YES Problem: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Proof: DP Processor: DPs: a__U11#(tt(),V1,V2) -> a__isNatKind#(V1) a__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2) a__U12#(tt(),V1,V2) -> a__isNatKind#(V2) a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2) a__U13#(tt(),V1,V2) -> a__isNatKind#(V2) a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2) a__U14#(tt(),V1,V2) -> a__isNat#(V1) a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2) a__U15#(tt(),V2) -> a__isNat#(V2) a__U15#(tt(),V2) -> a__U16#(a__isNat(V2)) a__U21#(tt(),V1) -> a__isNatKind#(V1) a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1) a__U22#(tt(),V1) -> a__isNat#(V1) a__U22#(tt(),V1) -> a__U23#(a__isNat(V1)) a__U31#(tt(),V2) -> a__isNatKind#(V2) a__U31#(tt(),V2) -> a__U32#(a__isNatKind(V2)) a__U51#(tt(),N) -> a__isNatKind#(N) a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) a__U61#(tt(),M,N) -> a__isNatKind#(M) a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N) a__U62#(tt(),M,N) -> a__isNat#(N) a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N) a__U63#(tt(),M,N) -> a__isNatKind#(N) a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N) a__U64#(tt(),M,N) -> mark#(M) a__U64#(tt(),M,N) -> mark#(N) a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__U41#(a__isNatKind(V1)) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__U51#(a__isNat(N),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> a__U13#(mark(X1),X2,X3) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> a__U14#(mark(X1),X2,X3) mark#(U15(X1,X2)) -> mark#(X1) mark#(U15(X1,X2)) -> a__U15#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> a__U16#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> a__U22#(mark(X1),X2) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> a__U23#(mark(X)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> a__U32#(mark(X)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> a__U41#(mark(X)) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2,X3)) -> mark#(X1) mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3) mark#(U63(X1,X2,X3)) -> mark#(X1) mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3) mark#(U64(X1,X2,X3)) -> mark#(X1) mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) TDG Processor: DPs: a__U11#(tt(),V1,V2) -> a__isNatKind#(V1) a__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2) a__U12#(tt(),V1,V2) -> a__isNatKind#(V2) a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2) a__U13#(tt(),V1,V2) -> a__isNatKind#(V2) a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2) a__U14#(tt(),V1,V2) -> a__isNat#(V1) a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2) a__U15#(tt(),V2) -> a__isNat#(V2) a__U15#(tt(),V2) -> a__U16#(a__isNat(V2)) a__U21#(tt(),V1) -> a__isNatKind#(V1) a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1) a__U22#(tt(),V1) -> a__isNat#(V1) a__U22#(tt(),V1) -> a__U23#(a__isNat(V1)) a__U31#(tt(),V2) -> a__isNatKind#(V2) a__U31#(tt(),V2) -> a__U32#(a__isNatKind(V2)) a__U51#(tt(),N) -> a__isNatKind#(N) a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) a__U61#(tt(),M,N) -> a__isNatKind#(M) a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N) a__U62#(tt(),M,N) -> a__isNat#(N) a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N) a__U63#(tt(),M,N) -> a__isNatKind#(N) a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N) a__U64#(tt(),M,N) -> mark#(M) a__U64#(tt(),M,N) -> mark#(N) a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__U41#(a__isNatKind(V1)) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__U51#(a__isNat(N),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> a__U13#(mark(X1),X2,X3) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> a__U14#(mark(X1),X2,X3) mark#(U15(X1,X2)) -> mark#(X1) mark#(U15(X1,X2)) -> a__U15#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U16(X)) -> mark#(X) mark#(U16(X)) -> a__U16#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> a__U22#(mark(X1),X2) mark#(U23(X)) -> mark#(X) mark#(U23(X)) -> a__U23#(mark(X)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> a__U32#(mark(X)) mark#(U41(X)) -> mark#(X) mark#(U41(X)) -> a__U41#(mark(X)) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2,X3)) -> mark#(X1) mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3) mark#(U63(X1,X2,X3)) -> mark#(X1) mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3) mark#(U64(X1,X2,X3)) -> mark#(X1) mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) graph: ... SCC Processor: #sccs: 3 #rules: 52 #arcs: 1152/6724 DPs: