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) Arctic Interpretation Processor: dimension: 1 usable rules: 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) interpretation: [a__plus#](x0, x1) = x0 + x1 + 0, [a__U42#](x0, x1, x2) = x0 + x1 + x2 + 0, [a__U41#](x0, x1, x2) = x0 + x1 + x2 + 0, [mark#](x0) = x0 + 0, [a__U31#](x0, x1) = 1x0 + x1 + 0, [U42](x0, x1, x2) = x0 + x1 + x2 + 0, [U41](x0, x1, x2) = x0 + x1 + x2 + 0, [U31](x0, x1) = 1x0 + x1 + 1, [U21](x0) = x0, [isNat](x0) = 0, [U12](x0) = x0, [U11](x0, x1) = x0 + 0, [plus](x0, x1) = x0 + x1 + 0, [0] = 1, [s](x0) = x0 + 0, [a__plus](x0, x1) = x0 + x1 + 0, [a__U42](x0, x1, x2) = x0 + x1 + x2 + 0, [a__U41](x0, x1, x2) = x0 + x1 + x2 + 0, [mark](x0) = x0 + 0, [a__U31](x0, x1) = 1x0 + x1 + 1, [a__U21](x0) = x0 + 0, [a__U12](x0) = x0 + 0, [a__isNat](x0) = 0, [a__U11](x0, x1) = x0 + 0, [tt] = 0 orientation: a__plus#(N,0()) = N + 1 >= N + 1 = a__U31#(a__isNat(N),N) a__U31#(tt(),N) = N + 1 >= N + 0 = mark#(N) mark#(U11(X1,X2)) = X1 + 0 >= X1 + 0 = mark#(X1) mark#(U12(X)) = X + 0 >= X + 0 = mark#(X) mark#(U21(X)) = X + 0 >= X + 0 = mark#(X) mark#(U31(X1,X2)) = 1X1 + X2 + 1 >= X1 + 0 = mark#(X1) mark#(U31(X1,X2)) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = a__U31#(mark(X1),X2) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + 0 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) = M + N + 0 >= M + N + 0 = a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) = M + N + 0 >= M + 0 = mark#(M) mark#(U42(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + 0 = mark#(X1) mark#(U42(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__U42#(mark(X1),X2,X3) a__U42#(tt(),M,N) = M + N + 0 >= N + 0 = mark#(N) mark#(plus(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = M + N + 0 >= M + N + 0 = a__U41#(a__isNat(M),M,N) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__U42#(tt(),M,N) = M + N + 0 >= M + N + 0 = 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 + 1 >= N + 0 = mark(N) a__U41(tt(),M,N) = M + N + 0 >= M + N + 0 = a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) = M + N + 0 >= M + N + 0 = 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 >= N + 1 = a__U31(a__isNat(N),N) a__plus(N,s(M)) = M + N + 0 >= M + N + 0 = a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) = X1 + 0 >= X1 + 0 = a__U11(mark(X1),X2) mark(U12(X)) = X + 0 >= X + 0 = a__U12(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U21(X)) = X + 0 >= X + 0 = a__U21(mark(X)) mark(U31(X1,X2)) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 1 >= 1 = 0() a__U11(X1,X2) = X1 + 0 >= X1 + 0 = U11(X1,X2) a__U12(X) = X + 0 >= X = U12(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U21(X) = X + 0 >= X = U21(X) a__U31(X1,X2) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = U31(X1,X2) a__U41(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = U41(X1,X2,X3) a__U42(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = U42(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = plus(X1,X2) problem: 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)) -> 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) Restore Modifier: 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)) -> 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=3 