1 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 1 >= 1 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 1 >= 1 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 1 >= 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 1 >= 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 1 >= 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 1 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N + 1 >= 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 1 >= 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 1 >= 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 1 >= 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + 1 >= X1 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 1 >= 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 1 >= 1 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U61(tt(),M,N)) -> mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 0, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 1, [U52](x0, x1) = 1, [U51](x0, x1) = 0, [U41](x0) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = 0 >= 0 = mark#(X) mark#(U16(X)) = 0 >= 0 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U21(X1,X2)) = 0 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U22(X1,X2)) = 0 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = 0 >= 0 = mark#(X) mark#(U23(X)) = 0 >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U31(X1,X2)) = 0 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) active#(U61(tt(),M,N)) = 1 >= 0 = mark#(U62(isNatKind(M),M,N)) mark#(U41(X)) = 0 >= 0 = mark#(X) mark#(U41(X)) = 0 >= 0 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 0 >= 0 = mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) = 0 >= 0 = mark#(U64(isNatKind(N),M,N)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = 0 >= 0 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 0 >= 0 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 0 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 0 >= 0 = U16(X) U16(active(X)) = 0 >= 0 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 0 >= 0 = U41(X) U41(active(X)) = 0 >= 0 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2,X3)) -> active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= 0 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= 0 = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= 0 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 0 >= 0 = mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) = 0 >= 0 = mark#(U64(isNatKind(N),M,N)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2,X3)) = 1 >= 0 = active#(U62(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 0 >= 0 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 0 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 1 >= 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 1 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N + 1 >= 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 1 >= 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + 1 >= X1 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U62(tt(),M,N)) -> mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(isNat(0())) -> mark#(tt()) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 0, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 0, [U15](x0, x1) = 1, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = 0 >= 0 = mark#(X) mark#(U16(X)) = 0 >= 0 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U21(X1,X2)) = 0 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U22(X1,X2)) = 0 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = 0 >= 0 = mark#(X) mark#(U23(X)) = 0 >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U31(X1,X2)) = 0 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) mark#(U41(X)) = 0 >= 0 = mark#(X) mark#(U41(X)) = 0 >= 0 = active#(U41(mark(X))) active#(U62(tt(),M,N)) = 1 >= 0 = mark#(U63(isNat(N),M,N)) active#(U63(tt(),M,N)) = 0 >= 0 = mark#(U64(isNatKind(N),M,N)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 0 >= 0 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 0 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 0 >= 0 = U16(X) U16(active(X)) = 0 >= 0 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 0 >= 0 = U41(X) U41(active(X)) = 0 >= 0 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) active#(isNat(0())) -> mark#(tt()) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) -> active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= 0 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= 0 = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= 0 = active#(U41(mark(X))) active#(U63(tt(),M,N)) = 0 >= 0 = mark#(U64(isNatKind(N),M,N)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) mark#(U63(X1,X2,X3)) = 1 >= 0 = active#(U63(mark(X1),X2,X3)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 0 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 1 >= 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 1 >= 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 1 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N + 1 >= 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 1 >= 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 1 >= 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + 1 >= X1 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(U63(tt(),M,N)) -> mark#(U64(isNatKind(N),M,N)) 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)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 0, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 1, [U41](x0) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 0, [U15](x0, x1) = 1, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = 0 >= 0 = mark#(X) mark#(U16(X)) = 0 >= 0 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U21(X1,X2)) = 0 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U22(X1,X2)) = 0 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = 0 >= 0 = mark#(X) mark#(U23(X)) = 0 >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U31(X1,X2)) = 0 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) mark#(U41(X)) = 0 >= 0 = mark#(X) mark#(U41(X)) = 0 >= 0 = active#(U41(mark(X))) active#(U63(tt(),M,N)) = 1 >= 0 = mark#(U64(isNatKind(N),M,N)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 0 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 0 >= 0 = U16(X) U16(active(X)) = 0 >= 0 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 0 >= 0 = U41(X) U41(active(X)) = 0 >= 0 = U41(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) 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)) mark#(U64(X1,X2,X3)) -> active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [U64](x0, x1, x2) = 1, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 0 >= 0 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 0 >= 0 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= 0 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= 0 = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= 0 = active#(U41(mark(X))) active#(isNat(0())) = 0 >= 0 = mark#(tt()) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U64(X1,X2,X3)) = 1 >= 0 = active#(U64(mark(X1),X2,X3)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 1 >= 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 1 >= 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 1 >= 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 1 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N + 1 >= 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 >= X1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 1 >= 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 1 >= 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 1 >= 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + 1 >= X1 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,mark(X2)) = X1 >= X1 = U22(X1,X2) U22(active(X1),X2) = X1 >= X1 = U22(X1,X2) U22(X1,active(X2)) = X1 >= X1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 1 >= 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 1 >= 1 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) 