a__plus#(N,0()) -> a__U51#(a__isNat(N),N) a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U15(X1,X2)) -> mark#(X1) mark#(U16(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> mark#(X1) mark#(U23(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U32(X)) -> mark#(X) mark#(U41(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N) a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N) a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N) a__U64#(tt(),M,N) -> mark#(M) mark#(U62(X1,X2,X3)) -> mark#(X1) mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3) mark#(U63(X1,X2,X3)) -> mark#(X1) mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3) mark#(U64(X1,X2,X3)) -> mark#(X1) mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3) a__U64#(tt(),M,N) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N) mark#(s(X)) -> mark#(X) a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) interpretation: [a__plus#](x0, x1) = x0 + x1, [a__U64#](x0, x1, x2) = x0 + x1 + x2, [a__U63#](x0, x1, x2) = x0 + x1 + x2, [a__U62#](x0, x1, x2) = x0 + x1 + x2, [a__U61#](x0, x1, x2) = x0 + x1 + x2, [mark#](x0) = x0, [a__U52#](x0, x1) = x0 + x1, [a__U51#](x0, x1) = x0 + x1 + 0, [U64](x0, x1, x2) = x0 + 5x1 + x2, [U63](x0, x1, x2) = x0 + 5x1 + x2 + 0, [U62](x0, x1, x2) = x0 + 5x1 + x2, [U61](x0, x1, x2) = x0 + 5x1 + x2, [U52](x0, x1) = x0 + x1 + 0, [U51](x0, x1) = x0 + x1 + 0, [U41](x0) = x0, [U32](x0) = 1x0 + -6, [U31](x0, x1) = x0 + 4x1, [U23](x0) = x0, [U22](x0, x1) = x0 + x1, [U21](x0, x1) = x0 + x1, [U16](x0) = x0, [isNat](x0) = x0, [U15](x0, x1) = x0 + x1 + 0, [U14](x0, x1, x2) = x0 + x1 + x2 + 0, [U13](x0, x1, x2) = x0 + x1 + x2, [isNatKind](x0) = x0, [U12](x0, x1, x2) = x0 + x1 + x2 + 0, [U11](x0, x1, x2) = x0 + x1 + x2, [plus](x0, x1) = x0 + 5x1, [0] = 0, [s](x0) = x0, [a__plus](x0, x1) = x0 + 5x1, [a__U64](x0, x1, x2) = x0 + 5x1 + x2, [a__U63](x0, x1, x2) = x0 + 5x1 + x2 + 0, [a__U62](x0, x1, x2) = x0 + 5x1 + x2, [a__U61](x0, x1, x2) = x0 + 5x1 + x2, [mark](x0) = x0, [a__U52](x0, x1) = x0 + x1 + 0, [a__U51](x0, x1) = x0 + x1 + 0, [a__U41](x0) = x0, [a__U32](x0) = 1x0 + -6, [a__U31](x0, x1) = x0 + 4x1, [a__U23](x0) = x0, [a__U22](x0, x1) = x0 + x1, [a__U21](x0, x1) = x0 + x1, [a__U16](x0) = x0, [a__U15](x0, x1) = x0 + x1 + 0, [a__isNat](x0) = x0, [a__U14](x0, x1, x2) = x0 + x1 + x2 + 0, [a__U13](x0, x1, x2) = x0 + x1 + x2, [a__U12](x0, x1, x2) = x0 + x1 + x2 + 0, [a__isNatKind](x0) = x0, [a__U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 0 orientation: a__plus#(N,0()) = N + 0 >= N + 0 = a__U51#(a__isNat(N),N) a__U51#(tt(),N) = N + 0 >= N = a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) = N + 0 >= N = mark#(N) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U12(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 = mark#(X1) mark#(U13(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U14(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 = mark#(X1) mark#(U15(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(U16(X)) = X >= X = mark#(X) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U22(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U23(X)) = X >= X = mark#(X) mark#(U31(X1,X2)) = X1 + 4X2 >= X1 = mark#(X1) mark#(U32(X)) = 1X + -6 >= X = mark#(X) mark#(U41(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(U51(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U51#(mark(X1),X2) mark#(U52(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 + 0 >= X1 + X2 = a__U52#(mark(X1),X2) mark#(U61(X1,X2,X3)) = X1 + 5X2 + X3 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + 5X2 + X3 >= X1 + X2 + X3 = a__U61#(mark(X1),X2,X3) a__U61#(tt(),M,N) = M + N + 0 >= M + N = a__U62#(a__isNatKind(M),M,N) a__U62#(tt(),M,N) = M + N + 0 >= M + N = a__U63#(a__isNat(N),M,N) a__U63#(tt(),M,N) = M + N + 0 >= M + N = a__U64#(a__isNatKind(N),M,N) a__U64#(tt(),M,N) = M + N + 0 >= M = mark#(M) mark#(U62(X1,X2,X3)) = X1 + 5X2 + X3 >= X1 = mark#(X1) mark#(U62(X1,X2,X3)) = X1 + 5X2 + X3 >= X1 + X2 + X3 = a__U62#(mark(X1),X2,X3) mark#(U63(X1,X2,X3)) = X1 + 5X2 + X3 + 0 >= X1 = mark#(X1) mark#(U63(X1,X2,X3)) = X1 + 5X2 + X3 + 0 >= X1 + X2 + X3 = a__U63#(mark(X1),X2,X3) mark#(U64(X1,X2,X3)) = X1 + 5X2 + X3 >= X1 = mark#(X1) mark#(U64(X1,X2,X3)) = X1 + 5X2 + X3 >= X1 + X2 + X3 = a__U64#(mark(X1),X2,X3) a__U64#(tt(),M,N) = M + N + 0 >= N = mark#(N) mark#(plus(X1,X2)) = X1 + 5X2 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + 5X2 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + 5X2 >= X1 + X2 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = M + N >= M + N = a__U61#(a__isNat(M),M,N) mark#(s(X)) = X >= X = mark#(X) a__U64#(tt(),M,N) = M + N + 0 >= M + N = a__plus#(mark(N),mark(M)) a__U11(tt(),V1,V2) = V1 + V2 + 0 >= V1 + V2 + 0 = a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) = V1 + V2 + 0 >= V1 + V2 = a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) = V1 + V2 + 0 >= V1 + V2 + 0 = a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) = V1 + V2 + 0 >= V1 + V2 + 0 = a__U15(a__isNat(V1),V2) a__U15(tt(),V2) = V2 + 0 >= V2 = a__U16(a__isNat(V2)) a__U16(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = V1 + 0 >= V1 = a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) = V1 + 0 >= V1 = a__U23(a__isNat(V1)) a__U23(tt()) = 0 >= 0 = tt() a__U31(tt(),V2) = 4V2 + 0 >= 1V2 + -6 = a__U32(a__isNatKind(V2)) a__U32(tt()) = 1 >= 0 = tt() a__U41(tt()) = 0 >= 0 = tt() a__U51(tt(),N) = N + 0 >= N + 0 = a__U52(a__isNatKind(N),N) a__U52(tt(),N) = N + 0 >= N = mark(N) a__U61(tt(),M,N) = 5M + N + 0 >= 5M + N = a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) = 5M + N + 0 >= 5M + N + 0 = a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) = 5M + N + 0 >= 5M + N = a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) = 5M + N + 0 >= 5M + N = s(a__plus(mark(N),mark(M))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(plus(V1,V2)) = V1 + 5V2 >= V1 + V2 = a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(plus(V1,V2)) = V1 + 5V2 >= V1 + 4V2 = a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) = V1 >= V1 = a__U41(a__isNatKind(V1)) a__plus(N,0()) = N + 5 >= N + 0 = a__U51(a__isNat(N),N) a__plus(N,s(M)) = 5M + N >= 5M + N = a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__U12(mark(X1),X2,X3) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(U13(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U15(mark(X1),X2) mark(isNat(X)) = X >= X = a__isNat(X) mark(U16(X)) = X >= X = a__U16(mark(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2) mark(U22(X1,X2)) = X1 + X2 >= X1 + X2 = a__U22(mark(X1),X2) mark(U23(X)) = X >= X = a__U23(mark(X)) mark(U31(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = a__U31(mark(X1),X2) mark(U32(X)) = 1X + -6 >= 1X + -6 = a__U32(mark(X)) mark(U41(X)) = X >= X = a__U41(mark(X)) mark(U51(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U51(mark(X1),X2) mark(U52(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) = X1 + 5X2 + X3 >= X1 + 5X2 + X3 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) = X1 + 5X2 + X3 >= X1 + 5X2 + X3 = a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) = X1 + 5X2 + X3 + 0 >= X1 + 5X2 + X3 + 0 = a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) = X1 + 5X2 + X3 >= X1 + 5X2 + X3 = a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + 5X2 >= X1 + 5X2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) a__U12(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = U12(X1,X2,X3) a__isNatKind(X) = X >= X = isNatKind(X) a__U13(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U13(X1,X2,X3) a__U14(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = U14(X1,X2,X3) a__U15(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U15(X1,X2) a__isNat(X) = X >= X = isNat(X) a__U16(X) = X >= X = U16(X) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X1,X2) = X1 + X2 >= X1 + X2 = U22(X1,X2) a__U23(X) = X >= X = U23(X) a__U31(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U31(X1,X2) a__U32(X) = 1X + -6 >= 1X + -6 = U32(X) a__U41(X) = X >= X = U41(X) a__U51(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U51(X1,X2) a__U52(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U52(X1,X2) a__U61(X1,X2,X3) = X1 + 5X2 + X3 >= X1 + 5X2 + X3 = U61(X1,X2,X3) a__U62(X1,X2,X3) = X1 + 5X2 + X3 >= X1 + 5X2 + X3 = U62(X1,X2,X3) a__U63(X1,X2,X3) = X1 + 5X2 + X3 + 0 >= X1 + 5X2 + X3 + 0 = U63(X1,X2,X3) a__U64(X1,X2,X3) = X1 + 5X2 + X3 >= X1 + 5X2 + X3 = U64(X1,X2,X3) a__plus(X1,X2) = X1 + 5X2 >= X1 + 5X2 = plus(X1,X2) problem: DPs: a__plus#(N,0()) -> a__U51#(a__isNat(N),N) a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U15(X1,X2)) -> mark#(X1) mark#(U16(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> mark#(X1) mark#(U23(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U41(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N) a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N) a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N) a__U64#(tt(),M,N) -> mark#(M) mark#(U62(X1,X2,X3)) -> mark#(X1) mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3) mark#(U63(X1,X2,X3)) -> mark#(X1) mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3) mark#(U64(X1,X2,X3)) -> mark#(X1) mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3) a__U64#(tt(),M,N) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N) mark#(s(X)) -> mark#(X) a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Restore Modifier: DPs: a__plus#(N,0()) -> a__U51#(a__isNat(N),N) a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U15(X1,X2)) -> mark#(X1) mark#(U16(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> mark#(X1) mark#(U23(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U41(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N) a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N) a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N) a__U64#(tt(),M,N) -> mark#(M) mark#(U62(X1,X2,X3)) -> mark#(X1) mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3) mark#(U63(X1,X2,X3)) -> mark#(X1) mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3) mark#(U64(X1,X2,X3)) -> mark#(X1) mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3) a__U64#(tt(),M,N) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N) mark#(s(X)) -> mark#(X) a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dim=3 