usable rules: 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) interpretation: [a__plus#](x0, x1) = [0 1 0]x0 + [0 1 0]x1, [a__U42#](x0, x1, x2) = [0 1 1]x1 + [0 1 0]x2, [a__U41#](x0, x1, x2) = [0 1 1]x1 + [0 1 0]x2, [mark#](x0) = [0 1 0]x0, [a__U31#](x0, x1) = [0 1 0]x1, [0 0 0] [0 0 0] [0 0 0] [U42](x0, x1, x2) = [0 1 0]x0 + [0 1 1]x1 + [0 1 0]x2 [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [0 0 0] [0 0 0] [U41](x0, x1, x2) = [1 1 0]x0 + [0 1 1]x1 + [0 1 0]x2 [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [0 0 0] [U31](x0, x1) = [0 1 1]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0 0 0] [U21](x0) = [0 1 0]x0 [0 0 0] , [0] [isNat](x0) = [1] [0], [0 0 0] [U12](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [U11](x0, x1) = [0 1 0]x0 [0 0 0] , [0 0 0] [0 0 0] [0] [plus](x0, x1) = [0 1 1]x0 + [0 1 0]x1 + [1] [0 0 0] [1 0 0] [0], [0] [0] = [0] [0], [1 0 1] [s](x0) = [0 1 1]x0 [0 0 0] , [0 0 0] [0 0 0] [0] [a__plus](x0, x1) = [0 1 1]x0 + [0 1 0]x1 + [1] [0 0 0] [1 0 0] [0], [0 0 0] [0 0 0] [0 0 0] [a__U42](x0, x1, x2) = [0 1 0]x0 + [0 1 1]x1 + [0 1 0]x2 [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [0 0 0] [0 0 0] [a__U41](x0, x1, x2) = [1 1 1]x0 + [0 1 1]x1 + [0 1 0]x2 [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [0 0 0] [a__U31](x0, x1) = [0 1 1]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0 0 0] [a__U21](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [a__U12](x0) = [0 1 0]x0 [0 0 0] , [0] [a__isNat](x0) = [1] [0], [0 0 0] [a__U11](x0, x1) = [0 1 0]x0 [0 0 0] , [0] [tt] = [1] [0] orientation: a__plus#(N,0()) = [0 1 0]N >= [0 1 0]N = a__U31#(a__isNat(N),N) a__U31#(tt(),N) = [0 1 0]N >= [0 1 0]N = mark#(N) mark#(U11(X1,X2)) = [0 1 0]X1 >= [0 1 0]X1 = mark#(X1) mark#(U12(X)) = [0 1 0]X >= [0 1 0]X = mark#(X) mark#(U21(X)) = [0 1 0]X >= [0 1 0]X = mark#(X) mark#(U31(X1,X2)) = [0 1 1]X1 + [0 1 0]X2 >= [0 1 0]X2 = a__U31#(mark(X1),X2) mark#(U41(X1,X2,X3)) = [1 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 >= [0 1 0]X1 = mark#(X1) mark#(U41(X1,X2,X3)) = [1 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 >= [0 1 1]X2 + [0 1 0]X3 = a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) = [0 1 1]M + [0 1 0]N >= [0 1 1]M + [0 1 0]N = a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) = [0 1 1]M + [0 1 0]N >= [0 1 0]M = mark#(M) mark#(U42(X1,X2,X3)) = [0 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 >= [0 1 0]X1 = mark#(X1) mark#(U42(X1,X2,X3)) = [0 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 >= [0 1 1]X2 + [0 1 0]X3 = a__U42#(mark(X1),X2,X3) a__U42#(tt(),M,N) = [0 1 1]M + [0 1 0]N >= [0 1 0]N = mark#(N) mark#(plus(X1,X2)) = [0 1 1]X1 + [0 1 0]X2 + [1] >= [0 1 0]X2 = mark#(X2) mark#(plus(X1,X2)) = [0 1 1]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 = mark#(X1) mark#(plus(X1,X2)) = [0 1 1]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = [0 1 1]M + [0 1 0]N >= [0 1 1]M + [0 1 0]N = a__U41#(a__isNat(M),M,N) mark#(s(X)) = [0 1 1]X >= [0 1 0]X = mark#(X) a__U42#(tt(),M,N) = [0 1 1]M + [0 1 0]N >= [0 1 0]M + [0 1 0]N = a__plus#(mark(N),mark(M)) [0] [0] a__U11(tt(),V2) = [1] >= [1] = a__U12(a__isNat(V2)) [0] [0] [0] [0] a__U12(tt()) = [1] >= [1] = tt() [0] [0] [0] [0] a__U21(tt()) = [1] >= [1] = tt() [0] [0] [0 