YES Problem: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) Proof: DP Processor: DPs: a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) -> a__isNat#(V2) a__U12#(tt(),V2) -> a__U13#(a__isNat(V2)) a__U21#(tt(),V1) -> a__isNat#(V1) a__U21#(tt(),V1) -> a__U22#(a__isNat(V1)) a__U31#(tt(),N) -> mark#(N) a__U41#(tt(),M,N) -> mark#(M) a__U41#(tt(),M,N) -> mark#(N) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__and#(tt(),X) -> mark#(X) a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),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__and#(a__isNatKind(V1),isNatKind(V2)) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M)) a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> a__U13#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> a__U22#(mark(X)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(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#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) TDG Processor: DPs: a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) -> a__isNat#(V2) a__U12#(tt(),V2) -> a__U13#(a__isNat(V2)) a__U21#(tt(),V1) -> a__isNat#(V1) a__U21#(tt(),V1) -> a__U22#(a__isNat(V1)) a__U31#(tt(),N) -> mark#(N) a__U41#(tt(),M,N) -> mark#(M) a__U41#(tt(),M,N) -> mark#(N) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__and#(tt(),X) -> mark#(X) a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),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__and#(a__isNatKind(V1),isNatKind(V2)) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M)) a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> a__U13#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> a__U22#(mark(X)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(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#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) graph: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) a__and#(tt(),X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) a__and#(tt(),X) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__and#(tt(),X) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U22(X)) -> a__U22#(mark(X)) a__and#(tt(),X) -> mark#(X) -> mark#(U22(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U13(X)) -> a__U13#(mark(X)) a__and#(tt(),X) -> mark#(X) -> mark#(U13(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(U12(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__and#(tt(),X) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> mark#(X1) a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) -> a__and#(tt(),X) -> mark#(X) a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) -> a__U31#(tt(),N) -> mark#(N) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) -> a__and#(tt(),X) -> mark#(X) a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M)) -> a__and#(tt(),X) -> mark#(X) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) -> a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) -> a__U41#(tt(),M,N) -> mark#(N) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) -> a__U41#(tt(),M,N) -> mark#(M) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M)) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__isNat#(M) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,0()) -> a__isNat#(N) a__U41#(tt(),M,N) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> mark#(N) -> mark#(isNatKind(X)) -> a__isNatKind#(X) a__U41#(tt(),M,N) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U41#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U31(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U22(X)) -> a__U22#(mark(X)) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U22(X)) -> mark#(X) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U21(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U13(X)) -> a__U13#(mark(X)) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U13(X)) -> mark#(X) a__U41#(tt(),M,N) -> mark#(N) -> mark#(isNat(X)) -> a__isNat#(X) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U12(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> mark#(M) -> mark#(isNatKind(X)) -> a__isNatKind#(X) a__U41#(tt(),M,N) -> mark#(M) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(M) -> mark#(and(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U41#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X2) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U31(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U22(X)) -> a__U22#(mark(X)) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U22(X)) -> mark#(X) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U21(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U13(X)) -> a__U13#(mark(X)) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U13(X)) -> mark#(X) a__U41#(tt(),M,N) -> mark#(M) -> mark#(isNat(X)) -> a__isNat#(X) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U12(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) -> a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) -> a__U41#(tt(),M,N) -> mark#(N) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) -> a__U41#(tt(),M,N) -> mark#(M) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U13(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U13(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) -> a__U31#(tt(),N) -> mark#(N) mark#(U22(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U22(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U22(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U22(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U22(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U22(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(U22(X)) -> mark#(X) -> mark#(U13(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U22(X)) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U13(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) -> a__U21#(tt(),V1) -> a__U22#(a__isNat(V1)) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) -> a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U13(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U13(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U13(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U13(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U13(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U13(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U13(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U13(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U13(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U13(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U13(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U13(X)) -> mark#(X) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U13(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(U13(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U13(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U13(X)) -> mark#(X) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(U13(X)) -> mark#(X) -> mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U13(X)) -> mark#(X) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U13(X)) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U13(X)) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U13(X)) -> mark#(X) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) -> a__U12#(tt(),V2) -> a__U13#(a__isNat(V2)) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) -> a__U12#(tt(),V2) -> a__isNat#(V2) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U13(X)) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) -> a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) -> a__U11#(tt(),V1,V2) -> a__isNat#(V1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) -> a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U13(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) mark#(isNatKind(X)) -> a__isNatKind#(X) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) mark#(isNatKind(X)) -> a__isNatKind#(X) -> a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) mark#(isNatKind(X)) -> a__isNatKind#(X) -> a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M)) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__isNat#(M) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,0()) -> a__isNat#(N) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U22(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U13(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> a__isNat#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U13(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U13(X)) -> a__U13#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U13(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> mark#(X1) a__U31#(tt(),N) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__U31#(tt(),N) -> mark#(N) -> mark#(isNatKind(X)) -> a__isNatKind#(X) a__U31#(tt(),N) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__U31#(tt(),N) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U31#(tt(),N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__U31#(tt(),N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__U31#(tt(),N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U31#(tt(),N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__U31#(tt(),N) -> mark#(N) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) -> mark#(U31(X1,X2)) -> mark#(X1) a__U31#(tt(),N) -> mark#(N) -> mark#(U22(X)) -> a__U