YES Problem: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Proof: DP Processor: DPs: a__U11#(tt(),V2) -> a__isNat#(V2) a__U11#(tt(),V2) -> a__U12#(a__isNat(V2)) a__U31#(tt(),N) -> mark#(N) a__U41#(tt(),M,N) -> a__isNat#(N) a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) -> mark#(M) a__U42#(tt(),M,N) -> mark#(N) a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__U31#(a__isNat(N),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> a__U12#(mark(X)) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> a__U21#(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#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) TDG Processor: DPs: a__U11#(tt(),V2) -> a__isNat#(V2) a__U11#(tt(),V2) -> a__U12#(a__isNat(V2)) a__U31#(tt(),N) -> mark#(N) a__U41#(tt(),M,N) -> a__isNat#(N) a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) -> mark#(M) a__U42#(tt(),M,N) -> mark#(N) a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__U31#(a__isNat(N),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> a__U12#(mark(X)) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> a__U21#(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#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) graph: a__plus#(N,0()) -> a__U31#(a__isNat(N),N) -> a__U31#(tt(),N) -> mark#(N) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) -> a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) -> a__U41#(tt(),M,N) -> a__isNat#(N) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__isNat#(M) a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,0()) -> a__U31#(a__isNat(N),N) a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,0()) -> a__isNat#(N) a__U42#(tt(),M,N) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__U42#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U42#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__U42#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U42(X1,X2,X3)) -> mark#(X1) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U31(X1,X2)) -> mark#(X1) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U21(X)) -> a__U21#(mark(X)) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U21(X)) -> mark#(X) a__U42#(tt(),M,N) -> mark#(N) -> mark#(isNat(X)) -> a__isNat#(X) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U12(X)) -> a__U12#(mark(X)) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U12(X)) -> mark#(X) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U42#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2)) -> mark#(X1) a__U42#(tt(),M,N) -> mark#(M) -> mark#(s(X)) -> mark#(X) a__U42#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U42#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X1) a__U42#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X2) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U42(X1,X2,X3)) -> mark#(X1) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U31(X1,X2)) -> mark#(X1) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U21(X)) -> a__U21#(mark(X)) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U21(X)) -> mark#(X) a__U42#(tt(),M,N) -> mark#(M) -> mark#(isNat(X)) -> a__isNat#(X) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U12(X)) -> a__U12#(mark(X)) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U12(X)) -> mark#(X) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U42#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) -> a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) -> a__U42#(tt(),M,N) -> mark#(N) a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) -> a__U42#(tt(),M,N) -> mark#(M) a__U41#(tt(),M,N) -> a__isNat#(N) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__U41#(tt(),M,N) -> a__isNat#(N) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__U41#(tt(),M,N) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) a__U41#(tt(),M,N) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) -> a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) -> a__U42#(tt(),M,N) -> mark#(N) mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) -> a__U42#(tt(),M,N) -> mark#(M) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) -> a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) -> a__U41#(tt(),M,N) -> a__isNat#(N) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) 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#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> mark#(X1) 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#(U21(X)) -> a__U21#(mark(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U21(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(X)) -> a__U12#(mark(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) 