usable rules: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) interpretation: [a__plus#](x0, x1) = [1 0 1]x0 + [1 0 1]x1, [a__U64#](x0, x1, x2) = [1 0 1]x1 + [1 0 1]x2, [a__U63#](x0, x1, x2) = [1 0 1]x1 + [1 0 1]x2, [a__U62#](x0, x1, x2) = [1 0 1]x1 + [1 0 1]x2, [a__U61#](x0, x1, x2) = [1 0 1]x1 + [1 0 1]x2, [mark#](x0) = [1 0 0]x0, [a__U52#](x0, x1) = [1 0 0]x1, [a__U51#](x0, x1) = [1 0 0]x1, [1 0 0] [1 0 1] [1 0 1] [1] [U64](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [1 0 1] [1 0 1] [1], [1 0 0] [1 0 1] [1 0 1] [1] [U63](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [1 0 1] [1 0 1] [1], [1 0 0] [1 0 1] [1 0 1] [1] [U62](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [1 0 1] [1 0 1] [1], [1 1 0] [1 0 1] [1 0 1] [1] [U61](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [1 0 1] [1 0 1] [1], [1 1 0] [1 1 1] [U52](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [1 1 1] , [1 0 0] [1 1 1] [U51](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [1 1 1] , [1 0 0] [U41](x0) = [0 0 0]x0 [0 0 1] , [0] [U32](x0) = [0] [0], [1 0 0] [U31](x0, x1) = [0 0 0]x0 [0 1 1] , [1 0 0] [U23](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [U22](x0, x1) = [0 0 0]x0 [0 0 1] , [1 0 0] [U21](x0, x1) = [0 0 0]x0 [0 0 1] , [1 1 0] [U16](x0) = [0 0 0]x0 [0 0 1] , [0] [isNat](x0) = [0] [0], [1 0 0] [U15](x0, x1) = [0 0 0]x0 [0 0 1] , [1 0 1] [U14](x0, x1, x2) = [0 0 0]x0 [1 0 1] , [1 0 1] [U13](x0, x1, x2) = [0 0 0]x0 [1 0 1] , [0] [isNatKind](x0) = [0] [0], [1 0 0] [U12](x0, x1, x2) = [0 0 0]x0 [0 0 1] , [1 0 1] [U11](x0, x1, x2) = [0 0 0]x0 [1 0 1] , [1 1 1] [1 1 1] [plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 0 1] [1 0 1] , [0] [0] = [0] [0], [1] [s](x0) = x0 + [0] [1], [1 1 1] [1 1 1] [a__plus](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [1 1 1] [1 1 1] , [1 1 0] [1 0 1] [1 0 1] [1] [a__U64](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [1 0 1] [1 0 1] [1], [1 0 0] [1 0 1] [1 0 1] [1] [a__U63](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [1 0 1] [1 0 1] [1], [1 0 0] [1 0 1] [1 0 1] [1] [a__U62](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [1 0 1] [1 0 1] [1], [1 1 0] [1 0 1] [1 0 1] [1] [a__U61](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [1 0 1] [1 0 1] [1], [0 0 1] [mark](x0) = [0 0 0]x0 [1 0 0] , [1 1 0] [1 1 1] [a__U52](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [1 1 1] , [1 0 0] [1 1 1] [a__U51](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [1 1 1] , [1 0 0] [a__U41](x0) = [0 0 0]x0 [0 0 1] , [0] [a__U32](x0) = [0] [0], [1 0 0] [a__U31](x0, x1) = [0 0 0]x0 [0 1 1] , [1 0 0] [a__U23](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [a__U22](x0, x1) = [0 0 0]x0 [0 0 1] , [1 0 0] [a__U21](x0, x1) = [0 0 0]x0 [0 0 1] , [1 1 0] [a__U16](x0) = [0 0 0]x0 [0 0 1] , [a__U15](x0, x1) = x0 , [0] [a__isNat](x0) = [0] [0], [1 0 1] [a__U14](x0, x1, x2) = [0 0 0]x0 [1 0 1] , [1 0 1] [a__U13](x0, x1, x2) = [0 0 0]x0 [1 0 1] , [1 0 0] [a__U12](x0, x1, x2) = [0 0 0]x0 [0 0 1] , [0] [a__isNatKind](x0) = [0] [0], [1 0 1] [a__U11](x0, x1, x2) = [0 0 0]x0 [1 0 1] , [0] [tt] = [0] [0] orientation: a__plus#(N,0()) = [1 0 1]N >= [1 0 0]N = a__U51#(a__isNat(N),N) a__U51#(tt(),N) = [1 0 0]N >= [1 0 0]N = a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) = [1 0 0]N >= [1 0 0]N = mark#(N) mark#(U11(X1,X2,X3)) = [1 0 1]X1 >= [1 0 0]X1 = mark#(X1) mark#(U12(X1,X2,X3)) = [1 0 0]X1 >= [1 0 0]X1 = mark#(X1) mark#(U13(X1,X2,X3)) = [1 0 1]X1 >= [1 0 0]X1 = mark#(X1) mark#(U14(X1,X2,X3)) = [1 0 1]X1 >= [1 0 0]X1 = mark#(X1) mark#(U15(X1,X2)) = [1 0 0]X1 >= [1 0 0]X1 = mark#(X1) mark#(U16(X)) = [1 1 0]X >= [1 0 0]X = mark#(X) mark#(U21(X1,X2)) = [1 0 0]X1 >= [1 0 0]X1 = mark#(X1) mark#(U22(X1,X2)) = [1 0 0]X1 >= [1 0 0]X1 = mark#(X1) mark#(U23(X)) = [1 0 0]X >= [1 0 0]X = mark#(X) mark#(U31(X1,X2)) = [1 0 0]X1 >= [1 0 0]X1 = mark#(X1) mark#(U41(X)) = [1 0 0]X >= [1 0 0]X = mark#(X) mark#(U51(X1,X2)) = [1 0 0]X1 + [1 1 1]X2 >= [1 0 0]X1 = mark#(X1) mark#(U51(X1,X2)) = [1 0 0]X1 + [1 1 1]X2 >= [1 0 0]X2 = a__U51#(mark(X1),X2) mark#(U52(X1,X2)) = [1 1 0]X1 + [1 1 1]X2 >= [1 0 0]X1 = mark#(X1) mark#(U52(X1,X2)) = [1 1 0]X1 + [1 1 1]X2 >= [1 0 0]X2 = a__U52#(mark(X1),X2) mark#(U61(X1,X2,X3)) = [1 1 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 0]X1 = mark#(X1) mark#(U61(X1,X2,X3)) = [1 1 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 1]X2 + [1 0 1]X3 = a__U61#(mark(X1),X2,X3) a__U61#(tt(),M,N) = [1 0 1]M + [1 0 1]N >= [1 0 1]M + [1 0 1]N = a__U62#(a__isNatKind(M),M,N) a__U62#(tt(),M,N) = [1 0 1]M + [1 0 1]N >= [1 0 1]M + [1 0 1]N = a__U63#(a__isNat(N),M,N) a__U63#(tt(),M,N) = [1 0 1]M + [1 0 1]N >= [1 0 1]M + [1 0 1]N = a__U64#(a__isNatKind(N),M,N) a__U64#(tt(),M,N) = [1 