0 0] [0] [0 0 0] a__U31(tt(),N) = [0 1 0]N + [1] >= [0 1 0]N = mark(N) [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] a__U41(tt(),M,N) = [0 1 1]M + [0 1 0]N + [1] >= [0 1 1]M + [0 1 0]N + [1] = a__U42(a__isNat(N),M,N) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] a__U42(tt(),M,N) = [0 1 1]M + [0 1 0]N + [1] >= [0 1 0]M + [0 1 0]N + [1] = s(a__plus(mark(N),mark(M))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] a__isNat(0()) = [1] >= [1] = tt() [0] [0] [0] [0] a__isNat(plus(V1,V2)) = [1] >= [1] = a__U11(a__isNat(V1),V2) [0] [0] [0] [0] a__isNat(s(V1)) = [1] >= [1] = a__U21(a__isNat(V1)) [0] [0] [0 0 0] [0] [0 0 0] [0] a__plus(N,0()) = [0 1 1]N + [1] >= [0 1 0]N + [1] = a__U31(a__isNat(N),N) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] a__plus(N,s(M)) = [0 1 1]M + [0 1 1]N + [1] >= [0 1 1]M + [0 1 0]N + [1] = a__U41(a__isNat(M),M,N) [1 0 1] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] mark(U11(X1,X2)) = [0 1 0]X1 >= [0 1 0]X1 = a__U11(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U12(X)) = [0 1 0]X >= [0 1 0]X = a__U12(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(isNat(X)) = [1] >= [1] = a__isNat(X) [0] [0] [0 0 0] [0 0 0] mark(U21(X)) = [0 1 0]X >= [0 1 0]X = a__U21(mark(X)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U31(X1,X2)) = [0 1 1]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = a__U31(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U41(X1,X2,X3)) = [1 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 = a__U41(mark(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U42(X1,X2,X3)) = [0 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 = a__U42(mark(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] mark(plus(X1,X2)) = [0 1 1]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = a__plus(mark(X1),mark(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(tt()) = [1] >= [1] = tt() [0] [0] [0 0 0] [0 0 0] mark(s(X)) = [0 1 1]X >= [0 1 0]X = s(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [0 0 0] [0 0 0] a__U11(X1,X2) = [0 1 0]X1 >= [0 1 0]X1 = U11(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U12(X) = [0 1 0]X >= [0 1 0]X = U12(X) [0 0 0] [0 0 0] [0] [0] a__isNat(X) = [1] >= [1] = isNat(X) [0] [0] [0 0 0] [0 0 0] a__U21(X) = [0 1 0]X >= [0 1 0]X = U21(X) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U31(X1,X2) = [0 1 1]X1 + [0 1 0]X2 >= [0 1 1]X1 + [0 1 0]X2 = U31(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U41(X1,X2,X3) = [1 1 1]X1 + [0 1 1]X2 + [0 1 0]X3 >= [1 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 = U41(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U42(X1,X2,X3) = [0 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 1]X2 + [0 1 0]X3 = U42(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] a__plus(X1,X2) = [0 1 1]X1 + [0 1 0]X2 + [1] >= [0 1 1]X1 + [0 1 0]X2 + [1] = plus(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] problem: 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)) -> 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) 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) Restore Modifier: 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)) -> 