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#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> mark#(X1) 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#(U21(X)) -> a__U21#(mark(X)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) -> a__U31#(tt(),N) -> mark#(N) mark#(U21(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U21(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U21(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(U21(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U21(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U21(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U21(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X)) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(U21(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U21(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__isNat#(V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) mark#(U12(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U12(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U12(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(U12(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U12(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U12(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U12(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U12(X)) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(U12(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U12(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) -> a__U11#(tt(),V2) -> a__U12#(a__isNat(V2)) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) -> a__U11#(tt(),V2) -> a__isNat#(V2) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) 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__isNat(N),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#(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#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U42(X1,X2,X3)) -> mark#(X1) 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#(U21(X)) -> a__U21#(mark(X)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> a__isNat#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U12(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) 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#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> mark#(X1) 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#(U21(X)) -> a__U21#(mark(X)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) 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#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> mark#(X1) 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#(U21(X)) -> a__U21#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) a__U31#(tt(),N) -> mark#(N) -> mark#(s(X)) -> mark#(X) 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#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) a__U31#(tt(),N) -> mark#(N) -> mark#(U42(X1,X2,X3)) -> mark#(X1) 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#(U21(X)) -> a__U21#(mark(X)) a__U31#(tt(),N) -> mark#(N) -> mark#(U21(X)) -> mark#(X) a__U31#(tt(),N) -> mark#(N) -> mark#(isNat(X)) -> a__isNat#(X) a__U31#(tt(),N) -> mark#(N) -> mark#(U12(X)) -> a__U12#(mark(X)) a__U31#(tt(),N) -> mark#(N) -> mark#(U12(X)) -> mark#(X) a__U31#(tt(),N) -> mark#(N) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) -> mark#(U11(X1,X2)) -> mark#(X1) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) -> a__U11#(tt(),V2) -> a__U12#(a__isNat(V2)) a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) -> a__U11#(tt(),V2) -> a__isNat#(V2) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__U11#(tt(),V2) -> a__isNat#(V2) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__U11#(tt(),V2) -> a__isNat#(V2) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__U11#(tt(),V2) -> a__isNat#(V2) -> a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) a__U11#(tt(),V2) -> a__isNat#(V2) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) SCC Processor: #sccs: 2 #rules: 24 #arcs: 256/1089 DPs: a__plus#(N,0()) -> a__U31#(a__isNat(N),N) a__U31#(tt(),N) -> mark#(N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> 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) a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) -> mark#(M) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) a__U42#(tt(),M,N) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) mark#(s(X)) -> mark#(X) a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [a__plus#](x0, x1) = 2x0 + 2x1, [a__U42#](x0, x1, x2) = 2x0 + 2x1 + 2x2, [a__U41#](x0, x1, x2) = 3/2x0 + 2x1 + 2x2, [mark#](x0) = 2x0, [a__U31#](x0, x1) = 2x0 + 2x1, [U42](x0, x1, x2) = x0 + x1 + x2, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U21](x0) = 