YES Problem: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Proof: DP Processor: DPs: active#(U11(tt(),V2)) -> isNat#(V2) active#(U11(tt(),V2)) -> U12#(isNat(V2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) active#(U12(tt())) -> mark#(tt()) active#(U21(tt())) -> mark#(tt()) active#(U31(tt(),N)) -> mark#(N) active#(U41(tt(),M,N)) -> isNat#(N) active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) active#(U42(tt(),M,N)) -> plus#(N,M) active#(U42(tt(),M,N)) -> s#(plus(N,M)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) active#(isNat(0())) -> mark#(tt()) active#(isNat(plus(V1,V2))) -> isNat#(V1) active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> U21#(isNat(V1)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> U31#(isNat(N),N) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> U41#(isNat(M),M,N) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> U12#(mark(X)) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNat(X)) -> active#(isNat(X)) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> U21#(mark(X)) mark#(U21(X)) -> active#(U21(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> U31#(mark(X1),X2) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> U41#(mark(X1),X2,X3) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> U42#(mark(X1),X2,X3) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(0()) -> active#(0()) U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) U12#(mark(X)) -> U12#(X) U12#(active(X)) -> U12#(X) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3) U42#(X1,mark(X2),X3) -> U42#(X1,X2,X3) U42#(X1,X2,mark(X3)) -> U42#(X1,X2,X3) U42#(active(X1),X2,X3) -> U42#(X1,X2,X3) U42#(X1,active(X2),X3) -> U42#(X1,X2,X3) U42#(X1,X2,active(X3)) -> U42#(X1,X2,X3) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) TDG Processor: DPs: active#(U11(tt(),V2)) -> isNat#(V2) active#(U11(tt(),V2)) -> U12#(isNat(V2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) active#(U12(tt())) -> mark#(tt()) active#(U21(tt())) -> mark#(tt()) active#(U31(tt(),N)) -> mark#(N) active#(U41(tt(),M,N)) -> isNat#(N) active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) active#(U42(tt(),M,N)) -> plus#(N,M) active#(U42(tt(),M,N)) -> s#(plus(N,M)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) active#(isNat(0())) -> mark#(tt()) active#(isNat(plus(V1,V2))) -> isNat#(V1) active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> U21#(isNat(V1)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> U31#(isNat(N),N) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> U41#(isNat(M),M,N) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> U12#(mark(X)) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNat(X)) -> active#(isNat(X)) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> U21#(mark(X)) mark#(U21(X)) -> active#(U21(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> U31#(mark(X1),X2) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> U41#(mark(X1),X2,X3) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> U42#(mark(X1),X2,X3) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(0()) -> active#(0()) U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) U12#(mark(X)) -> U12#(X) U12#(active(X)) -> U12#(X) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3) U42#(X1,mark(X2),X3) -> U42#(X1,X2,X3) U42#(X1,X2,mark(X3)) -> U42#(X1,X2,X3) U42#(active(X1),X2,X3) -> U42#(X1,X2,X3) U42#(X1,active(X2),X3) -> U42#(X1,X2,X3) U42#(X1,X2,active(X3)) -> U42#(X1,X2,X3) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) graph: ... SCC Processor: #sccs: 10 #rules: 63 #arcs: 1043/7225 DPs: mark#(0()) -> active#(0()) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(tt()) -> active#(tt()) active#(U21(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U31(tt(),N)) -> mark#(N) mark#(isNat(X)) -> active#(isNat(X)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> active#(U21(mark(X))) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(isNat(0())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) EDG Processor: DPs: mark#(0()) -> active#(0()) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(tt()) -> active#(tt()) active#(U21(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U31(tt(),N)) -> mark#(N) mark#(isNat(X)) -> active#(isNat(X)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> active#(U21(mark(X))) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(isNat(0())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) graph: mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(s(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(s(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(s(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(s(X)) -> active#(s(mark(X))) -> active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(s(X)) -> active#(s(mark(X))) -> active#(U12(tt())) -> mark#(tt()) mark#(s(X)) -> active#(s(mark(X))) -> active#(U21(tt())) -> mark#(tt()) mark#(s(X)) -> active#(s(mark(X))) -> active#(U31(tt(),N)) -> mark#(N) mark#(s(X)) -> active#(s(mark(X))) -> active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(s(X)) -> active#(s(mark(X))) -> active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(s(X)) -> active#(s(mark(X))) -> active#(isNat(0())) -> mark#(tt()) mark#(s(X)) -> active#(s(mark(X))) -> active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(s(X)) -> active#(s(mark(X))) -> active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U12(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(0()) -> active#(0()) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(U12(tt())) -> mark#(tt()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(U21(tt())) -> mark#(tt()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(U31(tt(),N)) -> mark#(N) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(isNat(0())) -> mark#(tt()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> active#(U41(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#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> 