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) 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) Arctic Interpretation Processor: dimension: 1 usable rules: 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) interpretation: [a__plus#](x0, x1) = 7x0 + 1x1 + 0, [a__U42#](x0, x1, x2) = x0 + 1x1 + 7x2, [a__U41#](x0, x1, x2) = x0 + 1x1 + 7x2, [mark#](x0) = x0, [a__U31#](x0, x1) = x1 + 0, [U42](x0, x1, x2) = x0 + 1x1 + 7x2 + 0, [U41](x0, x1, x2) = 1x0 + 1x1 + 7x2, [U31](x0, x1) = 7x1 + 0, [U21](x0) = x0, [isNat](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = 2x0 + x1, [plus](x0, x1) = 7x0 + 1x1, [0] = 1, [s](x0) = x0, [a__plus](x0, x1) = 7x0 + 1x1, [a__U42](x0, x1, x2) = x0 + 1x1 + 7x2 + 0, [a__U41](x0, x1, x2) = 1x0 + 1x1 + 7x2, [mark](x0) = x0, [a__U31](x0, x1) = 7x1 + 0, [a__U21](x0) = x0, [a__U12](x0) = x0, [a__isNat](x0) = x0, [a__U11](x0, x1) = 2x0 + x1, [tt] = 0 orientation: a__plus#(N,0()) = 7N + 2 >= N + 0 = a__U31#(a__isNat(N),N) a__U31#(tt(),N) = N + 0 >= N = mark#(N) mark#(U11(X1,X2)) = 2X1 + X2 >= X1 = mark#(X1) mark#(U12(X)) = X >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) mark#(U31(X1,X2)) = 7X2 + 0 >= X2 + 0 = a__U31#(mark(X1),X2) mark#(U41(X1,X2,X3)) = 1X1 + 1X2 + 7X3 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = 1X1 + 1X2 + 7X3 >= X1 + 1X2 + 7X3 = a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) = 1M + 7N + 0 >= 1M + 7N = a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) = 1M + 7N + 0 >= M = mark#(M) mark#(U42(X1,X2,X3)) = X1 + 1X2 + 7X3 + 0 >= X1 = mark#(X1) mark#(U42(X1,X2,X3)) = X1 + 1X2 + 7X3 + 0 >= X1 + 1X2 + 7X3 = a__U42#(mark(X1),X2,X3) a__U42#(tt(),M,N) = 1M + 7N + 0 >= N = mark#(N) a__plus#(N,s(M)) = 1M + 7N + 0 >= 1M + 7N = a__U41#(a__isNat(M),M,N) mark#(s(X)) = X >= X = mark#(X) a__U42#(tt(),M,N) = 1M + 7N + 0 >= 1M + 7N + 0 = a__plus#(mark(N),mark(M)) a__U11(tt(),V2) = V2 + 2 >= V2 = a__U12(a__isNat(V2)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt(),N) = 7N + 0 >= N = mark(N) a__U41(tt(),M,N) = 1M + 7N + 1 >= 1M + 7N + 0 = a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) = 1M + 7N + 0 >= 1M + 7N = s(a__plus(mark(N),mark(M))) a__isNat(0()) = 1 >= 0 = tt() a__isNat(plus(V1,V2)) = 7V1 + 1V2 >= 2V1 + V2 = a__U11(a__isNat(V1),V2) a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNat(V1)) a__plus(N,0()) = 7N + 2 >= 7N + 0 = a__U31(a__isNat(N),N) a__plus(N,s(M)) = 1M + 7N >= 1M + 7N = a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U21(X)) = X >= X = a__U21(mark(X)) mark(U31(X1,X2)) = 7X2 + 0 >= 7X2 + 0 = a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) = 1X1 + 1X2 + 7X3 >= 1X1 + 1X2 + 7X3 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) = X1 + 1X2 + 7X3 + 0 >= X1 + 1X2 + 7X3 + 0 = a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) = 7X1 + 1X2 >= 7X1 + 1X2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 1 >= 1 = 0() a__U11(X1,X2) = 2X1 + X2 >= 2X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNat(X) = X >= X = isNat(X) a__U21(X) = X >= X = U21(X) a__U31(X1,X2) = 7X2 + 0 >= 7X2 + 0 = U31(X1,X2) a__U41(X1,X2,X3) = 1X1 + 1X2 + 7X3 >= 1X1 + 1X2 + 7X3 = U41(X1,X2,X3) a__U42(X1,X2,X3) = X1 + 1X2 + 7X3 + 0 >= X1 + 1X2 + 7X3 + 0 = U42(X1,X2,X3) a__plus(X1,X2) = 7X1 + 