0 1]M + [1 0 1]N >= [1 0 0]M = mark#(M) mark#(U62(X1,X2,X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 0]X1 = mark#(X1) mark#(U62(X1,X2,X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 1]X2 + [1 0 1]X3 = a__U62#(mark(X1),X2,X3) mark#(U63(X1,X2,X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 0]X1 = mark#(X1) mark#(U63(X1,X2,X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 1]X2 + [1 0 1]X3 = a__U63#(mark(X1),X2,X3) mark#(U64(X1,X2,X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 0]X1 = mark#(X1) mark#(U64(X1,X2,X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 1]X2 + [1 0 1]X3 = a__U64#(mark(X1),X2,X3) a__U64#(tt(),M,N) = [1 0 1]M + [1 0 1]N >= [1 0 0]N = mark#(N) mark#(plus(X1,X2)) = [1 1 1]X1 + [1 1 1]X2 >= [1 0 0]X1 = mark#(X1) mark#(plus(X1,X2)) = [1 1 1]X1 + [1 1 1]X2 >= [1 0 1]X1 + [1 0 1]X2 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = [1 0 1]M + [1 0 1]N + [2] >= [1 0 1]M + [1 0 1]N = a__U61#(a__isNat(M),M,N) mark#(s(X)) = [1 0 0]X + [1] >= [1 0 0]X = mark#(X) a__U64#(tt(),M,N) = [1 0 1]M + [1 0 1]N >= [1 0 1]M + [1 0 1]N = a__plus#(mark(N),mark(M)) [0] [0] a__U11(tt(),V1,V2) = [0] >= [0] = a__U12(a__isNatKind(V1),V1,V2) [0] [0] [0] [0] a__U12(tt(),V1,V2) = [0] >= [0] = a__U13(a__isNatKind(V2),V1,V2) [0] [0] [0] [0] a__U13(tt(),V1,V2) = [0] >= [0] = a__U14(a__isNatKind(V2),V1,V2) [0] [0] [0] [0] a__U14(tt(),V1,V2) = [0] >= [0] = a__U15(a__isNat(V1),V2) [0] [0] [0] [0] a__U15(tt(),V2) = [0] >= [0] = a__U16(a__isNat(V2)) [0] [0] [0] [0] a__U16(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] a__U21(tt(),V1) = [0] >= [0] = a__U22(a__isNatKind(V1),V1) [0] [0] [0] [0] a__U22(tt(),V1) = [0] >= [0] = a__U23(a__isNat(V1)) [0] [0] [0] [0] a__U23(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] a__U31(tt(),V2) = [0] >= [0] = a__U32(a__isNatKind(V2)) [0] [0] [0] [0] a__U32(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] a__U41(tt()) = [0] >= [0] = tt() [0] [0] [1 1 1] [1 1 1] a__U51(tt(),N) = [0 0 0]N >= [0 0 0]N = a__U52(a__isNatKind(N),N) [1 1 1] [1 1 1] [1 1 1] [0 0 1] a__U52(tt(),N) = [0 0 0]N >= [0 0 0]N = mark(N) [1 1 1] [1 0 0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] a__U61(tt(),M,N) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = a__U62(a__isNatKind(M),M,N) [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] a__U62(tt(),M,N) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = a__U63(a__isNat(N),M,N) [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] a__U63(tt(),M,N) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = a__U64(a__isNatKind(N),M,N) [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] a__U64(tt(),M,N) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = s(a__plus(mark(N),mark(M))) [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] [0] [0] a__isNat(0()) = [0] >= [0] = tt() [0] [0] [0] [0] a__isNat(plus(V1,V2)) = [0] >= [0] = a__U11(a__isNatKind(V1),V1,V2) [0] [0] [0] [0] a__isNat(s(V1)) = [0] >= [0] = a__U21(a__isNatKind(V1),V1) [0] [0] [0] [0] a__isNatKind(0()) = [0] >= [0] = tt() [0] [0] [0] [0] a__isNatKind(plus(V1,V2)) = [0] >= [0] = a__U31(a__isNatKind(V1),V2) [0] [0] [0] [0] a__isNatKind(s(V1)) = [0] >= [0] = a__U41(a__isNatKind(V1)) [0] [0] [1 1 1] [1 1 1] a__plus(N,0()) = [0 1 0]N >= [0 0 0]N = a__U51(a__isNat(N),N) [1 1 1] [1 1 1] [1 1 1] [1 1 1] [2] [1 0 1] [1 0 1] [1] a__plus(N,s(M)) = [0 0 0]M + [0 1 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = a__U61(a__isNat(M),M,N) [1 1 1] [1 1 1] [2] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] mark(U11(X1,X2,X3)) = [0 0 0]X1 >= [0 0 0]X1 = a__U11(mark(X1),X2,X3) [1 0 1] [1 0 1] [0 0 1] [0 0 1] mark(U12(X1,X2,X3)) = [0 0 0]X1 >= [0 0 0]X1 = a__U12(mark(X1),X2,X3) [1 0 0] [1 0 0] [0] [0] mark(isNatKind(X)) = [0] >= [0] = a__isNatKind(X) [0] [0] [1 0 1] [1 0 1] mark(U13(X1,X2,X3)) = [0 0 0]X1 >= [0 0 0]X1 = a__U13(mark(X1),X2,X3) [1 0 1] [1 0 1] [1 0 1] [1 0 1] mark(U14(X1,X2,X3)) = [0 0 0]X1 >= [0 0 0]X1 = a__U14(mark(X1),X2,X3) [1 0 1] [1 0 1] [0 0 1] [0 0 1] mark(U15(X1,X2)) = [0 0 0]X1 >= [0 0 0]X1 = a__U15(mark(X1),X2) [1 0 0] [1 0 0] [0] [0] mark(isNat(X)) = [0] >= [0] = a__isNat(X) [0] [0] [0 0 1] [0 0 1] mark(U16(X)) = [0 0 0]X >= [0 0 0]X = a__U16(mark(X)) [1 1 0] [1 0 0] [0 0 1] [0 0 1] mark(U21(X1,X2)) = [0 0 0]X1 >= [0 0 0]X1 = a__U21(mark(X1),X2) [1 0 0] [1 0 0] [0 0 1] [0 0 1] mark(U22(X1,X2)) = [0 0 0]X1 >= [0 0 0]X1 = a__U22(mark(X1),X2) [1 0 0] [1 0 0] [0 0 1] [0 0 1] mark(U23(X)) = [0 0 0]X >= [0 0 0]X = a__U23(mark(X)) [1 0 0] [1 0 0] [0 1 1] [0 0 1] mark(U31(X1,X2)) = [0 0 0]X1 >= [0 0 0]X1 = a__U31(mark(X1),X2) [1 0 0] [1 0 0] [0] [0] mark(U32(X)) = [0] >= [0] = a__U32(mark(X)) [0] [0] [0 0 1] [0 0 1] mark(U41(X)) = [0 0 0]X >= [0 0 0]X = a__U41(mark(X)) [1 0 0] [1 0 0] [0 0 1] [1 1 1] [0 0 1] [1 1 1] mark(U51(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = a__U51(mark(X1),X2) [1 