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#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = x0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x0 + 1, [U21](x0, x1) = x0 + 1, [U16](x0) = x0, [U15](x0, x1) = 1, [isNat](x0) = 1, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = x0, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 1 >= 1 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = 1 >= 1 = mark#(U23(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(isNat(0())) = 1 >= 0 = mark#(tt()) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(U41(isNatKind(V1))) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 1 >= 1 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 1 >= 1 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 1 >= 1 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 1 >= 1 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 1 >= 1 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 1 >= 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 1 >= 1 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 1 >= 1 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 1 >= 1 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 1 >= 1 = active(U15(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 + 1 >= X1 + 1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X1 + 1 >= X1 + 1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 >= X1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = U21(X1,X2) U21(X1,mark(X2)) = X1 + 1 >= X1 + 1 = U21(X1,X2) U21(active(X1),X2) = X1 + 1 >= X1 + 1 = U21(X1,X2) U21(X1,active(X2)) = X1 + 1 >= X1 + 1 = U21(X1,X2) U22(mark(X1),X2) = X1 + 1 >= X1 + 1 = U22(X1,X2) U22(X1,mark(X2)) = X1 + 1 >= X1 + 1 = U22(X1,X2) U22(active(X1),X2) = X1 + 1 >= X1 + 1 = U22(X1,X2) U22(X1,active(X2)) = X1 + 1 >= X1 + 1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 >= X1 = plus(X1,X2) plus(X1,mark(X2)) = X1 >= X1 = plus(X1,X2) plus(active(X1),X2) = X1 >= X1 = plus(X1,X2) plus(X1,active(X2)) = X1 >= X1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) -> mark#(U23(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) -> mark#(U41(isNatKind(V1))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U23](x0) = x0, [U22](x0, x1) = x1 + 1, [U21](x0, x1) = x1 + 1, [U16](x0) = x0, [U15](x0, x1) = x1, [isNat](x0) = x0, [U14](x0, x1, x2) = x2, [U13](x0, x1, x2) = x2, [mark](x0) = x0, [U12](x0, x1, x2) = x2, [isNatKind](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = V2 >= V2 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X3 >= X3 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = V2 >= V2 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = V2 >= V2 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = X3 >= X3 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = V2 >= V2 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = X >= X = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = X3 >= X3 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X3 >= X3 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = V1 + 1 >= V1 + 1 = mark#(U22(isNatKind(V1),V1)) active#(U22(tt(),V1)) = V1 + 1 >= V1 = mark#(U23(isNat(V1))) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = V2 >= V2 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(U22(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(isNat(plus(V1,V2))) = V1 + V2 >= V2 = mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) = V1 + 1 >= V1 + 1 = mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U31(isNatKind(V1),V2)) active#(isNatKind(s(V1))) = V1 + 1 >= V1 = mark#(U41(isNatKind(V1))) active(U11(tt(),V1,V2)) = V2 >= V2 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = V2 >= V2 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = V2 >= V2 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = V2 >= V2 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = V2 >= V2 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= V1 + 1 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = V1 + 1 >= V1 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = V2 >= V2 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 >= V2 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= V1 + 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = V1 + 1 >= V1 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X3 >= X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X3 >= X3 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X3 >= X3 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X3 >= X3 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X2 >= X2 = active(U15(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X2 + 1 >= X2 + 1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X2 + 1 >= X2 + 1 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X3 >= X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X3 >= X3 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X3 >= X3 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X3 >= X3 = U12(X1,X2,X3) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) U13(mark(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X3 >= X3 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X3 >= X3 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X3 >= X3 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X3 >= X3 = U14(X1,X2,X3) U15(mark(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,mark(X2)) = X2 >= X2 = U15(X1,X2) U15(active(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,active(X2)) = X2 >= X2 = U15(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(active(X1),X2) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(X1,active(X2)) = X2 + 1 >= X2 + 1 = U21(X1,X2) U22(mark(X1),X2) = X2 + 1 >= X2 + 1 = U22(X1,X2) U22(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U22(X1,X2) U22(active(X1),X2) = X2 + 1 >= X2 + 1 = U22(X1,X2) U22(X1,active(X2)) = X2 + 1 >= X2 + 1 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(active(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,active(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2)) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 1, [U64](x0, x1, x2) = 1, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 1, [U51](x0, x1) = 0, [U41](x0) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U23](x0) = 1, [U22](x0, x1) = 0, [U21](x0, x1) = 1, [U16](x0) = 1, [U15](x0, x1) = 1, [isNat](x0) = 1, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 1 >= 1 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNatKind(V1),V1)) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U23(tt())) = 1 >= 1 = mark#(tt()) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 1 >= 1 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(U22(X1,X2)) = 1 >= 0 = active#(U22(mark(X1),X2)) active#(U41(tt())) = 1 >= 1 = mark#(tt()) mark#(U23(X)) = 1 >= 1 = mark#(X) mark#(U23(X)) = 1 >= 1 = active#(U23(mark(X))) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) mark#(U41(X)) = 1 >= 1 = mark#(X) mark#(U41(X)) = 1 >= 1 = active#(U41(mark(X))) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(U31(isNatKind(V1),V2)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 1 >= 1 = U23(X) U23(active(X)) = 1 >= 1 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 1 >= 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 1 >= 1 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1 + 1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x1, [U21](x0, x1) = x1, [U16](x0) = x0, [U15](x0, x1) = x1, [isNat](x0) = x0, [U14](x0, x1, x2) = x2, [U13](x0, x1, x2) = x2, [mark](x0) = x0, [U12](x0, x1, x2) = x2, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x1 + x2, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = V1 + V2 >= V2 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X2 + X3 >= X2 + X3 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = V2 >= V2 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = V2 >= V2 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = X3 >= X3 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = V2 >= V2 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = X3 >= X3 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X3 >= X3 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNatKind(V1),V1)) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X2 >= X2 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U11(isNatKind(V1),V1,V2)) active#(isNat(s(V1))) = V1 + 1 >= V1 = mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active(U11(tt(),V1,V2)) = V1 + V2 >= V2 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = V2 >= V2 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = V2 >= V2 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = V2 >= V2 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = V2 >= V2 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = V1 >= V1 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N + 1 >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N + 1 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X2 + X3 >= X2 + X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X3 >= X3 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X3 >= X3 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X3 >= X3 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X2 >= X2 = active(U15(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X2 >= X2 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X2 >= X2 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 + 1 >= X2 + 1 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) U11(mark(X1),X2,X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X3 >= X3 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X3 >= X3 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X3 >= X3 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X3 >= X3 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X3 >= X3 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X3 >= X3 = U14(X1,X2,X3) U15(mark(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,mark(X2)) = X2 >= X2 = U15(X1,X2) U15(active(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,active(X2)) = X2 >= X2 = U15(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X2 >= X2 = U21(X1,X2) U21(X1,mark(X2)) = X2 >= X2 = U21(X1,X2) U21(active(X1),X2) = X2 >= X2 = U21(X1,X2) U21(X1,active(X2)) = X2 >= X2 = U21(X1,X2) U22(mark(X1),X2) = X2 >= X2 = U22(X1,X2) U22(X1,mark(X2)) = X2 >= X2 = U22(X1,X2) U22(active(X1),X2) = X2 >= X2 = U22(X1,X2) U22(X1,active(X2)) = X2 >= X2 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(active(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,active(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNatKind(V1),V1,V2)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1 + 1, [U51](x0, x1) = x1 + 1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x1, [U21](x0, x1) = x1, [U16](x0) = x0, [U15](x0, x1) = x1, [isNat](x0) = x0, [U14](x0, x1, x2) = x2, [U13](x0, x1, x2) = x2, [mark](x0) = x0, [U12](x0, x1, x2) = x2, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = V2 >= V2 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X3 >= X3 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = V2 >= V2 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = V2 >= V2 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = X3 >= X3 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = V2 >= V2 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = X3 >= X3 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X3 >= X3 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNatKind(V1),V1)) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X2 >= X2 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V2 = mark#(U11(isNatKind(V1),V1,V2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(U31(isNatKind(V1),V2)) active(U11(tt(),V1,V2)) = V2 >= V2 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = V2 >= V2 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = V2 >= V2 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = V2 >= V2 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = V2 >= V2 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = V1 >= V1 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N + 1 >= N + 1 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N + 1 >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V2 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N + 1 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X3 >= X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X3 >= X3 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X3 >= X3 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X3 >= X3 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X2 >= X2 = active(U15(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X2 >= X2 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X2 >= X2 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 + 1 >= X2 + 1 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 + 1 >= X2 + 1 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X3 >= X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X3 >= X3 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X3 >= X3 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X3 >= X3 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X3 >= X3 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X3 >= X3 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X3 >= X3 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X3 >= X3 = U14(X1,X2,X3) U15(mark(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,mark(X2)) = X2 >= X2 = U15(X1,X2) U15(active(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,active(X2)) = X2 >= X2 = U15(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X2 >= X2 = U21(X1,X2) U21(X1,mark(X2)) = X2 >= X2 = U21(X1,X2) U21(active(X1),X2) = X2 >= X2 = U21(X1,X2) U21(X1,active(X2)) = X2 >= X2 = U21(X1,X2) U22(mark(X1),X2) = X2 >= X2 = U22(X1,X2) U22(X1,mark(X2)) = X2 >= X2 = U22(X1,X2) U22(active(X1),X2) = X2 >= X2 = U22(X1,X2) U22(X1,active(X2)) = X2 >= X2 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(active(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,active(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U52(mark(X1),X2) = X2 + 1 >= X2 + 1 = U52(X1,X2) U52(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U52(X1,X2) U52(active(X1),X2) = X2 + 1 >= X2 + 1 = U52(X1,X2) U52(X1,active(X2)) = X2 + 1 >= X2 + 1 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) mark#(isNat(X)) -> active#(isNat(X)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 1, [U41](x0) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U23](x0) = 1, [U22](x0, x1) = 0, [U21](x0, x1) = 1, [U16](x0) = 1, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 1 >= 1 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNatKind(V1),V1)) mark#(isNat(X)) = 1 >= 0 = active#(isNat(X)) active#(U23(tt())) = 1 >= 1 = mark#(tt()) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 1 >= 1 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 1 >= 1 = mark#(tt()) active#(U41(tt())) = 1 >= 1 = mark#(tt()) mark#(U23(X)) = 1 >= 1 = mark#(X) mark#(U23(X)) = 1 >= 1 = active#(U23(mark(X))) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) mark#(U41(X)) = 1 >= 1 = mark#(X) mark#(U41(X)) = 1 >= 1 = active#(U41(mark(X))) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(U31(isNatKind(V1),V2)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 1 >= 1 = U23(X) U23(active(X)) = 1 >= 1 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> mark#(U31(isNatKind(V1),V2)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U23](x0) = x0, [U22](x0, x1) = x1, [U21](x0, x1) = x1, [U16](x0) = x0, [U15](x0, x1) = x1 + 1, [isNat](x0) = x0, [U14](x0, x1, x2) = x2 + 1, [U13](x0, x1, x2) = x2 + 1, [mark](x0) = x0, [U12](x0, x1, x2) = x2 + 1, [isNatKind](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2 + 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = X >= X = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNatKind(V1),V1)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = V2 >= V2 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X2 >= X2 