1X2 >= 7X1 + 1X2 = plus(X1,X2) problem: DPs: a__U31#(tt(),N) -> mark#(N) mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) 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) Restore Modifier: DPs: a__U31#(tt(),N) -> mark#(N) mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) 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: 9 #arcs: 170/121 DPs: mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) a__U31#(tt(),N) -> mark#(N) 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: 26/1 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 usable rules: 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) interpretation: [a__plus#](x0, x1) = 5x1 + 2, [a__U42#](x0, x1, x2) = 2x0 + 5x1 + 5, [a__U41#](x0, x1, x2) = 4x0 + 5x1 + 1, [U42](x0, x1, x2) = x0 + 2x1 + 2x2, [U41](x0, x1, x2) = x0 + 2x1 + 2x2 + 2, [U31](x0, x1) = 2x1 + 3, [U21](x0) = 3, [isNat](x0) = 4, [U12](x0) = x0, [U11](x0, x1) = 4, [plus](x0, x1) = 2x0 + 2x1, [0] = 4, [s](x0) = x0 + 3, [a__plus](x0, x1) = 2x0 + 2x1, [a__U42](x0, x1, x2) = x0 + 2x1 + 2x2, [a__U41](x0, x1, x2) = x0 + 2x1 + 2x2 + 2, [mark](x0) = x0, [a__U31](x0, x1) = 2x1 + 3, [a__U21](x0) = 3, [a__U12](x0) = x0, [a__isNat](x0) = 4, [a__U11](x0, x1) = 4, [tt] = 3 orientation: a__plus#(N,s(M)) = 5M + 17 >= 5M + 17 = a__U41#(a__isNat(M),M,N) a__U41#(tt(),M,N) = 5M + 13 >= 5M + 13 = a__U42#(a__isNat(N),M,N) a__U42#(tt(),M,N) = 5M + 11 >= 5M + 2 = a__plus#(mark(N),mark(M)) a__U11(tt(),V2) = 4 >= 4 = a__U12(a__isNat(V2)) a__U12(tt()) = 3 >= 3 = tt() a__U21(tt()) = 3 >= 3 = tt() a__U31(tt(),N) = 2N + 3 >= N = mark(N) a__U41(tt(),M,N) = 2M + 2N + 5 >= 2M + 2N + 4 = a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) = 2M + 2N + 3 >= 2M + 2N + 3 = s(a__plus(mark(N),mark(M))) a__isNat(0()) = 4 >= 3 = tt() a__isNat(plus(V1,V2)) = 4 >= 4 = a__U11(a__isNat(V1),V2) a__isNat(s(V1)) = 4 >= 3 = a__U21(a__isNat(V1)) a__plus(N,0()) = 2N + 8 >= 2N + 3 = a__U31(a__isNat(N),N) a__plus(N,s(M)) = 2M + 2N + 6 >= 2M + 2N + 6 = a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) = 4 >= 4 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNat(X)) = 4 >= 4 = a__isNat(X) mark(U21(X)) = 3 >= 3 = a__U21(mark(X)) mark(U31(X1,X2)) = 2X2 + 3 >= 2X2 + 3 = a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) = X1 + 2X2 + 2X3 + 2 >= X1 + 2X2 + 2X3 + 2 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) = X1 + 2X2 + 2X3 >= X1 + 2X2 + 2X3 = a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 3 >= 3 = tt() mark(s(X)) = X + 3 >= X + 3 = s(mark(X)) mark(0()) = 4 >= 4 = 0() a__U11(X1,X2) = 4 >= 4 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNat(X) = 4 >= 4 = isNat(X) a__U21(X) = 3 >= 3 = U21(X) a__U31(X1,X2) = 2X2 + 3 >= 2X2 + 3 = U31(X1,X2) a__U41(X1,X2,X3) = X1 + 2X2 + 2X3 + 2 >= X1 + 2X2 + 2X3 + 2 = U41(X1,X2,X3) a__U42(X1,X2,X3) = X1 + 2X2 + 2X3 >= X1 + 2X2 + 2X3 = U42(X1,X2,X3) a__plus(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = plus(X1,X2) problem: 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) 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) Restore Modifier: 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) 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: 3/4 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