0 0] [1 1 1] [1 0 0] [1 1 1] [0 0 1] [1 1 1] [0 0 1] [1 1 1] mark(U52(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = a__U52(mark(X1),X2) [1 1 0] [1 1 1] [1 0 0] [1 1 1] [0 0 1] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] mark(U61(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = a__U61(mark(X1),X2,X3) [1 1 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] mark(U62(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = a__U62(mark(X1),X2,X3) [1 0 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] mark(U63(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = a__U63(mark(X1),X2,X3) [1 0 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] mark(U64(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = a__U64(mark(X1),X2,X3) [1 0 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] mark(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = a__plus(mark(X1),mark(X2)) [1 1 1] [1 1 1] [1 0 1] [1 0 1] [0] [0] mark(tt()) = [0] >= [0] = tt() [0] [0] [0 0 1] [1] [0 0 1] [1] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(mark(X)) [1 0 0] [1] [1 0 0] [1] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 1] [1 0 1] a__U11(X1,X2,X3) = [0 0 0]X1 >= [0 0 0]X1 = U11(X1,X2,X3) [1 0 1] [1 0 1] [1 0 0] [1 0 0] a__U12(X1,X2,X3) = [0 0 0]X1 >= [0 0 0]X1 = U12(X1,X2,X3) [0 0 1] [0 0 1] [0] [0] a__isNatKind(X) = [0] >= [0] = isNatKind(X) [0] [0] [1 0 1] [1 0 1] a__U13(X1,X2,X3) = [0 0 0]X1 >= [0 0 0]X1 = U13(X1,X2,X3) [1 0 1] [1 0 1] [1 0 1] [1 0 1] a__U14(X1,X2,X3) = [0 0 0]X1 >= [0 0 0]X1 = U14(X1,X2,X3) [1 0 1] [1 0 1] [1 0 0] a__U15(X1,X2) = X1 >= [0 0 0]X1 = U15(X1,X2) [0 0 1] [0] [0] a__isNat(X) = [0] >= [0] = isNat(X) [0] [0] [1 1 0] [1 1 0] a__U16(X) = [0 0 0]X >= [0 0 0]X = U16(X) [0 0 1] [0 0 1] [1 0 0] [1 0 0] a__U21(X1,X2) = [0 0 0]X1 >= [0 0 0]X1 = U21(X1,X2) [0 0 1] [0 0 1] [1 0 0] [1 0 0] a__U22(X1,X2) = [0 0 0]X1 >= [0 0 0]X1 = U22(X1,X2) [0 0 1] [0 0 1] [1 0 0] [1 0 0] a__U23(X) = [0 0 0]X >= [0 0 0]X = U23(X) [0 0 1] [0 0 1] [1 0 0] [1 0 0] a__U31(X1,X2) = [0 0 0]X1 >= [0 0 0]X1 = U31(X1,X2) [0 1 1] [0 1 1] [0] [0] a__U32(X) = [0] >= [0] = U32(X) [0] [0] [1 0 0] [1 0 0] a__U41(X) = [0 0 0]X >= [0 0 0]X = U41(X) [0 0 1] [0 0 1] [1 0 0] [1 1 1] [1 0 0] [1 1 1] a__U51(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U51(X1,X2) [0 0 1] [1 1 1] [0 0 1] [1 1 1] [1 1 0] [1 1 1] [1 1 0] [1 1 1] a__U52(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U52(X1,X2) [0 0 1] [1 1 1] [0 0 1] [1 1 1] [1 1 0] [1 0 1] [1 0 1] [1] [1 1 0] [1 0 1] [1 0 1] [1] a__U61(X1,X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U61(X1,X2,X3) [0 0 1] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] a__U62(X1,X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U62(X1,X2,X3) [0 0 1] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] a__U63(X1,X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U63(X1,X2,X3) [0 0 1] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] [1 1 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] a__U64(X1,X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U64(X1,X2,X3) [0 0 1] [1 0 1] [1 0 1] [1] [0 0 1] [1 0 1] [1 0 1] [1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] a__plus(X1,X2) = [0 1 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,X2) [1 1 1] [1 1 1] [1 0 1] [1 0 1] problem: DPs: a__plus#(N,0()) -> a__U51#(a__isNat(N),N) a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U15(X1,X2)) -> mark#(X1) mark#(U16(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> mark#(X1) mark#(U23(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U41(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N) a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N) a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N) a__U64#(tt(),M,N) -> mark#(M) a__U64#(tt(),M,N) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Restore Modifier: DPs: a__plus#(N,0()) -> a__U51#(a__isNat(N),N) a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U15(X1,X2)) -> mark#(X1) mark#(U16(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> mark#(X1) mark#(U23(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U41(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N) a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N) a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N) a__U64#(tt(),M,N) -> mark#(M) a__U64#(tt(),M,N) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) SCC Processor: #sccs: 1 #rules: 20 #arcs: 692/676 DPs: a__plus#(N,0()) -> a__U51#(a__isNat(N),N) a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U15(X1,X2)) -> mark#(X1) mark#(U16(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> mark#(X1) mark#(U23(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U41(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 usable rules: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) interpretation: [a__plus#](x0, x1) = 