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active#(isNatKind(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 = mark#(U31(isNatKind(V1),V2)) active(U11(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = V2 + 1 >= V2 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = V1 >= V1 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = V2 >= V2 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V2 + 1 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = V1 >= V1 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X2 + 1 >= X2 + 1 = active(U15(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X2 >= X2 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X2 >= X2 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) U13(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U15(mark(X1),X2) = X2 + 1 >= X2 + 1 = U15(X1,X2) U15(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U15(X1,X2) U15(active(X1),X2) = X2 + 1 >= X2 + 1 = U15(X1,X2) U15(X1,active(X2)) = X2 + 1 >= X2 + 1 = U15(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X2 >= X2 = U21(X1,X2) U21(X1,mark(X2)) = X2 >= X2 = U21(X1,X2) U21(active(X1),X2) = X2 >= X2 = U21(X1,X2) U21(X1,active(X2)) = X2 >= X2 = U21(X1,X2) U22(mark(X1),X2) = X2 >= X2 = U22(X1,X2) U22(X1,mark(X2)) = X2 >= X2 = U22(X1,X2) U22(active(X1),X2) = X2 >= X2 = U22(X1,X2) U22(X1,active(X2)) = X2 >= X2 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(active(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,active(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNatKind(0())) -> mark#(tt()) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1 + 1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0, [U22](x0, x1) = x1, [U21](x0, x1) = x1 + 1, [U16](x0) = x0, [U15](x0, x1) = x1, [isNat](x0) = x0, [U14](x0, x1, x2) = x2, [U13](x0, x1, x2) = x2, [mark](x0) = x0, [U12](x0, x1, x2) = x2, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x1 + x2, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = V1 + V2 >= V2 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X2 + X3 >= X2 + X3 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = V2 >= V2 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = V2 >= V2 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = X3 >= X3 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = V2 >= V2 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = X3 >= X3 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X3 >= X3 = active#(U14(mark(X1),X2,X3)) active#(U21(tt(),V1)) = V1 + 1 >= V1 = mark#(U22(isNatKind(V1),V1)) active#(U23(tt())) = 0 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 0 >= 0 = mark#(tt()) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X >= X = mark#(X) mark#(U23(X)) = X >= X = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active(U11(tt(),V1,V2)) = V1 + V2 >= V2 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = V2 >= V2 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = V2 >= V2 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = V2 >= V2 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = V2 >= V2 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= V1 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = V1 >= V1 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N + 1 >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= V1 + 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N + 1 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X2 + X3 >= X2 + X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X3 >= X3 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X3 >= X3 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X3 >= X3 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X2 >= X2 = active(U15(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X2 + 1 >= X2 + 1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X2 >= X2 = active(U22(mark(X1),X2)) mark(U23(X)) = X >= X = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 + 1 >= X2 + 1 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) U11(mark(X1),X2,X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X3 >= X3 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X3 >= X3 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X3 >= X3 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X3 >= X3 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X3 >= X3 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X3 >= X3 = U14(X1,X2,X3) U15(mark(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,mark(X2)) = X2 >= X2 = U15(X1,X2) U15(active(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,active(X2)) = X2 >= X2 = U15(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(active(X1),X2) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(X1,active(X2)) = X2 + 1 >= X2 + 1 = U21(X1,X2) U22(mark(X1),X2) = X2 >= X2 = U22(X1,X2) U22(X1,mark(X2)) = X2 >= X2 = U22(X1,X2) U22(active(X1),X2) = X2 >= X2 = U22(X1,X2) U22(X1,active(X2)) = X2 >= X2 = U22(X1,X2) U23(mark(X)) = X >= X = U23(X) U23(active(X)) = X >= X = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(active(X1),X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) U51(X1,active(X2)) = X2 + 1 >= X2 + 1 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNatKind(0())) -> mark#(tt()) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 1, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U23](x0) = 1, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 1, [U15](x0, x1) = 1, [isNat](x0) = 0, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 1 >= 1 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) active#(U23(tt())) = 1 >= 1 = mark#(tt()) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 1 >= 1 = mark#(U32(isNatKind(V2))) mark#(U21(X1,X2)) = 1 >= 0 = active#(U21(mark(X1),X2)) active#(U32(tt())) = 1 >= 1 = mark#(tt()) active#(U41(tt())) = 1 >= 1 = mark#(tt()) mark#(U23(X)) = 1 >= 1 = mark#(X) mark#(U23(X)) = 1 >= 1 = active#(U23(mark(X))) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) mark#(U41(X)) = 1 >= 1 = mark#(X) mark#(U41(X)) = 1 >= 1 = active#(U41(mark(X))) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 1 >= 1 = U23(X) U23(active(X)) = 1 >= 1 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 1 >= 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 1 >= 1 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) active#(U23(tt())) -> mark#(tt()) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNatKind(0())) -> mark#(tt()) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U23](x0) = x0 + 1, [U22](x0, x1) = x1 + 1, [U21](x0, x1) = x1 + 1, [U16](x0) = x0, [U15](x0, x1) = x1, [isNat](x0) = x0, [U14](x0, x1, x2) = x2, [U13](x0, x1, x2) = x2, [mark](x0) = x0, [U12](x0, x1, x2) = x2, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x1 + x2, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = V1 + V2 >= V2 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X2 + X3 >= X2 + X3 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = V2 >= V2 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = V2 >= V2 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = X3 >= X3 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = V2 >= V2 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = X3 >= X3 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X3 >= X3 = active#(U14(mark(X1),X2,X3)) active#(U23(tt())) = 1 >= 0 = mark#(tt()) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = 0 >= 0 = mark#(U32(isNatKind(V2))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U23(X)) = X + 1 >= X = mark#(X) mark#(U23(X)) = X + 1 >= X + 1 = active#(U23(mark(X))) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active(U11(tt(),V1,V2)) = V1 + V2 >= V2 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = V2 >= V2 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = V2 >= V2 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = V2 >= V2 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = V2 >= V2 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= V1 + 1 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = V1 + 1 >= V1 + 1 = mark(U23(isNat(V1))) active(U23(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= V1 + 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X2 + X3 >= X2 + X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X3 >= X3 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X3 >= X3 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X3 >= X3 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X2 >= X2 = active(U15(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X2 + 1 >= X2 + 1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = X2 + 1 >= X2 + 1 = active(U22(mark(X1),X2)) mark(U23(X)) = X + 1 >= X + 1 = active(U23(mark(X))) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X3 >= X3 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X3 >= X3 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X3 >= X3 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X3 >= X3 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X3 >= X3 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X3 >= X3 = U14(X1,X2,X3) U15(mark(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,mark(X2)) = X2 >= X2 = U15(X1,X2) U15(active(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,active(X2)) = X2 >= X2 = U15(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(active(X1),X2) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(X1,active(X2)) = X2 + 1 >= X2 + 1 = U21(X1,X2) U22(mark(X1),X2) = X2 + 1 >= X2 + 1 = U22(X1,X2) U22(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U22(X1,X2) U22(active(X1),X2) = X2 + 1 >= X2 + 1 = U22(X1,X2) U22(X1,active(X2)) = X2 + 1 >= X2 + 1 = U22(X1,X2) U23(mark(X)) = X + 1 >= X + 1 = U23(X) U23(active(X)) = X + 1 >= X + 1 = U23(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U23(X)) -> active#(U23(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNatKind(0())) -> mark#(tt()) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 1, [U15](x0, x1) = 1, [isNat](x0) = 0, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = 1 >= 1 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active#(U31(tt(),V2)) = 1 >= 1 = mark#(U32(isNatKind(V2))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) active#(U41(tt())) = 1 >= 1 = mark#(tt()) mark#(U23(X)) = 1 >= 0 = active#(U23(mark(X))) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) mark#(U41(X)) = 1 >= 1 = mark#(X) mark#(U41(X)) = 1 >= 1 = active#(U41(mark(X))) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) -> mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U31(tt(),V2)) -> mark#(U32(isNatKind(V2))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNatKind(0())) -> mark#(tt()) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = x0 + x1 + 1, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = x0, [U16](x0) = x0, [U15](x0, x1) = x1, [isNat](x0) = x0, [U14](x0, x1, x2) = x2, [U13](x0, x1, x2) = x2, [mark](x0) = x0, [U12](x0, x1, x2) = x1 + x2 + 1, [isNatKind](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x1 + x2 + 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V1,V2)) = V1 + V2 + 1 >= V2 = mark#(U13(isNatKind(V2),V1,V2)) active#(U13(tt(),V1,V2)) = V2 >= V2 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = V2 >= V2 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = X >= X = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = X3 >= X3 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X3 >= X3 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U31(tt(),V2)) = V2 + 1 >= V2 = mark#(U32(isNatKind(V2))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U31(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) active(U11(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = V1 + V2 + 1 >= V2 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = V2 >= V2 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = V2 >= V2 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = V2 >= V2 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = V2 + 1 >= V2 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = V1 >= V1 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X3 >= X3 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X3 >= X3 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X2 >= X2 = active(U15(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U12(X1,X2,X3) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) U13(mark(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X3 >= X3 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X3 >= X3 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X3 >= X3 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X3 >= X3 = U14(X1,X2,X3) U15(mark(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,mark(X2)) = X2 >= X2 = U15(X1,X2) U15(active(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,active(X2)) = X2 >= X2 = U15(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U31(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U31(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U31(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) active#(isNatKind(0())) -> mark#(tt()) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = x0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0, [U31](x0, x1) = 1, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 1, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 0 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active#(isNatKind(0())) = 1 >= 0 = mark#(tt()) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 1 >= 1 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 1 >= 1 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 >= X1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 >= X1 = plus(X1,X2) plus(X1,mark(X2)) = X1 >= X1 = plus(X1,X2) plus(active(X1),X2) = X1 >= X1 = plus(X1,X2) plus(X1,active(X2)) = X1 >= X1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 1, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 1, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 1, [U32](x0) = 1, [U31](x0, x1) = 0, [U23](x0) = 1, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 1, [U15](x0, x1) = 1, [isNat](x0) = 0, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 1 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) active#(U41(tt())) = 1 >= 1 = mark#(tt()) mark#(U31(X1,X2)) = 1 >= 0 = active#(U31(mark(X1),X2)) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) mark#(U41(X)) = 1 >= 1 = mark#(X) mark#(U41(X)) = 1 >= 1 = active#(U41(mark(X))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 1 >= 1 = U23(X) U23(active(X)) = 1 >= 1 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 1 >= 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 1 >= 1 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) -> active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 0, [s](x0) = 1, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 1, [U51](x0, x1) = 1, [U41](x0) = 1, [U32](x0) = 1, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 1, [U21](x0, x1) = 0, [U16](x0) = 1, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U12(X1,X2,X3)) = 1 >= 0 = active#(U12(mark(X1),X2,X3)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(isNatKind(X)) = 1 >= 0 = active#(isNatKind(X)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) active#(U41(tt())) = 1 >= 1 = mark#(tt()) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) mark#(U41(X)) = 1 >= 1 = mark#(X) mark#(U41(X)) = 1 >= 1 = active#(U41(mark(X))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,mark(X2)) = 1 >= 1 = U22(X1,X2) U22(active(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,active(X2)) = 1 >= 1 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0, [U32](x0) = x0 + 1, [U31](x0, x1) = x1 + 1, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= 0 = active#(U16(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U32(X)) = X + 1 >= X = mark#(X) mark#(U32(X)) = X + 1 >= 0 = active#(U32(mark(X))) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= 0 = active#(U41(mark(X))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = V2 + 1 >= V2 + 1 = mark(U32(isNatKind(V2))) active(U32(tt())) = 