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)) -> active#(plus(mark(X1),mark(X2))) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(U12(tt())) -> mark#(tt()) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(U21(tt())) -> mark#(tt()) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(U31(tt(),N)) -> mark#(N) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(isNat(0())) -> mark#(tt()) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> active#(U41(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#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> 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)) -> active#(plus(mark(X1),mark(X2))) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(U12(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(U21(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(U31(tt(),N)) -> mark#(N) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(isNat(0())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(U12(tt())) -> mark#(tt()) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(U21(tt())) -> mark#(tt()) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(U31(tt(),N)) -> mark#(N) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(isNat(0())) -> mark#(tt()) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(U21(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U21(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(U21(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U21(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U21(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(U21(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U21(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U21(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(U21(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(U21(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U21(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(U21(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(U12(tt())) -> mark#(tt()) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(U21(tt())) -> mark#(tt()) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(U31(tt(),N)) -> mark#(N) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(isNat(0())) -> mark#(tt()) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(U12(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U12(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(U12(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U12(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U12(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(U12(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U12(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U12(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(U12(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(U12(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U12(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(U12(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(U12(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(U21(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(U31(tt(),N)) -> mark#(N) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(isNat(0())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(isNat(X)) -> active#(isNat(X)) -> active#(U12(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) -> active#(U21(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) -> active#(U31(tt(),N)) -> mark#(N) mark#(isNat(X)) -> active#(isNat(X)) -> active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(0())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) -> active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U12(tt())) -> mark#(tt()) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U21(tt())) -> mark#(tt()) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U31(tt(),N)) -> mark#(N) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(isNat(0())) -> mark#(tt()) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U11(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U12(X)) -> mark#(X) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U12(X)) -> active#(U12(mark(X))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(isNat(X)) -> active#(isNat(X)) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U21(X)) -> mark#(X) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U31(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U41(X1,X2,X3)) -> mark#(X1) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U42(X1,X2,X3)) -> mark#(X1) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(s(X)) -> mark#(X) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(s(X)) -> active#(s(mark(X))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(plus(X1,X2)) -> mark#(X2) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