2x0 + 2x1, [mark#](x0) = 2x0, [a__U52#](x0, x1) = 2x0 + 2x1, [a__U51#](x0, x1) = 2x1, [U64](x0, x1, x2) = 3, [U63](x0, x1, x2) = 2, [U62](x0, x1, x2) = x2 + 5, [U61](x0, x1, x2) = 7x0 + x2 + 3, [U52](x0, x1) = 4x0 + 2x1, [U51](x0, x1) = 4x0 + 2x1, [U41](x0) = 4x0, [U32](x0) = 0, [U31](x0, x1) = x0, [U23](x0) = 3x0, [U22](x0, x1) = x0, [U21](x0, x1) = x0, [U16](x0) = 5x0, [isNat](x0) = 0, [U15](x0, x1) = x0, [U14](x0, x1, x2) = x0, [U13](x0, x1, x2) = x0, [isNatKind](x0) = 0, [U12](x0, x1, x2) = x0, [U11](x0, x1, x2) = 2x0, [plus](x0, x1) = 3x0 + 4x1, [0] = 1, [s](x0) = 4, [a__plus](x0, x1) = 3x0 + 4x1, [a__U64](x0, x1, x2) = 4, [a__U63](x0, x1, x2) = 4, [a__U62](x0, x1, x2) = 2x2 + 6, [a__U61](x0, x1, x2) = 7x0 + 2x2 + 6, [mark](x0) = 3x0, [a__U52](x0, x1) = 4x0 + 3x1, [a__U51](x0, x1) = 4x0 + 3x1, [a__U41](x0) = 4x0, [a__U32](x0) = 0, [a__U31](x0, x1) = x0, [a__U23](x0) = 3x0, [a__U22](x0, x1) = x0, [a__U21](x0, x1) = x0, [a__U16](x0) = 5x0, [a__U15](x0, x1) = x0, [a__isNat](x0) = 0, [a__U14](x0, x1, x2) = x0, [a__U13](x0, x1, x2) = x0, [a__U12](x0, x1, x2) = x0, [a__isNatKind](x0) = 0, [a__U11](x0, x1, x2) = 2x0, [tt] = 0 orientation: a__plus#(N,0()) = 2N + 2 >= 2N = a__U51#(a__isNat(N),N) a__U51#(tt(),N) = 2N >= 2N = a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) = 2N >= 2N = mark#(N) mark#(U11(X1,X2,X3)) = 4X1 >= 2X1 = mark#(X1) mark#(U12(X1,X2,X3)) = 2X1 >= 2X1 = mark#(X1) mark#(U13(X1,X2,X3)) = 2X1 >= 2X1 = mark#(X1) mark#(U14(X1,X2,X3)) = 2X1 >= 2X1 = mark#(X1) mark#(U15(X1,X2)) = 2X1 >= 2X1 = mark#(X1) mark#(U16(X)) = 10X >= 2X = mark#(X) mark#(U21(X1,X2)) = 2X1 >= 2X1 = mark#(X1) mark#(U22(X1,X2)) = 2X1 >= 2X1 = mark#(X1) mark#(U23(X)) = 6X >= 2X = mark#(X) mark#(U31(X1,X2)) = 2X1 >= 2X1 = mark#(X1) mark#(U41(X)) = 8X >= 2X = mark#(X) mark#(U51(X1,X2)) = 8X1 + 4X2 >= 2X1 = mark#(X1) mark#(U51(X1,X2)) = 8X1 + 4X2 >= 2X2 = a__U51#(mark(X1),X2) mark#(U52(X1,X2)) = 8X1 + 4X2 >= 2X1 = mark#(X1) mark#(U52(X1,X2)) = 8X1 + 4X2 >= 6X1 + 2X2 = a__U52#(mark(X1),X2) mark#(plus(X1,X2)) = 6X1 + 8X2 >= 2X1 = mark#(X1) mark#(plus(X1,X2)) = 6X1 + 8X2 >= 6X1 + 6X2 = a__plus#(mark(X1),mark(X2)) a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2) a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2)) a__U16(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1)) a__U23(tt()) = 0 >= 0 = tt() a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt()) = 0 >= 0 = tt() a__U51(tt(),N) = 3N >= 3N = a__U52(a__isNatKind(N),N) a__U52(tt(),N) = 3N >= 3N = mark(N) a__U61(tt(),M,N) = 2N + 6 >= 2N + 6 = a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) = 2N + 6 >= 4 = a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) = 4 >= 4 = a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) = 4 >= 4 = s(a__plus(mark(N),mark(M))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1)) a__plus(N,0()) = 3N + 4 >= 3N = a__U51(a__isNat(N),N) a__plus(N,s(M)) = 3N + 16 >= 2N + 6 = a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) = 6X1 >= 6X1 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = 3X1 >= 3X1 = a__U12(mark(X1),X2,X3) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(U13(X1,X2,X3)) = 3X1 >= 3X1 = a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) = 3X1 >= 3X1 = a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) = 3X1 >= 3X1 = a__U15(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U16(X)) = 15X >= 15X = a__U16(mark(X)) mark(U21(X1,X2)) = 3X1 >= 3X1 = a__U21(mark(X1),X2) mark(U22(X1,X2)) = 3X1 >= 3X1 = a__U22(mark(X1),X2) mark(U23(X)) = 9X >= 9X = a__U23(mark(X)) mark(U31(X1,X2)) = 3X1 >= 3X1 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X)) = 12X >= 12X = a__U41(mark(X)) mark(U51(X1,X2)) = 12X1 + 6X2 >= 12X1 + 3X2 = a__U51(mark(X1),X2) mark(U52(X1,X2)) = 12X1 + 6X2 >= 12X1 + 3X2 = a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) = 21X1 + 3X3 + 9 >= 21X1 + 2X3 + 6 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) = 3X3 + 15 >= 2X3 + 6 = a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) = 6 >= 4 = a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) = 9 >= 4 = a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) = 9X1 + 12X2 >= 9X1 + 12X2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 12 >= 4 = s(mark(X)) mark(0()) = 3 >= 1 = 0() a__U11(X1,X2,X3) = 2X1 >= 2X1 = U11(X1,X2,X3) a__U12(X1,X2,X3) = X1 >= X1 = U12(X1,X2,X3) a__isNatKind(X) = 0 >= 0 = isNatKind(X) a__U13(X1,X2,X3) = X1 >= X1 = U13(X1,X2,X3) a__U14(X1,X2,X3) = X1 >= X1 = U14(X1,X2,X3) a__U15(X1,X2) = X1 >= X1 = U15(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__U16(X) = 5X >= 5X = U16(X) a__U21(X1,X2) = X1 >= X1 = U21(X1,X2) a__U22(X1,X2) = X1 >= X1 = U22(X1,X2) a__U23(X) = 3X >= 3X = U23(X) a__U31(X1,X2) = X1 >= X1 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X) = 4X >= 4X = U41(X) a__U51(X1,X2) = 4X1 + 3X2 >= 4X1 + 2X2 = U51(X1,X2) a__U52(X1,X2) = 4X1 + 3X2 >= 