22#(mark(X)) a__U31#(tt(),N) -> mark#(N) -> mark#(U22(X)) -> mark#(X) a__U31#(tt(),N) -> mark#(N) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) -> mark#(U21(X1,X2)) -> mark#(X1) a__U31#(tt(),N) -> mark#(N) -> mark#(U13(X)) -> a__U13#(mark(X)) a__U31#(tt(),N) -> mark#(N) -> mark#(U13(X)) -> mark#(X) a__U31#(tt(),N) -> mark#(N) -> mark#(isNat(X)) -> a__isNat#(X) a__U31#(tt(),N) -> mark#(N) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) -> mark#(U12(X1,X2)) -> mark#(X1) a__U31#(tt(),N) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U31#(tt(),N) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> mark#(X1) a__U21#(tt(),V1) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U21#(tt(),V1) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U21#(tt(),V1) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__U21#(tt(),V1) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) a__U12#(tt(),V2) -> a__isNat#(V2) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U12#(tt(),V2) -> a__isNat#(V2) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U12#(tt(),V2) -> a__isNat#(V2) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U12#(tt(),V2) -> a__isNat#(V2) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__U12#(tt(),V2) -> a__isNat#(V2) -> a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) -> a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) -> a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) -> a__U21#(tt(),V1) -> a__U22#(a__isNat(V1)) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) -> a__U21#(tt(),V1) -> a__isNat#(V1) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) -> a__U12#(tt(),V2) -> a__U13#(a__isNat(V2)) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) -> a__U12#(tt(),V2) -> a__isNat#(V2) a__U11#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U11#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U11#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__U11#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) SCC Processor: #sccs: 1 #rules: 44 #arcs: 415/2304 DPs: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) -> a__isNat#(V2) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(M) 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,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M)) a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) a__U41#(tt(),M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) interpretation: [a__isNatKind#](x0) = x0, [a__and#](x0, x1) = x1, [a__plus#](x0, x1) = x0 + x1, [a__U41#](x0, x1, x2) = x0 + x1 + x2 + 0, [mark#](x0) = x0, [a__U31#](x0, x1) = x1, [a__U21#](x0, x1) = x0 + x1, [a__U12#](x0, x1) = x0 + x1, [a__isNat#](x0) = x0 + 0, [a__U11#](x0, x1, x2) = x0 + 1x1 + x2 + 0, [U41](x0, x1, x2) = x0 + x1 + 1x2 + 7, [U31](x0, x1) = x0 + x1, [U22](x0) = x0 + 0, [U21](x0, x1) = x0 + x1, [U13](x0) = x0 + 0, [U12](x0, x1) = x0 + x1, [U11](x0, x1, x2) = x0 + 1x1 + x2 + 0, [and](x0, x1) = x0 + x1 + 7, [isNat](x0) = x0 + 0, [isNatKind](x0) = x0, [a__isNatKind](x0) = x0, [plus](x0, x1) = 1x0 + x1 + 7, [0] = 3, [a__and](x0, x1) = x0 + x1 + 7, [s](x0) = x0 + 7, [a__plus](x0, x1) = 1x0 + x1 + 7, [a__U41](x0, x1, x2) = x0 + x1 + 1x2 + 7, [mark](x0) = x0, [a__U31](x0, x1) = x0 + x1, [a__U22](x0) = x0 + 0, [a__U21](x0, x1) = x0 + x1, [a__U13](x0) = x0 + 0, [a__U12](x0, x1) = x0 + x1, [a__isNat](x0) = x0 + 0, [a__U11](x0, x1, x2) = x0 + 1x1 + x2 + 0, [tt] = 0 orientation: a__isNatKind#(plus(V1,V2)) = 1V1 + V2 + 7 >= V1 = a__isNatKind#(V1) a__isNatKind#(plus(V1,V2)) = 1V1 + V2 + 7 >= V2 = a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2,X3)) = X1 + 1X2 + X3 + 0 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 + 1X2 + X3 + 0 >= X1 + 1X2 + X3 + 0 = a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) = 1V1 + V2 + 0 >= V1 + 0 = a__isNat#(V1) a__isNat#(plus(V1,V2)) = 1V1 + V2 + 7 >= V1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = V1 + 7 >= V1 = a__isNatKind#(V1) a__isNat#(plus(V1,V2)) = 1V1 + V2 + 7 >= V2 = a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) = 1V1 + V2 + 7 >= 1V1 + V2 + 7 = a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) = 1V1 + V2 + 0 >= V1 + V2 + 0 = a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) = V2 + 0 >= V2 + 0 = a__isNat#(V2) a__isNat#(s(V1)) = V1 + 7 >= V1 = a__isNatKind#(V1) a__isNat#(s(V1)) = V1 + 7 >= V1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = V1 + 0 >= V1 + 0 = a__isNat#(V1) mark#(U12(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 + X2 >= X1 + X2 = a__U12#(mark(X1),X2) mark#(isNat(X)) = X + 0 >= X + 0 = a__isNat#(X) mark#(U13(X)) = X + 0 >= X = mark#(X) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21#(mark(X1),X2) mark#(U22(X)) = X + 0 >= X = mark#(X) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X2 = a__U31#(mark(X1),X2) a__U31#(tt(),N) = N >= N = mark#(N) mark#(U41(X1,X2,X3)) = X1 + X2 + 1X3 + 7 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + 1X3 + 7 >= X1 + X2 + X3 + 0 = a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) = M + N + 0 >= M = mark#(M) mark#(plus(X1,X2)) = 1X1 + X2 + 7 >= X2 = mark#(X2) mark#(plus(X1,X2)) = 1X1 + X2 + 7 >= X1 = mark#(X1) mark#(plus(X1,X2)) = 1X1 + X2 + 7 >= X1 + X2 = a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) = N + 3 >= N + 0 = a__isNat#(N) a__plus#(N,0()) = N + 3 >= N = a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) = N + 3 >= N = a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) = M + N + 7 >= M + 0 = a__isNat#(M) a__plus#(N,s(M)) = M + N + 7 >= M = a__and#(a__isNat(M),isNatKind(M)) a__plus#(N,s(M)) = M + N + 7 >= N + 7 = a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) = M + N + 7 >= M + N + 7 = a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N) a__U41#(tt(),M,N) = M + N + 0 >= N = mark#(N) mark#(and(X1,X2)) = X1 + X2 + 7 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 7 >= X2 = a__and#(mark(X1),X2) mark#(isNatKind(X)) = X >= X = a__isNatKind#(X) mark#(s(X)) = X + 7 >= X = mark#(X) a__U41#(tt(),M,N) = M + N + 0 >= M + N = a__plus#(mark(N),mark(M)) a__U11(tt(),V1,V2) = 1V1 + V2 + 0 >= V1 + V2 + 0 = a__U12(a__isNat(V1),V2) a__U12(tt(),V2) = V2 + 0 >= V2 + 0 = a__U13(a__isNat(V2)) a__U13(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = V1 + 0 >= V1 + 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),N) = N + 0 >= N = mark(N) a__U41(tt(),M,N) = M + 1N + 7 >= M + 1N + 7 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X + 7 >= X = mark(X) a__isNat(0()) = 3 >= 0 = tt() a__isNat(plus(V1,V2)) = 1V1 + V2 + 7 >= 1V1 + V2 + 7 = a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) = V1 + 7 >= V1 = a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) = 3 >= 0 = tt() a__isNatKind(plus(V1,V2)) = 1V1 + V2 + 7 >= V1 + V2 + 7 = a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) = V1 + 7 >= V1 = a__isNatKind(V1) a__plus(N,0()) = 1N + 7 >= N + 7 = a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) = M + 1N + 7 >= M + 1N + 7 = a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N) mark(U11(X1,X2,X3)) = X1 + 1X2 + X3 + 0 >= X1 + 1X2 + X3 + 0 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) = X1 + X2 >= X1 + X2 = a__U12(mark(X1),X2) mark(isNat(X)) = X + 0 >= X + 0 = a__isNat(X) mark(U13(X)) = X + 0 >= X + 0 = a__U13(mark(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2) mark(U22(X)) = X + 0 >= X + 0 = a__U22(mark(X)) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) = X1 + X2 + 1X3 + 7 >= X1 + X2 + 1X3 + 7 = a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) = 1X1 + X2 + 7 >= 1X1 + X2 + 7 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 + 7 >= X1 + X2 + 7 = a__and(mark(X1),X2) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 7 >= X + 7 = s(mark(X)) mark(0()) = 3 >= 3 = 0() a__U11(X1,X2,X3) = X1 + 1X2 + X3 + 0 >= X1 + 1X2 + X3 + 0 = U11(X1,X2,X3) a__U12(X1,X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) a__isNat(X) = X + 0 >= X + 0 = isNat(X) a__U13(X) = X + 0 >= X + 0 = U13(X) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X) = X + 0 >= X + 0 = U22(X) a__U31(X1,X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) a__U41(X1,X2,X3) = X1 + X2 + 1X3 + 7 >= X1 + X2 + 1X3 + 7 = U41(X1,X2,X3) a__plus(X1,X2) = 1X1 + X2 + 7 >= 1X1 + X2 + 7 = plus(X1,X2) a__and(X1,X2) = X1 + X2 + 7 >= X1 + X2 + 7 = and(X1,X2) a__isNatKind(X) = X >= X = isNatKind(X) problem: DPs: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) -> a__isNat#(V2) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(M) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M)) a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) a__U41#(tt(),M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) Restore Modifier: DPs: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) -> a__isNat#(V2) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(M) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M)) a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) a__U41#(tt(),M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) interpretation: [a__isNatKind#](x0) = 1x0 + 6, [a__and#](x0, x1) = x0 + 1x1 + 0, [a__plus#](x0, x1) = 1x0 + 2x1 + 0, [a__U41#](x0, x1, x2) = 1x0 + 2x1 + 1x2 + 0, [mark#](x0) = 1x0 + 0, [a__U31#](x0, x1) = 1x1 + 6, [a__U21#](x0, x1) = 1x0 + 1x1 + 0, [a__U12#](x0, x1) = 2x1 + 0, [a__isNat#](x0) = 1x0 + 0, [a__U11#](x0, x1, x2) = x0 + 1x1 + 2x2 + 0, [U41](x0, x1, x2) = x0 + 2x1 + x2 + 5, [U31](x0, x1) = x0 + x1 + 5, [U22](x0) = x0 + 0, [U21](x0, x1) = x0 + x1, [U13](x0) = 1x0, [U12](x0, x1) = x0 + 1x1, [U11](x0, x1, x2) = x0 + x1 + 1x2 + 0, [and](x0, x1) = x0 + x1, [isNat](x0) = x0, [isNatKind](x0) = x0 + 5, [a__isNatKind](x0) = x0 + 5, [plus](x0, x1) = x0 + 2x1 + 5, [0] = 4, [a__and](x0, x1) = x0 + x1, [s](x0) = x0 + 5, [a__plus](x0, x1) = x0 + 2x1 + 5, [a__U41](x0, x1, x2) = x0 + 2x1 + x2 + 5, [mark](x0) = x0, [a__U31](x0, x1) = x0 + x1 + 5, [a__U22](x0) = x0 + 0, [a__U21](x0, x1) = x0 + x1, [a__U13](x0) = 1x0, [a__U12](x0, x1) = x0 + 1x1, [a__isNat](x0) = x0, [a__U11](x0, x1, x2) = x0 + x1 + 1x2 + 0, [tt] = 0 orientation: a__isNatKind#(plus(V1,V2)) = 1V1 + 3V2 + 6 >= V1 + 1V2 + 6 = a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) = 1X + 0 >= 1X + 0 = mark#(X) mark#(U11(X1,X2,X3)) = 1X1 + 1X2 + 2X3 + 1 >= 1X1 + 0 = mark#(X1) mark#(U11(X1,X2,X3)) = 1X1 + 1X2 + 2X3 + 1 >= X1 + 1X2 + 2X3 + 0 = a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) = 1V1 + 2V2 + 0 >= 1V1 + 0 = a__isNat#(V1) a__isNatKind#(s(V1)) = 1V1 + 6 >= 1V1 + 6 = a__isNatKind#(V1) a__isNat#(plus(V1,V2)) = 1V1 + 3V2 + 6 >= V1 + 1V2 + 6 = a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) = 1V1 + 3V2 + 6 >= 1V1 + 2V2 + 5 = a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) = 1V1 + 2V2 + 0 >= 2V2 + 0 = a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) = 2V2 + 0 >= 1V2 + 0 = a__isNat#(V2) a__isNat#(s(V1)) = 1V1 + 6 >= 1V1 + 6 = a__isNatKind#(V1) a__isNat#(s(V1)) = 1V1 + 6 >= 1V1 + 6 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 1V1 + 1 >= 1V1 + 0 = a__isNat#(V1) mark#(U12(X1,X2)) = 1X1 + 2X2 + 0 >= 1X1 + 0 = mark#(X1) mark#(U12(X1,X2)) = 1X1 + 2X2 + 0 >= 2X2 + 0 = a__U12#(mark(X1),X2) mark#(isNat(X)) = 1X + 0 >= 1X + 0 = a__isNat#(X) mark#(U13(X)) = 2X + 0 >= 1X + 0 = mark#(X) mark#(U21(X1,X2)) = 1X1 + 1X2 + 0 >= 1X1 + 0 = mark#(X1) mark#(U21(X1,X2)) = 1X1 + 1X2 + 0 >= 1X1 + 1X2 + 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = 1X + 1 >= 1X + 0 = mark#(X) mark#(U31(X1,X2)) = 1X1 + 1X2 + 6 >= 1X1 + 0 = mark#(X1) mark#(U31(X1,X2)) = 1X1 + 1X2 + 6 >= 1X2 + 6 = a__U31#(mark(X1),X2) a__U31#(tt(),N) = 1N + 6 >= 1N + 0 = mark#(N) mark#(U41(X1,X2,X3)) = 1X1 + 3X2 + 1X3 + 6 >= 1X1 + 0 = mark#(X1) mark#(U41(X1,X2,X3)) = 1X1 + 3X2 + 1X3 + 6 >= 1X1 + 2X2 + 1X3 + 0 = a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) = 2M + 1N + 1 >= 1M + 0 = mark#(M) mark#(plus(X1,X2)) = 1X1 + 3X2 + 6 >= 1X2 + 0 = mark#(X2) mark#(plus(X1,X2)) = 1X1 + 3X2 + 6 >= 1X1 + 2X2 + 0 = a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) = 1N + 6 >= 1N + 0 = a__isNat#(N) a__plus#(N,0()) = 1N + 6 >= 1N + 6 = a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) = 1N + 6 >= 1N + 6 = a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) = 2M + 1N + 7 >= 1M + 0 = a__isNat#(M) a__plus#(N,s(M)) = 2M + 1N + 7 >= 1M + 6 = a__and#(a__isNat(M),isNatKind(M)) a__plus#(N,s(M)) = 2M + 1N + 7 >= M + 1N + 6 = a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) = 2M + 1N + 7 >= 2M + 1N + 6 = a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N) a__U41#(tt(),M,N) = 2M + 1N + 1 >= 1N + 0 = mark#(N) mark#(and(X1,X2)) = 1X1 + 1X2 + 0 >= 1X1 + 0 = mark#(X1) mark#(and(X1,X2)) = 1X1 + 1X2 + 0 >= X1 + 1X2 + 0 = a__and#(mark(X1),X2) mark#(isNatKind(X)) = 1X + 6 >= 1X + 6 = a__isNatKind#(X) mark#(s(X)) = 1X + 6 >= 1X + 0 = mark#(X) a__U41#(tt(),M,N) = 2M + 1N + 1 >= 2M + 1N + 0 = a__plus#(mark(N),mark(M)) a__U11(tt(),V1,V2) = V1 + 1V2 + 0 >= V1 + 1V2 = a__U12(a__isNat(V1),V2) a__U12(tt(),V2) = 1V2 + 0 >= 1V2 = a__U13(a__isNat(V2)) a__U13(tt()) = 1 >= 0 = tt() a__U21(tt(),V1) = V1 + 0 >= V1 + 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),N) = N + 5 >= N = mark(N) a__U41(tt(),M,N) = 2M + N + 5 >= 2M + N + 5 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X + 0 >= X = mark(X) a__isNat(0()) = 4 >= 0 = tt() a__isNat(plus(V1,V2)) = V1 + 2V2 + 5 >= V1 + 1V2 + 5 = a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) = V1 + 5 >= V1 + 5 = a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) = 5 >= 0 = tt() a__isNatKind(plus(V1,V2)) = V1 + 2V2 + 5 >= V1 + V2 + 5 = a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) = V1 + 5 >= V1 + 5 = a__isNatKind(V1) a__plus(N,0()) = N + 6 >= N + 5 = a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) = 2M + N + 7 >= 2M + N + 5 = a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N) mark(U11(X1,X2,X3)) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) = X1 + 1X2 >= X1 + 1X2 = a__U12(mark(X1),X2) mark(isNat(X)) = X >= X = a__isNat(X) mark(U13(X)) = 1X >= 1X = a__U13(mark(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2) mark(U22(X)) = X + 0 >= X + 0 = a__U22(mark(X)) mark(U31(X1,X2)) = X1 + X2 + 5 >= X1 + X2 + 5 = a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) = X1 + 2X2 + X3 + 5 >= X1 + 2X2 + X3 + 5 = a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + 2X2 + 5 >= X1 + 2X2 + 5 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatKind(X)) = X + 5 >= X + 5 = a__isNatKind(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 5 >= X + 5 = s(mark(X)) mark(0()) = 4 >= 4 = 0() a__U11(X1,X2,X3) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = U11(X1,X2,X3) a__U12(X1,X2) = X1 + 1X2 >= X1 + 1X2 = U12(X1,X2) a__isNat(X) = X >= X = isNat(X) a__U13(X) = 1X >= 1X = U13(X) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X) = X + 0 >= X + 0 = U22(X) a__U31(X1,X2) = X1 + X2 + 5 >= X1 + X2 + 5 = U31(X1,X2) a__U41(X1,X2,X3) = X1 + 2X2 + X3 + 5 >= X1 + 2X2 + X3 + 5 = U41(X1,X2,X3) a__plus(X1,X2) = X1 + 2X2 + 5 >= X1 + 2X2 + 5 = plus(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatKind(X) = X + 5 >= X + 5 = isNatKind(X) problem: DPs: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) -> a__isNat#(V2) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) a__U41#(tt(),M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) Restore Modifier: DPs: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) -> a__isNat#(V2) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) a__U41#(tt(),M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) interpretation: [a__isNatKind#](x0) = 4x0, [a__and#](x0, x1) = x0 + 4x1, [a__plus#](x0, x1) = 4x0 + x1, [a__U41#](x0, x1, x2) = x1 + 4x2, [mark#](x0) = 4x0, [a__U31#](x0, x1) = x0 + 4x1, [a__U21#](x0, x1) = 2x0 + 4x1 + 4, [a__U12#](x0, x1) = x0 + 4x1, [a__isNat#](x0) = 4x0, [a__U11#](x0, x1, x2) = 4x1 + 4x2, [U41](x0, x1, x2) = x0 + x1 + 2x2 + 0, [U31](x0, x1) = 1x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1 + 0, [U13](x0) = x0, [U12](x0, x1) = x0 + x1, [U11](x0, x1, x2) = x0 + 2x1 + x2, [and](x0, x1) = x0 + x1, [isNat](x0) = x0, [isNatKind](x0) = x0, [a__isNatKind](x0) = x0, [plus](x0, x1) = 2x0 + x1, [0] = 0, [a__and](x0, x1) = x0 + x1, [s](x0) = x0 + 0, [a__plus](x0, x1) = 2x0 + x1, [a__U41](x0, x1, x2) = x0 + x1 + 2x2 + 0, [mark](x0) = x0, [a__U31](x0, x1) = 1x0 + x1, [a__U22](x0) = x0, [a__U21](x0, x1) = x0 + x1 + 0, [a__U13](x0) = x0, [a__U12](x0, x1) = x0 + x1, [a__isNat](x0) = x0, [a__U11](x0, x1, x2) = x0 + 2x1 + x2, [tt] = 0 orientation: a__isNatKind#(plus(V1,V2)) = 6V1 + 4V2 >= V1 + 4V2 = a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) = 4X + 0 >= 4X = mark#(X) mark#(U11(X1,X2,X3)) = 4X1 + 6X2 + 4X3 >= 4X1 = mark#(X1) mark#(U11(X1,X2,X3)) = 4X1 + 6X2 + 4X3 >= 4X2 + 4X3 = a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) = 4V1 + 4V2 >= 4V1 = a__isNat#(V1) a__isNatKind#(s(V1)) = 4V1 + 4 >= 4V1 = a__isNatKind#(V1) a__isNat#(plus(V1,V2)) = 6V1 + 4V2 >= V1 + 4V2 = a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) = 6V1 + 4V2 >= 4V1 + 4V2 = a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) = 4V1 + 4V2 >= V1 + 4V2 = a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) = 4V2 + 0 >= 4V2 = a__isNat#(V2) a__isNat#(s(V1)) = 4V1 + 4 >= 4V1 = a__isNatKind#(V1) a__isNat#(s(V1)) = 4V1 + 4 >= 4V1 + 4 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 4V1 + 4 >= 4V1 = a__isNat#(V1) mark#(U12(X1,X2)) = 4X1 + 4X2 >= 4X1 = mark#(X1) mark#(U12(X1,X2)) = 4X1 + 4X2 >= X1 + 4X2 = a__U12#(mark(X1),X2) mark#(isNat(X)) = 4X >= 4X = a__isNat#(X) mark#(U13(X)) = 4X >= 4X = mark#(X) mark#(U21(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 = mark#(X1) mark#(U21(X1,X2)) = 4X1 + 4X2 + 4 >= 2X1 + 4X2 + 4 = a__U21#(mark(X1),X2) mark#(U22(X)) = 4X >= 4X = mark#(X) mark#(U31(X1,X2)) = 5X1 + 4X2 >= 4X1 = mark#(X1) mark#(U31(X1,X2)) = 5X1 + 4X2 >= X1 + 4X2 = a__U31#(mark(X1),X2) a__U31#(tt(),N) = 4N + 0 >= 4N = mark#(N) mark#(U41(X1,X2,X3)) = 4X1 + 4X2 + 6X3 + 4 >= 4X1 = mark#(X1) mark#(U41(X1,X2,X3)) = 4X1 + 4X2 + 6X3 + 4 >= X2 + 4X3 = a__U41#(mark(X1),X2,X3) mark#(plus(X1,X2)) = 6X1 + 4X2 >= 4X1 + X2 = a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) = 4N + 0 >= 4N = a__isNat#(N) a__plus#(N,0()) = 4N + 0 >= 4N = a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) = 4N + 0 >= 4N = a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) = M + 4N + 0 >= M + 4N = a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) = M + 4N + 0 >= M + 4N = a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N) a__U41#(tt(),M,N) = M + 4N >= 4N = mark#(N) mark#(and(X1,X2)) = 4X1 + 4X2 >= 4X1 = mark#(X1) mark#(and(X1,X2)) = 4X1 + 4X2 >= X1 + 4X2 = a__and#(mark(X1),X2) mark#(isNatKind(X)) = 4X >= 4X = a__isNatKind#(X) mark#(s(X)) = 4X + 4 >= 4X = mark#(X) a__U41#(tt(),M,N) = M + 4N >= M + 4N = a__plus#(mark(N),mark(M)) a__U11(tt(),V1,V2) = 2V1 + V2 + 0 >= V1 + V2 = a__U12(a__isNat(V1),V2) a__U12(tt(),V2) = V2 + 0 >= V2 = a__U13(a__isNat(V2)) a__U13(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = V1 + 0 >= V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),N) = N + 1 >= N = mark(N) a__U41(tt(),M,N) = M + 2N + 0 >= M + 2N + 0 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X + 0 >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(plus(V1,V2)) = 2V1 + V2 >= 2V1 + V2 = a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) = V1 + 0 >= V1 + 0 = a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(plus(V1,V2)) = 2V1 + V2 >= V1 + V2 = a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) = V1 + 0 >= V1 = a__isNatKind(V1) a__plus(N,0()) = 2N + 0 >= 1N = a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) = M + 2N + 0 >= M + 2N + 0 = a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N) mark(U11(X1,X2,X3)) = X1 + 2X2 + X3 >= X1 + 2X2 + X3 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) = X1 + X2 >= X1 + X2 = a__U12(mark(X1),X2) mark(isNat(X)) = X >= X = a__isNat(X) mark(U13(X)) = X >= X = a__U13(mark(X)) mark(U21(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(U31(X1,X2)) = 1X1 + X2 >= 1X1 + X2 = a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) = X1 + X2 + 2X3 + 0 >= X1 + X2 + 2X3 + 0 = a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2,X3) = X1 + 2X2 + X3 >= X1 + 2X2 + X3 = U11(X1,X2,X3) a__U12(X1,X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) a__isNat(X) = X >= X = isNat(X) a__U13(X) = X >= X = U13(X) a__U21(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__U31(X1,X2) = 1X1 + X2 >= 1X1 + X2 = U31(X1,X2) a__U41(X1,X2,X3) = X1 + X2 + 2X3 + 0 >= X1 + X2 + 2X3 + 0 = U41(X1,X2,X3) a__plus(X1,X2) = 2X1 + X2 >= 2X1 + X2 = plus(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatKind(X) = X >= X = isNatKind(X) problem: DPs: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) -> a__isNat#(V2) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) mark#(U41(X1,X2,X3)) -> mark#(X1) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) a__U41#(tt(),M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) Restore Modifier: DPs: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) -> a__isNat#(V2) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) mark#(U41(X1,X2,X3)) -> mark#(X1) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__and#(a__isNat(N),isNatKind(N)) a__plus#(N,0()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N) a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) a__U41#(tt(),M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) SCC Processor: #sccs: 2 #rules: 29 #arcs: 381/1156 DPs: a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=3 usable rules: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) interpretation: [a__plus#](x0, x1) = [0 1 0]x0 + [1 0 1]x1, [a__U41#](x0, x1, x2) = [1 0 1]x1 + [0 1 0]x2, [1 0 1] [1 1 1] [1] [U41](x0, x1, x2) = [0 0 0]x1 + [0 0 0]x2 + [0] [1 0 1] [1 1 1] [1], [1 0 1] [U31](x0, x1) = [0 0 0]x1 [1 0 1] , [0 0 0] [U22](x0) = [0 0 0]x0 [0 1 0] , [0 1 0] [U21](x0, x1) = [0 0 0]x0 [0 0 0] , [0] [U13](x0) = [0] [0], [0] [U12](x0, x1) = [0] [0], [0] [U11](x0, x1, x2) = [0] [0], [1 0 1] [and](x0, x1) = [0 0 0]x1 [1 0 1] , [0 0 1] [isNat](x0) = [0 0 0]x0 [0 0 1] , [0] [isNatKind](x0) = [0] [0], [0] [a__isNatKind](x0) = [0] [0], [1 0 1] [1 0 1] [plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 0 1] [1 0 1] , [0] [0] = [0] [0], [1 0 1] [a__and](x0, x1) = [0 0 0]x1 [1 0 1] , [1 0 0] [1] [s](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 1 1] [1 0 1] [a__plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 1 1] [1 0 1] , [0 0 0] [1 0 1] [1 1 1] [1] [a__U41](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 1 0] [1 0 1] [1 1 1] [1], [0 0 1] [mark](x0) = [0 0 0]x0 [1 0 0] , [0 1 0] [1 0 1] [a__U31](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [1 0 1] , [0 0 0] [a__U22](x0) = [0 0 0]x0 [0 1 0] , [0 1 0] [a__U21](x0, x1) = [0 0 0]x0 [0 0 0] , [0 0 0] [a__U13](x0) = [0 0 0]x0 [0 1 0] , [0 0 0] [a__U12](x0, x1) = [0 1 0]x0 [0 0 0] , [0 0 1] [a__isNat](x0) = [0 0 0]x0 [0 0 1] , [0 1 0] [a__U11](x0, x1, x2) = [0 1 0]x0 [0 0 0] , [0] [tt] = [0] [0] orientation: a__plus#(N,s(M)) = [1 0 1]M + [0 1 0]N + [2] >= [1 0 1]M + [0 1 0]N = a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N) a__U41#(tt(),M,N) = [1 0 1]M + [0 1 0]N >= [1 0 1]M = a__plus#(mark(N),mark(M)) [0] [0] a__U11(tt(),V1,V2) = [0] >= [0] = a__U12(a__isNat(V1),V2) [0] [0] [0] [0] a__U12(tt(),V2) = [0] >= [0] = a__U13(a__isNat(V2)) [0] [0] [0] [0] a__U13(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] a__U21(tt(),V1) = [0] >= [0] = a__U22(a__isNat(V1)) [0] [0] [0] [0] a__U22(tt()) = [0] >= [0] = tt() [0] [0] [1 