1 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 + 1 >= V2 + 1 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = V1 >= V1 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = X2 + 1 >= X2 + 1 = active(U31(mark(X1),X2)) mark(U32(X)) = X + 1 >= X + 1 = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = X2 + 1 >= X2 + 1 = U31(X1,X2) U31(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U31(X1,X2) U31(active(X1),X2) = X2 + 1 >= X2 + 1 = U31(X1,X2) U31(X1,active(X2)) = X2 + 1 >= X2 + 1 = U31(X1,X2) U32(mark(X)) = X + 1 >= X + 1 = U32(X) U32(active(X)) = X + 1 >= X + 1 = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt())) -> mark#(tt()) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 1, [s](x0) = 0, [plus](x0, x1) = x0 + x1 + 1, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 1, [U52](x0, x1) = x1, [U51](x0, x1) = x0 + x1 + 1, [U41](x0) = x0, [U32](x0) = 1, [U31](x0, x1) = 1, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 1, [U16](x0) = x0, [U15](x0, x1) = 1, [isNat](x0) = 1, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = x0, [U12](x0, x1, x2) = 1, [isNatKind](x0) = 1, [active](x0) = x0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active#(U32(tt())) = 1 >= 0 = mark#(tt()) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X)) = X >= X = mark#(X) mark#(U41(X)) = X >= X = active#(U41(mark(X))) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 1 >= 1 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 1 >= 1 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 1 >= 1 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 1 >= 1 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 1 >= 1 = mark(U32(isNatKind(V2))) active(U32(tt())) = 1 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N + 1 >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 1 >= 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 1 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 1 >= 1 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 2 >= N + 2 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N + 1 >= 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 1 >= 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 1 >= 1 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 1 >= 1 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 1 >= 1 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 1 >= 1 = active(U15(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X)) = X >= X = active(U41(mark(X))) mark(U51(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 1 >= 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 1 >= 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 1 >= 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 1 >= 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,mark(X2)) = 1 >= 1 = U15(X1,X2) U15(active(X1),X2) = 1 >= 1 = U15(X1,X2) U15(X1,active(X2)) = 1 >= 1 = U15(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X)) = X >= X = U41(X) U41(active(X)) = X >= X = U41(X) U51(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(X1,X2) U51(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(X1,X2) U51(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(X1,X2) U51(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U41(tt())) -> mark#(tt()) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> active#(U41(mark(X))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [0] = 1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = x0 + 1, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = x0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= 0 = active#(U16(mark(X))) active#(U41(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X)) = X + 1 >= X = mark#(X) mark#(U41(X)) = X + 1 >= 0 = active#(U41(mark(X))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 1 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = V1 + 1 >= V1 + 1 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = X + 1 >= X + 1 = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = X + 1 >= X + 1 = U41(X) U41(active(X)) = X + 1 >= X + 1 = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) active#(U41(tt())) -> mark#(tt()) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 0, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 1, [U32](x0) = 0, [U31](x0, x1) = 1, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = 0 >= 0 = mark#(X) mark#(U16(X)) = 0 >= 0 = active#(U16(mark(X))) active#(U41(tt())) = 1 >= 0 = mark#(tt()) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 0 >= 0 = U16(X) U16(active(X)) = 0 >= 0 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U11(tt(),V1,V2)) -> mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U64](x0, x1, x2) = x1 + x2 + 1, [U63](x0, x1, x2) = x1 + x2 + 1, [U62](x0, x1, x2) = x1 + x2 + 1, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = x1, [U16](x0) = x0, [U15](x0, x1) = x1, [isNat](x0) = x0, [U14](x0, x1, x2) = x2, [U13](x0, x1, x2) = x2, [mark](x0) = x0, [U12](x0, x1, x2) = x2, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x2 + 1, [tt] = 0 orientation: active#(U11(tt(),V1,V2)) = V2 + 1 >= V2 = mark#(U12(isNatKind(V1),V1,V2)) mark#(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) = V2 >= V2 = mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) = V2 >= V2 = mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) = X3 >= X3 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 0 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X3 >= X3 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = X >= X = mark#(X) mark#(U16(X)) = X >= X = active#(U16(mark(X))) active(U11(tt(),V1,V2)) = V2 + 1 >= V2 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = V2 >= V2 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = V2 >= V2 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = V2 >= V2 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = V2 >= V2 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V2 + 1 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X3 >= X3 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X3 >= X3 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X3 >= X3 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X2 >= X2 = active(U15(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U16(X)) = X >= X = active(U16(mark(X))) mark(U21(X1,X2)) = X2 >= X2 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X3 >= X3 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X3 >= X3 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X3 >= X3 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X3 >= X3 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X3 >= X3 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X3 >= X3 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X3 >= X3 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X3 >= X3 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X3 >= X3 = U14(X1,X2,X3) U15(mark(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,mark(X2)) = X2 >= X2 = U15(X1,X2) U15(active(X1),X2) = X2 >= X2 = U15(X1,X2) U15(X1,active(X2)) = X2 >= X2 = U15(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U16(mark(X)) = X >= X = U16(X) U16(active(X)) = X >= X = U16(X) U21(mark(X1),X2) = X2 >= X2 = U21(X1,X2) U21(X1,mark(X2)) = X2 >= X2 = U21(X1,X2) U21(active(X1),X2) = X2 >= X2 = U21(X1,X2) U21(X1,active(X2)) = X2 >= X2 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 0 >= 0 = U41(X) U41(active(X)) = 0 >= 0 = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 1, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 0, [U32](x0) = 0, [U31](x0, x1) = 1, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 1, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: mark#(U11(X1,X2,X3)) = 1 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) = 1 >= 1 = mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) = 1 >= 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 1 = mark#(tt()) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = 1 >= 1 = mark#(X) mark#(U16(X)) = 1 >= 1 = active#(U16(mark(X))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 0 >= 0 = U41(X) U41(active(X)) = 0 >= 0 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) problem: DPs: active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) active#(U16(tt())) -> mark#(tt()) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> active#(U16(mark(X))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = 1, [plus](x0, x1) = x0 + x1 + 1, [U64](x0, x1, x2) = 1, [U63](x0, x1, x2) = 1, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 1, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 1, [U22](x0, x1) = 1, [U21](x0, x1) = 1, [U16](x0) = x0 + 1, [U15](x0, x1) = x1 + 1, [isNat](x0) = x0, [U14](x0, x1, x2) = x2 + 1, [U13](x0, x1, x2) = x2 + 1, [mark](x0) = x0, [U12](x0, x1, x2) = x2 + 1, [isNatKind](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x2 + 1, [tt] = 0 orientation: active#(U13(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active#(U13(mark(X1),X2,X3)) active#(U16(tt())) = 1 >= 0 = mark#(tt()) mark#(U14(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = X + 1 >= X = mark#(X) mark#(U16(X)) = X + 1 >= X + 1 = active#(U16(mark(X))) active(U11(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = V2 + 1 >= V2 + 1 = mark(U16(isNat(V2))) active(U16(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 1 >= 1 = mark(U23(isNat(V1))) active(U23(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 1 >= 1 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 1 >= 1 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 1 >= 1 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 1 >= 1 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V2 + 1 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N + 1 >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N + 2 >= 1 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = X2 + 1 >= X2 + 1 = active(U15(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U16(X)) = X + 1 >= X + 1 = active(U16(mark(X))) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 1 >= 1 = active(U22(mark(X1),X2)) mark(U23(X)) = 1 >= 1 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 1 >= 1 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 1 >= 1 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 1 >= 1 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) U12(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) U12(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U14(X1,X2,X3) U15(mark(X1),X2) = X2 + 1 >= X2 + 1 = U15(X1,X2) U15(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U15(X1,X2) U15(active(X1),X2) = X2 + 1 >= X2 + 1 = U15(X1,X2) U15(X1,active(X2)) = X2 + 1 >= X2 + 1 = U15(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U16(mark(X)) = X + 1 >= X + 1 = U16(X) U16(active(X)) = X + 1 >= X + 1 = U16(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,mark(X2)) = 1 >= 1 = U22(X1,X2) U22(active(X1),X2) = 1 >= 1 = U22(X1,X2) U22(X1,active(X2)) = 1 >= 1 = U22(X1,X2) U23(mark(X)) = 1 >= 1 = U23(X) U23(active(X)) = 1 >= 1 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 0 >= 0 = U41(X) U41(active(X)) = 0 >= 0 = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 1 >= 1 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 1 >= 1 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 1 >= 1 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 1 >= 1 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 1 >= 1 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 1 >= 1 = U64(X1,X2,X3) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) mark#(U16(X)) -> active#(U16(mark(X))) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 1, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) = 0 >= 0 = active#(U13(mark(X1),X2,X3)) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) mark#(U16(X)) = 1 >= 0 = active#(U16(mark(X))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 0 >= 0 = U41(X) U41(active(X)) = 0 >= 0 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) -> active#(U13(mark(X1),X2,X3)) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 1, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 1, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 0, [U15](x0, x1) = 0, [isNat](x0) = 1, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U13(tt(),V1,V2)) = 0 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) = 0 >= 0 = mark#(U15(isNat(V1),V2)) mark#(U13(X1,X2,X3)) = 1 >= 0 = active#(U13(mark(X1),X2,X3)) mark#(U14(X1,X2,X3)) = 0 >= 0 = active#(U14(mark(X1),X2,X3)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U16(mark(X)) = 0 >= 0 = U16(X) U16(active(X)) = 0 >= 0 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 1 >= 1 = U23(X) U23(active(X)) = 1 >= 1 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) -> mark#(U15(isNat(V1),V2)) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 1, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 0, [U32](x0) = 0, [U31](x0, x1) = 1, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 1, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 1, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) active#(U14(tt(),V1,V2)) = 1 >= 0 = mark#(U15(isNat(V1),V2)) mark#(U14(X1,X2,X3)) = 1 >= 1 = active#(U14(mark(X1),X2,X3)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 1 >= 1 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 1 >= 1 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 1 >= 1 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 1 >= 1 = U16(X) U16(active(X)) = 1 >= 1 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 0 >= 0 = U41(X) U41(active(X)) = 0 >= 0 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 1 >= 1 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 1 >= 1 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 1 >= 1 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) mark#(U14(X1,X2,X3)) -> active#(U14(mark(X1),X2,X3)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [0] = 0, [s](x0) = 0, [plus](x0, x1) = 1, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = 0, [U51](x0, x1) = 0, [U41](x0) = 1, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 1, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(U13(tt(),V1,V2)) = 1 >= 1 = mark#(U14(isNatKind(V2),V1,V2)) mark#(U14(X1,X2,X3)) = 1 >= 0 = active#(U14(mark(X1),X2,X3)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),N)) = 0 >= 0 = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = 0 >= 0 = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 0 >= 0 = mark(U41(isNatKind(V1))) active(plus(N,0())) = 0 >= 0 = mark(U51(isNat(N),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 0 >= 0 = active(U41(mark(X))) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) U13(mark(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 1 >= 1 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 1 >= 1 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 1 >= 1 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U16(mark(X)) = 0 >= 0 = U16(X) U16(active(X)) = 0 >= 0 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) problem: DPs: active#(U13(tt(),V1,V2)) -> mark#(U14(isNatKind(V2),V1,V2)) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 0, [active#](x0) = 1, [0] = 0, [s](x0) = 0, [plus](x0, x1) = x0, [U64](x0, x1, x2) = 0, [U63](x0, x1, x2) = 0, [U62](x0, x1, x2) = 0, [U61](x0, x1, x2) = 0, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = 1, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 0, [U21](x0, x1) = 0, [U16](x0) = 0, [U15](x0, x1) = 0, [isNat](x0) = 1, [U14](x0, x1, x2) = 0, [U13](x0, x1, x2) = 0, [mark](x0) = x0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 1, [active](x0) = x0, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: active#(U13(tt(),V1,V2)) = 1 >= 0 = mark#(U14(isNatKind(V2),V1,V2)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNatKind(V1),V1,V2)) active(U12(tt(),V1,V2)) = 0 >= 0 = mark(U13(isNatKind(V2),V1,V2)) active(U13(tt(),V1,V2)) = 0 >= 0 = mark(U14(isNatKind(V2),V1,V2)) active(U14(tt(),V1,V2)) = 0 >= 0 = mark(U15(isNat(V1),V2)) active(U15(tt(),V2)) = 0 >= 