4X1 + 2X2 = U52(X1,X2) a__U61(X1,X2,X3) = 7X1 + 2X3 + 6 >= 7X1 + X3 + 3 = U61(X1,X2,X3) a__U62(X1,X2,X3) = 2X3 + 6 >= X3 + 5 = U62(X1,X2,X3) a__U63(X1,X2,X3) = 4 >= 2 = U63(X1,X2,X3) a__U64(X1,X2,X3) = 4 >= 3 = U64(X1,X2,X3) a__plus(X1,X2) = 3X1 + 4X2 >= 3X1 + 4X2 = plus(X1,X2) problem: DPs: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U15(X1,X2)) -> mark#(X1) mark#(U16(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> mark#(X1) mark#(U23(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U41(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Restore Modifier: DPs: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U15(X1,X2)) -> mark#(X1) mark#(U16(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> mark#(X1) mark#(U23(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U41(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) SCC Processor: #sccs: 1 #rules: 18 #arcs: 260/361 DPs: mark#(U52(X1,X2)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U13(X1,X2,X3)) -> mark#(X1) mark#(U14(X1,X2,X3)) -> mark#(X1) mark#(U15(X1,X2)) -> mark#(X1) mark#(U16(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X1,X2)) -> mark#(X1) mark#(U23(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U41(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(a__U51#) = 1 pi(a__U52#) = 1 pi(mark#) = 0 problem: DPs: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N) a__U52#(tt(),N) -> mark#(N) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 243/4 DPs: a__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2) a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2) a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2) a__U14#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1) a__U22#(tt(),V1) -> a__isNat#(V1) a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2) a__U15#(tt(),V2) -> a__isNat#(V2) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Size-Change Termination Processor: DPs: TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) The DP: a__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2) has the edges: 1 >= 1 2 >= 2 The DP: a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2) has the edges: 1 >= 1 2 >= 2 The DP: a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2) has the edges: 1 >= 1 2 >= 2 The DP: a__U14#(tt(),V1,V2) -> a__isNat#(V1) has the edges: 1 >= 0 The DP: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2) has the edges: 0 > 2 0 > 1 The DP: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) has the edges: 0 > 1 The DP: a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1) has the edges: 1 >= 1 The DP: a__U22#(tt(),V1) -> a__isNat#(V1) has the edges: 1 >= 0 The DP: a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2) has the edges: 2 >= 1 The DP: a__U15#(tt(),V2) -> a__isNat#(V2) has the edges: 1 >= 0 Qed DPs: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2) a__U31#(tt(),V2) -> a__isNatKind#(V2) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(a__isNatKind#) = 0 pi(a__U31#) = 1 problem: DPs: a__U31#(tt(),V2) -> a__isNatKind#(V2) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2) a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2) a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2) a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2) a__U15(tt(),V2) -> a__U16(a__isNat(V2)) a__U16(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1) a__U22(tt(),V1) -> a__U23(a__isNat(V1)) a__U23(tt()) -> tt() a__U31(tt(),V2) -> a__U32(a__isNatKind(V2)) a__U32(tt()) -> tt() a__U41(tt()) -> tt() a__U51(tt(),N) -> a__U52(a__isNatKind(N),N) a__U52(tt(),N) -> mark(N) a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N) a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N) a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N) a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2) a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) a__plus(N,0()) -> a__U51(a__isNat(N),N) a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3) mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3) mark(U15(X1,X2)) -> a__U15(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X)) -> a__U41(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3) mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3) mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__isNatKind(X) -> isNatKind(X) a__U13(X1,X2,X3) -> U13(X1,X2,X3) a__U14(X1,X2,X3) -> U14(X1,X2,X3) a__U15(X1,X2) -> U15(X1,X2) a__isNat(X) -> isNat(X) a__U16(X) -> U16(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X1,X2) -> U22(X1,X2) a__U23(X) -> U23(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X) -> U41(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X1,X2) -> U52(X1,X2) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2,X3) -> U62(X1,X2,X3) a__U63(X1,X2,X3) -> U63(X1,X2,X3) a__U64(X1,X2,X3) -> U64(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 10/1