(plus(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U11(X1,X2)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U12(X)) -> mark#(X) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U12(X)) -> active#(U12(mark(X))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(isNat(X)) -> active#(isNat(X)) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U21(X)) -> mark#(X) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U31(X1,X2)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U41(X1,X2,X3)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U42(X1,X2,X3)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(s(X)) -> mark#(X) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(s(X)) -> active#(s(mark(X))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(plus(X1,X2)) -> mark#(X2) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(plus(X1,X2)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U11(X1,X2)) -> mark#(X1) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U12(X)) -> mark#(X) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U12(X)) -> active#(U12(mark(X))) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(isNat(X)) -> active#(isNat(X)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U21(X)) -> mark#(X) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U31(X1,X2)) -> mark#(X1) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U41(X1,X2,X3)) -> mark#(X1) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U42(X1,X2,X3)) -> mark#(X1) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> mark#(X) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> active#(s(mark(X))) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> mark#(X2) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> mark#(X1) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U11(X1,X2)) -> mark#(X1) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U12(X)) -> mark#(X) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U12(X)) -> active#(U12(mark(X))) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(isNat(X)) -> active#(isNat(X)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U21(X)) -> mark#(X) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U31(X1,X2)) -> mark#(X1) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U41(X1,X2,X3)) -> mark#(X1) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U42(X1,X2,X3)) -> mark#(X1) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(s(X)) -> mark#(X) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(s(X)) -> active#(s(mark(X))) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(plus(X1,X2)) -> mark#(X2) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(plus(X1,X2)) -> mark#(X1) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(U31(tt(),N)) -> mark#(N) -> mark#(U11(X1,X2)) -> mark#(X1) active#(U31(tt(),N)) -> mark#(N) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U31(tt(),N)) -> mark#(N) -> mark#(tt()) -> active#(tt()) active#(U31(tt(),N)) -> mark#(N) -> mark#(U12(X)) -> mark#(X) active#(U31(tt(),N)) -> mark#(N) -> mark#(U12(X)) -> active#(U12(mark(X))) active#(U31(tt(),N)) -> mark#(N) -> mark#(isNat(X)) -> active#(isNat(X)) active#(U31(tt(),N)) -> mark#(N) -> mark#(U21(X)) -> mark#(X) active#(U31(tt(),N)) -> mark#(N) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(U31(tt(),N)) -> mark#(N) -> mark#(U31(X1,X2)) -> mark#(X1) active#(U31(tt(),N)) -> mark#(N) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U31(tt(),N)) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> mark#(X1) active#(U31(tt(),N)) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U31(tt(),N)) -> mark#(N) -> mark#(U42(X1,X2,X3)) -> mark#(X1) active#(U31(tt(),N)) -> mark#(N) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(U31(tt(),N)) -> mark#(N) -> mark#(s(X)) -> mark#(X) active#(U31(tt(),N)) -> mark#(N) -> mark#(s(X)) -> active#(s(mark(X))) active#(U31(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) active#(U31(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) active#(U31(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(U31(tt(),N)) -> mark#(N) -> mark#(0()) -> active#(0()) active#(U21(tt())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(U12(tt())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNat(0())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U11(X1,X2)) -> mark#(X1) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U12(X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U12(X)) -> active#(U12(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U21(X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U31(X1,X2)) -> mark#(X1) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U41(X1,X2,X3)) -> mark#(X1) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U42(X1,X2,X3)) -> mark#(X1) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(s(X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(plus(X1,X2)) -> mark#(X2) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(plus(X1,X2)) -> mark#(X1) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U11(X1,X2)) -> mark#(X1) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U12(X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U12(X)) -> active#(U12(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U21(X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U31(X1,X2)) -> mark#(X1) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U41(X1,X2,X3)) -> mark