0 = mark(U16(isNat(V2))) active(U16(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) = 0 >= 0 = mark(U23(isNat(V1))) active(U23(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V2)) = 0 >= 0 = mark(U32(isNatKind(V2))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt())) = 1 >= 0 = mark(tt()) active(U51(tt(),N)) = N >= N = mark(U52(isNatKind(N),N)) active(U52(tt(),N)) = N >= N = mark(N) active(U61(tt(),M,N)) = 0 >= 0 = mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) = 0 >= 0 = mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) = 0 >= 0 = mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) = 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 0 = mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) = 1 >= 1 = mark(U41(isNatKind(V1))) active(plus(N,0())) = N >= N = mark(U51(isNat(N),N)) active(plus(N,s(M))) = N >= 0 = mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) = X1 >= X1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2,X3)) = 0 >= 0 = active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) mark(U13(X1,X2,X3)) = 0 >= 0 = active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) = 0 >= 0 = active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) = 0 >= 0 = active(U15(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U16(X)) = 0 >= 0 = active(U16(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X1,X2)) = 0 >= 0 = active(U22(mark(X1),X2)) mark(U23(X)) = 0 >= 0 = active(U23(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X)) = 1 >= 1 = active(U41(mark(X))) mark(U51(X1,X2)) = X2 >= X2 = active(U51(mark(X1),X2)) mark(U52(X1,X2)) = X2 >= X2 = active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) = 0 >= 0 = active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) = 0 >= 0 = active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) = 0 >= 0 = active(U64(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 >= X1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 >= X1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 >= X1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 0 >= 0 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 0 >= 0 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 0 >= 0 = U12(X1,X2,X3) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) U13(mark(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,mark(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,mark(X3)) = 0 >= 0 = U13(X1,X2,X3) U13(active(X1),X2,X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,active(X2),X3) = 0 >= 0 = U13(X1,X2,X3) U13(X1,X2,active(X3)) = 0 >= 0 = U13(X1,X2,X3) U14(mark(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,mark(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,mark(X3)) = 0 >= 0 = U14(X1,X2,X3) U14(active(X1),X2,X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,active(X2),X3) = 0 >= 0 = U14(X1,X2,X3) U14(X1,X2,active(X3)) = 0 >= 0 = U14(X1,X2,X3) U15(mark(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,mark(X2)) = 0 >= 0 = U15(X1,X2) U15(active(X1),X2) = 0 >= 0 = U15(X1,X2) U15(X1,active(X2)) = 0 >= 0 = U15(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U16(mark(X)) = 0 >= 0 = U16(X) U16(active(X)) = 0 >= 0 = U16(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,mark(X2)) = 0 >= 0 = U22(X1,X2) U22(active(X1),X2) = 0 >= 0 = U22(X1,X2) U22(X1,active(X2)) = 0 >= 0 = U22(X1,X2) U23(mark(X)) = 0 >= 0 = U23(X) U23(active(X)) = 0 >= 0 = U23(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X)) = 1 >= 1 = U41(X) U41(active(X)) = 1 >= 1 = U41(X) U51(mark(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,mark(X2)) = X2 >= X2 = U51(X1,X2) U51(active(X1),X2) = X2 >= X2 = U51(X1,X2) U51(X1,active(X2)) = X2 >= X2 = U51(X1,X2) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 >= X2 = U52(X1,X2) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,mark(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,mark(X3)) = 0 >= 0 = U62(X1,X2,X3) U62(active(X1),X2,X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,active(X2),X3) = 0 >= 0 = U62(X1,X2,X3) U62(X1,X2,active(X3)) = 0 >= 0 = U62(X1,X2,X3) U63(mark(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,mark(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,mark(X3)) = 0 >= 0 = U63(X1,X2,X3) U63(active(X1),X2,X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,active(X2),X3) = 0 >= 0 = U63(X1,X2,X3) U63(X1,X2,active(X3)) = 0 >= 0 = U63(X1,X2,X3) U64(mark(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,mark(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,mark(X3)) = 0 >= 0 = U64(X1,X2,X3) U64(active(X1),X2,X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,active(X2),X3) = 0 >= 0 = U64(X1,X2,X3) U64(X1,X2,active(X3)) = 0 >= 0 = U64(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 >= X1 = plus(X1,X2) plus(X1,mark(X2)) = X1 >= X1 = plus(X1,X2) plus(active(X1),X2) = X1 >= X1 = plus(X1,X2) plus(X1,active(X2)) = X1 >= X1 = plus(X1,X2) problem: DPs: TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed DPs: U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: isNatKind#(mark(X)) -> isNatKind#(X) isNatKind#(active(X)) -> isNatKind#(X) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U13#(mark(X1),X2,X3) -> U13#(X1,X2,X3) U13#(X1,mark(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) U13#(active(X1),X2,X3) -> U13#(X1,X2,X3) U13#(X1,active(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U14#(mark(X1),X2,X3) -> U14#(X1,X2,X3) U14#(X1,mark(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) U14#(active(X1),X2,X3) -> U14#(X1,X2,X3) U14#(X1,active(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U15#(mark(X1),X2) -> U15#(X1,X2) U15#(X1,mark(X2)) -> U15#(X1,X2) U15#(active(X1),X2) -> U15#(X1,X2) U15#(X1,active(X2)) -> U15#(X1,X2) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U16#(mark(X)) -> U16#(X) U16#(active(X)) -> U16#(X) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U22#(mark(X1),X2) -> U22#(X1,X2) U22#(X1,mark(X2)) -> U22#(X1,X2) U22#(active(X1),X2) -> U22#(X1,X2) U22#(X1,active(X2)) -> U22#(X1,X2) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U23#(mark(X)) -> U23#(X) U23#(active(X)) -> U23#(X) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U32#(mark(X)) -> U32#(X) U32#(active(X)) -> U32#(X) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U52#(mark(X1),X2) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) U52#(active(X1),X2) -> U52#(X1,X2) U52#(X1,active(X2)) -> U52#(X1,X2) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U62#(mark(X1),X2,X3) -> U62#(X1,X2,X3) U62#(X1,mark(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) U62#(active(X1),X2,X3) -> U62#(X1,X2,X3) U62#(X1,active(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U63#(mark(X1),X2,X3) -> U63#(X1,X2,X3) U63#(X1,mark(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) U63#(active(X1),X2,X3) -> U63#(X1,X2,X3) U63#(X1,active(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U64#(mark(X1),X2,X3) -> U64#(X1,X2,X3) U64#(X1,mark(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) U64#(active(X1),X2,X3) -> U64#(X1,X2,X3) U64#(X1,active(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U41#(mark(X)) -> U41#(X) U41#(active(X)) -> U41#(X) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) U51#(active(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open DPs: U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) TRS: 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(),V2)) -> mark(U32(isNatKind(V2))) active(U32(tt())) -> mark(tt()) active(U41(tt())) -> mark(tt()) active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) active(U52(tt(),N)) -> mark(N) active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 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(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(plus(N,0())) -> mark(U51(isNat(N),N)) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U15(active(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X)) -> U41(X) U41(active(X)) -> U41(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Open