0 1] [0 0 1] a__U31(tt(),N) = [0 0 0]N >= [0 0 0]N = mark(N) [1 0 1] [1 0 0] [1 0 1] [1 1 1] [1] [1 0 1] [1 0 1] [1] a__U41(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 1 1] [1] [1 0 1] [1 0 1] [1] [1 0 1] [0 0 1] a__and(tt(),X) = [0 0 0]X >= [0 0 0]X = mark(X) [1 0 1] [1 0 0] [0] [0] a__isNat(0()) = [0] >= [0] = tt() [0] [0] [1 0 1] [1 0 1] [0] a__isNat(plus(V1,V2)) = [0 0 0]V1 + [0 0 0]V2 >= [0] = a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) [1 0 1] [1 0 1] [0] [0 0 1] [1] [0] a__isNat(s(V1)) = [0 0 0]V1 + [0] >= [0] = a__U21(a__isNatKind(V1),V1) [0 0 1] [1] [0] [0] [0] a__isNatKind(0()) = [0] >= [0] = tt() [0] [0] [0] [0] a__isNatKind(plus(V1,V2)) = [0] >= [0] = a__and(a__isNatKind(V1),isNatKind(V2)) [0] [0] [0] [0] a__isNatKind(s(V1)) = [0] >= [0] = a__isNatKind(V1) [0] [0] [1 1 1] [1 0 1] a__plus(N,0()) = [0 0 0]N >= [0 0 0]N = a__U31(a__and(a__isNat(N),isNatKind(N)),N) [1 1 1] [1 0 1] [1 0 1] [1 1 1] [2] [1 0 1] [1 1 1] [1] a__plus(N,s(M)) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N) [1 0 1] [1 1 1] [2] [1 0 1] [1 1 1] [1] [0] [0] mark(U11(X1,X2,X3)) = [0] >= [0] = a__U11(mark(X1),X2,X3) [0] [0] [0] [0] mark(U12(X1,X2)) = [0] >= [0] = a__U12(mark(X1),X2) [0] [0] [0 0 1] [0 0 1] mark(isNat(X)) = [0 0 0]X >= [0 0 0]X = a__isNat(X) [0 0 1] [0 0 1] [0] [0] mark(U13(X)) = [0] >= [0] = a__U13(mark(X)) [0] [0] [0 0 0] [0] mark(U21(X1,X2)) = [0 0 0]X1 >= [0] = a__U21(mark(X1),X2) [0 1 0] [0] [0 1 0] [0] mark(U22(X)) = [0 0 0]X >= [0] = a__U22(mark(X)) [0 0 0] [0] [1 0 1] [1 0 1] mark(U31(X1,X2)) = [0 0 0]X2 >= [0 0 0]X2 = a__U31(mark(X1),X2) [1 0 1] [1 0 1] [1 0 1] [1 1 1] [1] [1 0 1] [1 1 1] [1] mark(U41(X1,X2,X3)) = [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X2 + [0 0 0]X3 + [0] = a__U41(mark(X1),X2,X3) [1 0 1] [1 1 1] [1] [1 0 1] [1 1 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 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] mark(and(X1,X2)) = [0 0 0]X2 >= [0 0 0]X2 = a__and(mark(X1),X2) [1 0 1] [1 0 1] [0] [0] mark(isNatKind(X)) = [0] >= [0] = a__isNatKind(X) [0] [0] [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] [0 1 0] [0] a__U11(X1,X2,X3) = [0 1 0]X1 >= [0] = U11(X1,X2,X3) [0 0 0] [0] [0 0 0] [0] a__U12(X1,X2) = [0 1 0]X1 >= [0] = U12(X1,X2) [0 0 0] [0] [0 0 1] [0 0 1] a__isNat(X) = [0 0 0]X >= [0 0 0]X = isNat(X) [0 0 1] [0 0 1] [0 0 0] [0] a__U13(X) = [0 0 0]X >= [0] = U13(X) [0 1 0] [0] [0 1 0] [0 1 0] a__U21(X1,X2) = [0 0 0]X1 >= [0 0 0]X1 = U21(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U22(X) = [0 0 0]X >= [0 0 0]X = U22(X) [0 1 0] [0 1 0] [0 1 0] [1 0 1] [1 0 1] a__U31(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X2 = U31(X1,X2) [0 0 0] [1 0 1] [1 0 1] [0 0 0] [1 0 1] [1 1 1] [1] [1 0 1] [1 1 1] [1] a__U41(X1,X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X2 + [0 0 0]X3 + [0] = U41(X1,X2,X3) [0 1 0] [1 0 1] [1 1 1] [1] [1 0 1] [1 1 1] [1] [1 1 1] [1 0 1] [1 0 1] [1 0 1] a__plus(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,X2) [1 1 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] a__and(X1,X2) = [0 0 0]X2 >= [0 0 0]X2 = and(X1,X2) [1 0 1] [1 0 1] [0] [0] a__isNatKind(X) = [0] >= [0] = isNatKind(X) [0] [0] problem: DPs: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) Restore Modifier: DPs: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/1 DPs: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U11#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2) a__U12#(tt(),V2) -> a__isNat#(V2) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(U13(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) Subterm Criterion Processor: simple projection: pi(isNatKind) = [0,0] pi(U11) = [0,0,1,1,2] pi(U13) = [0,0] pi(a__U11#) = [1,2] pi(a__isNat#) = 0 pi(a__U12#) = 1 pi(a__U21#) = 1 pi(a__U31#) = 1 pi(mark#) = 0 pi(a__and#) = [1,1] pi(a__isNatKind#) = [0,0] problem: DPs: a__U12#(tt(),V2) -> a__isNat#(V2) a__U21#(tt(),V1) -> a__isNat#(V1) a__U31#(tt(),N) -> mark#(N) mark#(isNatKind(X)) -> a__isNatKind#(X) TRS: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2) a__U12(tt(),V2) -> a__U13(a__isNat(V2)) a__U13(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatKind(0()) -> tt() a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N) a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatKind(X)) -> a__isNatKind(X) 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) -> U12(X1,X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatKind(X) -> isNatKind(X) SCC Processor: #sccs: 0 #rules: 0 #arcs: 184/16