3x0, [isNat](x0) = 0, [U12](x0) = 2x0, [U11](x0, x1) = 2x0, [plus](x0, x1) = x0 + x1, [0] = 1/2, [s](x0) = x0, [a__plus](x0, x1) = x0 + x1, [a__U42](x0, x1, x2) = x0 + x1 + x2, [a__U41](x0, x1, x2) = x0 + x1 + x2, [mark](x0) = x0, [a__U31](x0, x1) = x0 + x1, [a__U21](x0) = 3x0, [a__U12](x0) = 2x0, [a__isNat](x0) = 0, [a__U11](x0, x1) = 2x0, [tt] = 0 orientation: a__plus#(N,0()) = 2N + 1 >= 2N = a__U31#(a__isNat(N),N) a__U31#(tt(),N) = 2N >= 2N = mark#(N) mark#(U11(X1,X2)) = 4X1 >= 2X1 = mark#(X1) mark#(U12(X)) = 4X >= 2X = mark#(X) mark#(U21(X)) = 6X >= 2X = mark#(X) mark#(U31(X1,X2)) = 2X1 + 2X2 >= 2X1 = mark#(X1) mark#(U31(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__U31#(mark(X1),X2) mark#(U41(X1,X2,X3)) = 2X1 + 2X2 + 2X3 >= 2X1 = mark#(X1) mark#(U41(X1,X2,X3)) = 2X1 + 2X2 + 2X3 >= 3/2X1 + 2X2 + 2X3 = a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) = 2M + 2N >= 2M + 2N = a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) = 2M + 2N >= 2M = mark#(M) mark#(U42(X1,X2,X3)) = 2X1 + 2X2 + 2X3 >= 2X1 = mark#(X1) mark#(U42(X1,X2,X3)) = 2X1 + 2X2 + 2X3 >= 2X1 + 2X2 + 2X3 = a__U42#(mark(X1),X2,X3) a__U42#(tt(),M,N) = 2M + 2N >= 2N = mark#(N) mark#(plus(X1,X2)) = 2X1 + 2X2 >= 2X2 = mark#(X2) mark#(plus(X1,X2)) = 2X1 + 2X2 >= 2X1 = mark#(X1) mark#(plus(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = 2M + 2N >= 2M + 2N = a__U41#(a__isNat(M),M,N) mark#(s(X)) = 2X >= 2X = mark#(X) a__U42#(tt(),M,N) = 2M + 2N >= 2M + 2N = a__plus#(mark(N),mark(M)) a__U11(tt(),V2) = 0 >= 0 = a__U12(a__isNat(V2)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt(),N) = N >= N = mark(N) a__U41(tt(),M,N) = M + N >= M + N = a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) = M + N >= M + N = s(a__plus(mark(N),mark(M))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNat(V1),V2) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNat(V1)) a__plus(N,0()) = N + 1/2 >= N = a__U31(a__isNat(N),N) a__plus(N,s(M)) = M + N >= M + N = a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) = 2X1 >= 2X1 = a__U11(mark(X1),X2) mark(U12(X)) = 2X >= 2X = a__U12(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U21(X)) = 3X >= 3X = a__U21(mark(X)) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 1/2 >= 1/2 = 0() a__U11(X1,X2) = 2X1 >= 2X1 = U11(X1,X2) a__U12(X) = 2X >= 2X = U12(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U21(X) = 3X >= 3X = U21(X) a__U31(X1,X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) a__U41(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) a__U42(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U42(X1,X2,X3) a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: a__U31#(tt(),N) -> mark#(N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> 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) a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) -> mark#(M) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) a__U42#(tt(),M,N) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) mark#(s(X)) -> mark#(X) a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [a__plus#](x0, x1) = x0 + x1 + 1, [a__U42#](x0, x1, x2) = 2x0 + x1 + x2, [a__U41#](x0, x1, x2) = 2x0 + x1 + x2, [mark#](x0) = x0, [a__U31#](x0, x1) = x1, [U42](x0, x1, x2) = 3x0 + x1 + x2, [U41](x0, x1, x2) = 3x0 + x1 + x2, [U31](x0, x1) = 2x0 + x1, [U21](x0) = x0, [isNat](x0) = 1/2, [U12](x0) = x0, [U11](x0, x1) = x0, [plus](x0, x1) = x0 + x1 + 3/2, [0] = 2, [s](x0) = x0, [a__plus](x0, x1) = x0 + x1 + 3/2, [a__U42](x0, x1, x2) = 3x0 + x1 + x2, [a__U41](x0, x1, x2) = 3x0 + x1 + x2, [mark](x0) = x0, [a__U31](x0, x1) = 2x0 + x1, [a__U21](x0) = x0, [a__U12](x0) = x0, [a__isNat](x0) = 1/2, [a__U11](x0, x1) = x0, [tt] = 1/2 orientation: a__U31#(tt(),N) = N >= N = mark#(N) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U12(X)) = X >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) mark#(U31(X1,X2)) = 2X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = 2X1 + X2 >= X2 = a__U31#(mark(X1),X2) mark#(U41(X1,X2,X3)) = 3X1 + X2 + X3 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = 3X1 + X2 + X3 >= 2X1 + X2 + X3 = a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) = M + N + 1 >= M = mark#(M) mark#(U42(X1,X2,X3)) = 3X1 + X2 + X3 >= X1 = mark#(X1) mark#(U42(X1,X2,X3)) = 3X1 + X2 + X3 >= 2X1 + X2 + X3 = a__U42#(mark(X1),X2,X3) a__U42#(tt(),M,N) = M + N + 1 >= N = mark#(N) mark#(plus(X1,X2)) = X1 + X2 + 3/2 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 + 3/2 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 3/2 >= X1 + X2 + 1 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = M + N + 1 >= M + N + 1 = a__U41#(a__isNat(M),M,N) mark#(s(X)) = X >= X = mark#(X) a__U42#(tt(),M,N) = M + N + 1 >= M + N + 1 = a__plus#(mark(N),mark(M)) a__U11(tt(),V2) = 1/2 >= 1/2 = a__U12(a__isNat(V2)) a__U12(tt()) = 1/2 >= 1/2 = tt() a__U21(tt()) = 1/2 >= 1/2 = tt() a__U31(tt(),N) = N + 1 >= N = mark(N) a__U41(tt(),M,N) = M + N + 3/2 >= M + N + 3/2 = a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) = M + N + 3/2 >= M + N + 3/2 = s(a__plus(mark(N),mark(M))) a__isNat(0()) = 1/2 >= 1/2 = tt() a__isNat(plus(V1,V2)) = 1/2 >= 1/2 = a__U11(a__isNat(V1),V2) a__isNat(s(V1)) = 1/2 >= 1/2 = a__U21(a__isNat(V1)) a__plus(N,0()) = N + 7/2 >= N + 1 = a__U31(a__isNat(N),N) a__plus(N,s(M)) = M + N + 3/2 >= M + N + 3/2 = a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) = X1 >= X1 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNat(X)) = 1/2 >= 1/2 = a__isNat(X) mark(U21(X)) = X >= X = a__U21(mark(X)) mark(U31(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) = 3X1 + X2 + X3 >= 3X1 + X2 + X3 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) = 3X1 + X2 + X3 >= 3X1 + X2 + X3 = a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 + 3/2 >= X1 + X2 + 3/2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 1/2 >= 1/2 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 2 >= 2 = 0() a__U11(X1,X2) = X1 >= X1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNat(X) = 1/2 >= 1/2 = isNat(X) a__U21(X) = X >= X = U21(X) a__U31(X1,X2) = 2X1 + X2 >= 2X1 + X2 = U31(X1,X2) a__U41(X1,X2,X3) = 3X1 + X2 + X3 >= 3X1 + X2 + X3 = U41(X1,X2,X3) a__U42(X1,X2,X3) = 3X1 + X2 + X3 >= 3X1 + X2 + X3 = U42(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 3/2 >= X1 + X2 + 3/2 = plus(X1,X2) problem: DPs: a__U31#(tt(),N) -> mark#(N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> 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) a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) mark#(s(X)) -> mark#(X) a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) SCC Processor: #sccs: 2 #rules: 12 #arcs: 170/196 DPs: mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) mark#(U21(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#(s(X)) -> mark#(X) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(a__U31#) = 1 pi(mark#) = 0 problem: DPs: a__U31#(tt(),N) -> mark#(N) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 65/1 DPs: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [a__plus#](x0, x1) = 4x1 + 5, [a__U42#](x0, x1, x2) = 4x1 + 5, [a__U41#](x0, x1, x2) = x0 + 4x1 + 6, [U42](x0, x1, x2) = 2x1 + x2 + 3, [U41](x0, x1, x2) = 2x1 + x2 + 4, [U31](x0, x1) = x1, [U21](x0) = x0, [isNat](x0) = 2, [U12](x0) = 0, [U11](x0, x1) = 0, [plus](x0, x1) = x0 + 2x1, [0] = 0, [s](x0) = x0 + 2, [a__plus](x0, x1) = x0 + 2x1, [a__U42](x0, x1, x2) = 2x1 + x2 + 3, [a__U41](x0, x1, x2) = 2x1 + x2 + 4, [mark](x0) = x0, [a__U31](x0, x1) = x1, [a__U21](x0) = x0, [a__U12](x0) = 0, [a__isNat](x0) = 2, [a__U11](x0, x1) = 0, [tt] = 0 orientation: a__plus#(N,s(M)) = 4M + 13 >= 4M + 8 = a__U41#(a__isNat(M),M,N) a__U41#(tt(),M,N) = 4M + 6 >= 4M + 5 = a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) = 4M + 5 >= 4M + 5 = a__plus#(mark(N),mark(M)) a__U11(tt(),V2) = 0 >= 0 = a__U12(a__isNat(V2)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt(),N) = N >= N = mark(N) a__U41(tt(),M,N) = 2M + N + 4 >= 2M + N + 3 = a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) = 2M + N + 3 >= 2M + N + 2 = s(a__plus(mark(N),mark(M))) a__isNat(0()) = 2 >= 0 = tt() a__isNat(plus(V1,V2)) = 2 >= 0 = a__U11(a__isNat(V1),V2) a__isNat(s(V1)) = 2 >= 2 = a__U21(a__isNat(V1)) a__plus(N,0()) = N >= N = a__U31(a__isNat(N),N) a__plus(N,s(M)) = 2M + N + 4 >= 2M + N + 4 = a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNat(X)) = 2 >= 2 = a__isNat(X) mark(U21(X)) = X >= X = a__U21(mark(X)) mark(U31(X1,X2)) = X2 >= X2 = a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) = 2X2 + X3 + 4 >= 2X2 + X3 + 4 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) = 2X2 + X3 + 3 >= 2X2 + X3 + 3 = a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 2 >= X + 2 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNat(X) = 2 >= 2 = isNat(X) a__U21(X) = X >= X = U21(X) a__U31(X1,X2) = X2 >= X2 = U31(X1,X2) a__U41(X1,X2,X3) = 2X2 + X3 + 4 >= 2X2 + X3 + 4 = U41(X1,X2,X3) a__U42(X1,X2,X3) = 2X2 + X3 + 3 >= 2X2 + X3 + 3 = U42(X1,X2,X3) a__plus(X1,X2) = X1 + 2X2 >= X1 + 2X2 = plus(X1,X2) problem: DPs: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 3/1 DPs: a__U11#(tt(),V2) -> a__isNat#(V2) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) a__isNat#(s(V1)) -> a__isNat#(V1) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(a__U11#) = 1 pi(a__isNat#) = 0 problem: DPs: a__U11#(tt(),V2) -> a__isNat#(V2) TRS: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 10/1