#(X1) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U42(X1,X2,X3)) -> mark#(X1) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(s(X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(s(X)) -> active#(s(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(plus(X1,X2)) -> mark#(X2) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(plus(X1,X2)) -> mark#(X1) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U11(X1,X2)) -> mark#(X1) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U12(X)) -> mark#(X) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U12(X)) -> active#(U12(mark(X))) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(isNat(X)) -> active#(isNat(X)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U21(X)) -> mark#(X) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U31(X1,X2)) -> mark#(X1) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U41(X1,X2,X3)) -> mark#(X1) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U42(X1,X2,X3)) -> mark#(X1) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(s(X)) -> mark#(X) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(s(X)) -> active#(s(mark(X))) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(plus(X1,X2)) -> mark#(X2) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(plus(X1,X2)) -> mark#(X1) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) CDG Processor: DPs: mark#(0()) -> active#(0()) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(tt()) -> active#(tt()) active#(U21(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U31(tt(),N)) -> mark#(N) mark#(isNat(X)) -> active#(isNat(X)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> active#(U21(mark(X))) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(isNat(0())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) graph: mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> active#(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#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> active#(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)) -> active#(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)) -> active#(U31(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(s(X)) -> mark#(X) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(0()) -> active#(0()) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> active#(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#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U42(X1,X2,X3)) -> active#(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)) -> active#(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)) -> active#(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)) -> active#(U21(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U12(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(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#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> active#(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)) -> active#(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)) -> active#(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)) -> active#(U21(mark(X))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(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#(s(X)) -> active#(s(mark(X))) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> active#(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)) -> active#(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)) -> active#(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)) -> active#(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)) -> active#(isNat(X)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U42(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) -> active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(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#(s(X)) -> active#(s(mark(X))) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> active#(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)) -> active#(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)) -> active#(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)) -> active#(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)) -> active#(isNat(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) -> active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(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#(s(X)) -> active#(s(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> active#(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)) -> active#(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)) -> active#(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)) -> active#(U21(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U31(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) -> active#(U31(tt(),N)) -> mark#(N) mark#(U21(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(U21(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> active#(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#(s(X)) -> active#(s(mark(X))) mark#(U21(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> active#(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)) -> active#(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)) -> active#(U31(mark(X1),X2)) mark#(U21(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(U21(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U21(X)) -> mark#(X) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U21(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(U21(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U21(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(U21(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(U12(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> active#(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#(s(X)) -> active#(s(mark(X))) mark#(U12(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(U42(X1,X2,X3)) -> active#(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)) -> active#(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)) -> active#(U31(mark(X1),X2)) mark#(U12(X)) -> mark#(X) -> mark#(U31(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(U12(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U12(X)) -> mark#(X) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U12(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(U12(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U12(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U12(X)) -> active#(U12(mark(X))) -> active#(U12(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(0())) -> mark#(tt()) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(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#(s(X)) -> active#(s(mark(X))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U42(X1,X2,X3)) -> active#(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)) -> active#(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)) -> active#(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)) -> active#(U21(mark(X))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> active#(U12(mark(X))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) -> mark#(U31(X1,X2)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) -> mark#(U41(X1,X2,X3)) -> mark#(X1) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> active#(s(mark(X))) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> mark#(X) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) -> mark#(U42(X1,X2,X3)) -> mark#(X1) active#(U31(tt(),N)) -> mark#(N) -> mark#(0()) -> active#(0()) active#(U31(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(U31(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) active#(U31(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) active#(U31(tt(),N)) -> mark#(N) -> mark#(s(X)) -> active#(s(mark(X))) active#(U31(tt(),N)) -> mark#(N) -> mark#(s(X)) -> mark#(X) active#(U31(tt(),N)) -> mark#(N) -> mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(U31(tt(),N)) -> mark#(N) -> mark#(U42(X1,X2,X3)) -> mark#(X1) active#(U31(tt(),N)) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U31(tt(),N)) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> mark#(X1) active#(U31(tt(),N)) -> mark#(N) -> mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U31(tt(),N)) -> mark#(N) -> mark#(U31(X1,X2)) -> mark#(X1) active#(U31(tt(),N)) -> mark#(N) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(U31(tt(),N)) -> mark#(N) -> mark#(U21(X)) -> mark#(X) active#(U31(tt(),N)) -> mark#(N) -> mark#(isNat(X)) -> active#(isNat(X)) active#(U31(tt(),N)) -> mark#(N) -> mark#(U12(X)) -> active#(U12(mark(X))) active#(U31(tt(),N)) -> mark#(N) -> mark#(U12(X)) -> mark#(X) active#(U31(tt(),N)) -> mark#(N) -> mark#(tt()) -> active#(tt()) active#(U31(tt(),N)) -> mark#(N) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U31(tt(),N)) -> mark#(N) -> mark#(U11(X1,X2)) -> mark#(X1) active#(U21(tt())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(U12(tt())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNat(0())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U21(X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) -> mark#(U11(X1,X2)) -> mark#(X1) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U12(X)) -> active#(U12(mark(X))) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) -> mark#(U12(X)) -> mark#(X) SCC Processor: #sccs: 1 #rules: 23 #arcs: 228/961 DPs: mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U21(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U31(tt(),N)) -> mark#(N) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 usable rules: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) interpretation: [mark#](x0) = x0, [active#](x0) = x0, [0] = 1/2, [s](x0) = x0 + 7/2, [plus](x0, x1) = x0 + x1 + 1/2, [U42](x0, x1, x2) = x0 + x1 + x2 + 3, [U41](x0, x1, x2) = 3/2x0 + x1 + x2 + 5/2, [U31](x0, x1) = x0 + x1, [U21](x0) = x0, [mark](x0) = x0, [U12](x0) = x0, [isNat](x0) = 1, [active](x0) = x0, [U11](x0, x1) = x0, [tt] = 1 orientation: mark#(s(X)) = X + 7/2 >= X = mark#(X) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= X1 = active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) = 1 >= 1 = mark#(U12(isNat(V2))) mark#(U12(X)) = X >= X = mark#(X) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(isNat(V1),V2)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNat(V1))) mark#(U21(X)) = X >= X = mark#(X) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U31(mark(X1),X2)) active#(U31(tt(),N)) = N + 1 >= N = mark#(N) mark#(U41(X1,X2,X3)) = 3/2X1 + X2 + X3 + 5/2 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = 3/2X1 + X2 + X3 + 5/2 >= 3/2X1 + X2 + X3 + 5/2 = active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) = M + N + 4 >= M + N + 4 = mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) = X1 + X2 + X3 + 3 >= X1 = mark#(X1) mark#(U42(X1,X2,X3)) = X1 + X2 + X3 + 3 >= X1 + X2 + X3 + 3 = active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) = M + N + 4 >= M + N + 4 = mark#(s(plus(N,M))) mark#(plus(X1,X2)) = X1 + X2 + 1/2 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 + 1/2 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 1/2 >= X1 + X2 + 1/2 = active#(plus(mark(X1),mark(X2))) active#(plus(N,0())) = N + 1 >= N + 1 = mark#(U31(isNat(N),N)) active#(plus(N,s(M))) = M + N + 4 >= M + N + 4 = mark#(U41(isNat(M),M,N)) active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N = mark(N) active(U41(tt(),M,N)) = M + N + 4 >= M + N + 4 = mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) = M + N + 4 >= M + N + 4 = mark(s(plus(N,M))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(isNat(N),N)) active(plus(N,s(M))) = M + N + 4 >= M + N + 4 = mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) = X1 >= X1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 3/2X1 + X2 + X3 + 5/2 >= 3/2X1 + X2 + X3 + 5/2 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) = X1 + X2 + X3 + 3 >= X1 + X2 + X3 + 3 = active(U42(mark(X1),X2,X3)) mark(s(X)) = X + 7/2 >= X + 7/2 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1/2 >= X1 + X2 + 1/2 = active(plus(mark(X1),mark(X2))) mark(0()) = 1/2 >= 1/2 = active(0()) U11(mark(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,mark(X2)) = X1 >= X1 = U11(X1,X2) U11(active(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,active(X2)) = X1 >= X1 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(active(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,active(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U41(mark(X1),X2,X3) = 3/2X1 + X2 + X3 + 5/2 >= 3/2X1 + X2 + X3 + 5/2 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 3/2X1 + X2 + X3 + 5/2 >= 3/2X1 + X2 + X3 + 5/2 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 3/2X1 + X2 + X3 + 5/2 >= 3/2X1 + X2 + X3 + 5/2 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 3/2X1 + X2 + X3 + 5/2 >= 3/2X1 + X2 + X3 + 5/2 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 3/2X1 + X2 + X3 + 5/2 >= 3/2X1 + X2 + X3 + 5/2 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 3/2X1 + X2 + X3 + 5/2 >= 3/2X1 + X2 + X3 + 5/2 = U41(X1,X2,X3) U42(mark(X1),X2,X3) = X1 + X2 + X3 + 3 >= X1 + X2 + X3 + 3 = U42(X1,X2,X3) U42(X1,mark(X2),X3) = X1 + X2 + X3 + 3 >= X1 + X2 + X3 + 3 = U42(X1,X2,X3) U42(X1,X2,mark(X3)) = X1 + X2 + X3 + 3 >= X1 + X2 + X3 + 3 = U42(X1,X2,X3) U42(active(X1),X2,X3) = X1 + X2 + X3 + 3 >= X1 + X2 + X3 + 3 = U42(X1,X2,X3) U42(X1,active(X2),X3) = X1 + X2 + X3 + 3 >= X1 + X2 + X3 + 3 = U42(X1,X2,X3) U42(X1,X2,active(X3)) = X1 + X2 + X3 + 3 >= X1 + X2 + X3 + 3 = U42(X1,X2,X3) s(mark(X)) = X + 7/2 >= X + 7/2 = s(X) s(active(X)) = X + 7/2 >= X + 7/2 = s(X) plus(mark(X1),X2) = X1 + X2 + 1/2 >= X1 + X2 + 1/2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1/2 >= X1 + X2 + 1/2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1/2 >= X1 + X2 + 1/2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1/2 >= X1 + X2 + 1/2 = plus(X1,X2) problem: DPs: mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U21(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Restore Modifier: DPs: mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U21(X)) -> mark#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) -> mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) active#(plus(N,s(M))) -> mark#(U41(isNat(M),M,N)) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) SCC Processor: #sccs: 1 #rules: 11 #arcs: 169/289 DPs: mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,0())) -> mark#(U31(isNat(N),N)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U21(X)) -> mark#(X) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 usable rules: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) interpretation: [mark#](x0) = 2x0 + 1, [active#](x0) = 2x0, [0] = 0, [s](x0) = x0 + 1, [plus](x0, x1) = 5/2x0 + 5/2x1 + 2, [U42](x0, x1, x2) = 5/2x1 + 5/2x2 + 7/2, [U41](x0, x1, x2) = 5/2x1 + 5/2x2 + 7/2, [U31](x0, x1) = 3/2x0 + x1, [U21](x0) = x0, [mark](x0) = x0, [U12](x0) = 2x0, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1) = 2x0 + 2x1 + 3/2, [tt] = 0 orientation: mark#(plus(X1,X2)) = 5X1 + 5X2 + 5 >= 5X1 + 5X2 + 4 = active#(plus(mark(X1),mark(X2))) active#(plus(N,0())) = 5N + 4 >= 5N + 1 = mark#(U31(isNat(N),N)) mark#(U31(X1,X2)) = 3X1 + 2X2 + 1 >= 2X1 + 1 = mark#(X1) mark#(U11(X1,X2)) = 4X1 + 4X2 + 4 >= 2X1 + 1 = mark#(X1) mark#(U11(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 3 = active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) = 4V2 + 3 >= 4V2 + 1 = mark#(U12(isNat(V2))) mark#(U12(X)) = 4X + 1 >= 2X + 1 = mark#(X) mark#(isNat(X)) = 2X + 1 >= 2X = active#(isNat(X)) active#(isNat(plus(V1,V2))) = 5V1 + 5V2 + 4 >= 4V1 + 4V2 + 4 = mark#(U11(isNat(V1),V2)) active#(isNat(s(V1))) = 2V1 + 2 >= 2V1 + 1 = mark#(U21(isNat(V1))) mark#(U21(X)) = 2X + 1 >= 2X + 1 = mark#(X) active(U11(tt(),V2)) = 2V2 + 3/2 >= 2V2 = mark(U12(isNat(V2))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = 5/2M + 5/2N + 7/2 >= 5/2M + 5/2N + 7/2 = mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) = 5/2M + 5/2N + 7/2 >= 5/2M + 5/2N + 3 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 5/2V1 + 5/2V2 + 2 >= 2V1 + 2V2 + 3/2 = mark(U11(isNat(V1),V2)) active(isNat(s(V1))) = V1 + 1 >= V1 = mark(U21(isNat(V1))) active(plus(N,0())) = 5/2N + 2 >= 5/2N = mark(U31(isNat(N),N)) active(plus(N,s(M))) = 5/2M + 5/2N + 9/2 >= 5/2M + 5/2N + 7/2 = mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) = 2X1 + 2X2 + 3/2 >= 2X1 + 2X2 + 3/2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 2X >= 2X = active(U12(mark(X))) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X1,X2)) = 3/2X1 + X2 >= 3/2X1 + X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = active(U42(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = 5/2X1 + 5/2X2 + 2 >= 5/2X1 + 5/2X2 + 2 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2) = 2X1 + 2X2 + 3/2 >= 2X1 + 2X2 + 3/2 = U11(X1,X2) U11(X1,mark(X2)) = 2X1 + 2X2 + 3/2 >= 2X1 + 2X2 + 3/2 = U11(X1,X2) U11(active(X1),X2) = 2X1 + 2X2 + 3/2 >= 2X1 + 2X2 + 3/2 = U11(X1,X2) U11(X1,active(X2)) = 2X1 + 2X2 + 3/2 >= 2X1 + 2X2 + 3/2 = U11(X1,X2) U12(mark(X)) = 2X >= 2X = U12(X) U12(active(X)) = 2X >= 2X = U12(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X1),X2) = 3/2X1 + X2 >= 3/2X1 + X2 = U31(X1,X2) U31(X1,mark(X2)) = 3/2X1 + X2 >= 3/2X1 + X2 = U31(X1,X2) U31(active(X1),X2) = 3/2X1 + X2 >= 3/2X1 + X2 = U31(X1,X2) U31(X1,active(X2)) = 3/2X1 + X2 >= 3/2X1 + X2 = U31(X1,X2) U41(mark(X1),X2,X3) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U41(X1,X2,X3) U42(mark(X1),X2,X3) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U42(X1,X2,X3) U42(X1,mark(X2),X3) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U42(X1,X2,X3) U42(X1,X2,mark(X3)) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U42(X1,X2,X3) U42(active(X1),X2,X3) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U42(X1,X2,X3) U42(X1,active(X2),X3) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U42(X1,X2,X3) U42(X1,X2,active(X3)) = 5/2X2 + 5/2X3 + 7/2 >= 5/2X2 + 5/2X3 + 7/2 = U42(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = 5/2X1 + 5/2X2 + 2 >= 5/2X1 + 5/2X2 + 2 = plus(X1,X2) plus(X1,mark(X2)) = 5/2X1 + 5/2X2 + 2 >= 5/2X1 + 5/2X2 + 2 = plus(X1,X2) plus(active(X1),X2) = 5/2X1 + 5/2X2 + 2 >= 5/2X1 + 5/2X2 + 2 = plus(X1,X2) plus(X1,active(X2)) = 5/2X1 + 5/2X2 + 2 >= 5/2X1 + 5/2X2 + 2 = plus(X1,X2) problem: DPs: mark#(U31(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U21(X)) -> mark#(X) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Restore Modifier: DPs: mark#(U31(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U21(X)) -> mark#(X) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) SCC Processor: #sccs: 1 #rules: 3 #arcs: 37/16 DPs: mark#(U31(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed DPs: U12#(mark(X)) -> U12#(X) U12#(active(X)) -> U12#(X) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U12#) = 0 problem: DPs: TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed DPs: isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(isNat#) = 0 problem: DPs: TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed DPs: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3) U42#(X1,mark(X2),X3) -> U42#(X1,X2,X3) U42#(X1,X2,mark(X3)) -> U42#(X1,X2,X3) U42#(active(X1),X2,X3) -> U42#(X1,X2,X3) U42#(X1,active(X2),X3) -> U42#(X1,X2,X3) U42#(X1,X2,active(X3)) -> U42#(X1,X2,X3) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U42#) = 2 problem: DPs: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3) U42#(X1,mark(X2),X3) -> U42#(X1,X2,X3) U42#(active(X1),X2,X3) -> U42#(X1,X2,X3) U42#(X1,active(X2),X3) -> U42#(X1,X2,X3) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U42#) = 1 problem: DPs: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3) U42#(active(X1),X2,X3) -> U42#(X1,X2,X3) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U42#) = 0 problem: DPs: TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed DPs: plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(plus#) = 1 problem: DPs: plus#(mark(X1),X2) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(plus#) = 0 problem: DPs: TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed DPs: s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed DPs: U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U11#) = 1 problem: DPs: U11#(mark(X1),X2) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U11#) = 0 problem: DPs: TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed DPs: U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U21#) = 0 problem: DPs: TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed DPs: U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U31#) = 1 problem: DPs: U31#(mark(X1),X2) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U31#) = 0 problem: DPs: TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed DPs: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U41#) = 2 problem: DPs: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U41#) = 1 problem: DPs: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Subterm Criterion Processor: simple projection: pi(U41#) = 0 problem: DPs: TRS: active(U11(tt(),V2)) -> mark(U12(isNat(V2))) active(U12(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) -> mark(s(plus(N,M))) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N,0())) -> mark(U31(isNat(N),N)) active(plus(N,s(M))) -> mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,mark(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,mark(X3)) -> U42(X1,X2,X3) U42(active(X1),X2,X3) -> U42(X1,X2,X3) U42(X1,active(X2),X3) -> U42(X1,X2,X3) U42(X1,X2,active(X3)) -> U42(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Qed