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#(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#(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#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) 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#(U41(tt(),M,N)) -> mark#(U42(isNat(N),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#(U31(tt(),N)) -> mark#(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#(U21(tt())) -> mark#(tt()) 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#(U12(tt())) -> mark#(tt()) 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#(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))) 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,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#(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#(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#(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#(U21(X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) -> mark#(U21(X)) -> active#(U21(mark(X))) 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#(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))) SCC Processor: #sccs: 1 #rules: 23 #arcs: 228/961 DPs: mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U31(tt(),N)) -> mark#(N) mark#(U31(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(U31(isNat(N),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) Arctic Interpretation Processor: dimension: 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: [U12](x0) = x0 + 0, [U41](x0, x1, x2) = 4x0 + 6x1 + 1x2 + 0, [U11](x0, x1) = x0 + 0, [isNat](x0) = 0, [mark#](x0) = x0 + 0, [mark](x0) = x0, [plus](x0, x1) = 1x0 + 6x1 + 0, [tt] = 0, [U31](x0, x1) = 4x0 + 1x1 + 4, [0] = 0, [s](x0) = x0 + 0, [U42](x0, x1, x2) = 4x0 + 6x1 + 1x2 + 0, [active#](x0) = x0 + 0, [U21](x0) = x0 + 0, [active](x0) = x0 + 0 orientation: mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(plus(X1,X2)) = 1X1 + 6X2 + 0 >= 1X1 + 6X2 + 0 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = 6M + 1N + 6 >= 6M + 1N + 4 = mark#(U41(isNat(M),M,N)) mark#(U41(X1,X2,X3)) = 4X1 + 6X2 + 1X3 + 0 >= 4X1 + 6X2 + 1X3 + 0 = active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) = 6M + 1N + 4 >= 6M + 1N + 4 = mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) = 4X1 + 6X2 + 1X3 + 0 >= 4X1 + 6X2 + 1X3 + 0 = active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) = 6M + 1N + 4 >= 6M + 1N + 0 = mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) = 4X1 + 6X2 + 1X3 + 0 >= X1 + 0 = mark#(X1) mark#(plus(X1,X2)) = 1X1 + 6X2 + 0 >= X1 + 0 = mark#(X1) mark#(plus(X1,X2)) = 1X1 + 6X2 + 0 >= X2 + 0 = mark#(X2) mark#(U41(X1,X2,X3)) = 4X1 + 6X2 + 1X3 + 0 >= X1 + 0 = mark#(X1) mark#(U31(X1,X2)) = 4X1 + 1X2 + 4 >= 4X1 + 1X2 + 4 = active#(U31(mark(X1),X2)) active#(U31(tt(),N)) = 1N + 4 >= N + 0 = mark#(N) mark#(U31(X1,X2)) = 4X1 + 1X2 + 4 >= X1 + 0 = mark#(X1) mark#(U21(X)) = X + 0 >= X + 0 = mark#(X) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNat(V1))) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(isNat(V1),V2)) mark#(U11(X1,X2)) = X1 + 0 >= X1 + 0 = active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) = 0 >= 0 = mark#(U12(isNat(V2))) mark#(U12(X)) = X + 0 >= X + 0 = mark#(X) mark#(U11(X1,X2)) = X1 + 0 >= X1 + 0 = mark#(X1) active#(plus(N,0())) = 1N + 6 >= 1N + 4 = mark#(U31(isNat(N),N)) active(U11(tt(),V2)) = 0 >= 0 = mark(U12(isNat(V2))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 1N + 4 >= N = mark(N) active(U41(tt(),M,N)) = 6M + 1N + 4 >= 6M + 1N + 4 = mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) = 6M + 1N + 4 >= 6M + 1N + 0 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(isNat(V1),V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNat(V1))) active(plus(N,0())) = 1N + 6 >= 1N + 4 = mark(U31(isNat(N),N)) active(plus(N,s(M))) = 6M + 1N + 6 >= 6M + 1N + 4 = mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) = X1 + 0 >= X1 + 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X + 0 >= X + 0 = active(U12(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U21(X)) = X + 0 >= X + 0 = active(U21(mark(X))) mark(U31(X1,X2)) = 4X1 + 1X2 + 4 >= 4X1 + 1X2 + 4 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 4X1 + 6X2 + 1X3 + 0 >= 4X1 + 6X2 + 1X3 + 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) = 4X1 + 6X2 + 1X3 + 0 >= 4X1 + 6X2 + 1X3 + 0 = active(U42(mark(X1),X2,X3)) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(plus(X1,X2)) = 1X1 + 6X2 + 0 >= 1X1 + 6X2 + 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) U11(mark(X1),X2) = X1 + 0 >= X1 + 0 = U11(X1,X2) U11(X1,mark(X2)) = X1 + 0 >= X1 + 0 = U11(X1,X2) U11(active(X1),X2) = X1 + 0 >= X1 + 0 = U11(X1,X2) U11(X1,active(X2)) = X1 + 0 >= X1 + 0 = U11(X1,X2) U12(mark(X)) = X + 0 >= X + 0 = U12(X) U12(active(X)) = X + 0 >= X + 0 = U12(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U21(mark(X)) = X + 0 >= X + 0 = U21(X) U21(active(X)) = X + 0 >= X + 0 = U21(X) U31(mark(X1),X2) = 4X1 + 1X2 + 4 >= 4X1 + 1X2 + 4 = U31(X1,X2) U31(X1,mark(X2)) = 4X1 + 1X2 + 4 >= 4X1 + 1X2 + 4 = U31(X1,X2) U31(active(X1),X2) = 4X1 + 1X2 + 4 >= 4X1 + 1X2 + 4 = U31(X1,X2) U31(X1,active(X2)) = 4X1 + 1X2 + 4 >= 4X1 + 1X2 + 4 = U31(X1,X2) U41(mark(X1),X2,X3) = 4X1 + 6X2 + 1X3 + 0 >= 4X1 + 6X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 4X1 + 6X2 + 1X3 + 0 >= 4X1 + 6X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 4X1 + 6X2 + 1X3 + 0 >= 4X1 + 6X2 + 1X3 + 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 4X1 + 6X2 + 1X3 + 4 >= 4X1 + 6X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 4X1 + 6X2 + 1X3 + 6 >= 4X1 + 6X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 4X1 + 6X2 + 1X3 + 1 >= 4X1 + 6X2 + 1X3 + 0 = U41(X1,X2,X3) U42(mark(X1),X2,X3) = 4X1 + 6X2 + 1X3 + 0 >= 4X1 + 6X2 + 1X3 + 0 = U42(X1,X2,X3) U42(X1,mark(X2),X3) = 4X1 + 6X2 + 1X3 + 0 >= 4X1 + 6X2 + 1X3 + 0 = U42(X1,X2,X3) U42(X1,X2,mark(X3)) = 4X1 + 6X2 + 1X3 + 0 >= 4X1 + 6X2 + 1X3 + 0 = U42(X1,X2,X3) U42(active(X1),X2,X3) = 4X1 + 6X2 + 1X3 + 4 >= 4X1 + 6X2 + 1X3 + 0 = U42(X1,X2,X3) U42(X1,active(X2),X3) = 4X1 + 6X2 + 1X3 + 6 >= 4X1 + 6X2 + 1X3 + 0 = U42(X1,X2,X3) U42(X1,X2,active(X3)) = 4X1 + 6X2 + 1X3 + 1 >= 4X1 + 6X2 + 1X3 + 0 = U42(X1,X2,X3) s(mark(X)) = X + 0 >= X + 0 = s(X) s(active(X)) = X + 0 >= X + 0 = s(X) plus(mark(X1),X2) = 1X1 + 6X2 + 0 >= 1X1 + 6X2 + 0 = plus(X1,X2) plus(X1,mark(X2)) = 1X1 + 6X2 + 0 >= 1X1 + 6X2 + 0 = plus(X1,X2) plus(active(X1),X2) = 1X1 + 6X2 + 1 >= 1X1 + 6X2 + 0 = plus(X1,X2) plus(X1,active(X2)) = 1X1 + 6X2 + 6 >= 1X1 + 6X2 + 0 = plus(X1,X2) problem: DPs: mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(U31(isNat(N),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#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(U31(isNat(N),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: 19 #arcs: 169/441 DPs: mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) 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) Arctic Interpretation Processor: dimension: 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: [U12](x0) = x0 + 0, [U41](x0, x1, x2) = x0 + x1 + 2x2 + 4, [U11](x0, x1) = x0 + x1, [isNat](x0) = x0 + 4, [mark#](x0) = x0, [mark](x0) = x0, [plus](x0, x1) = 2x0 + x1, [tt] = 6, [U31](x0, x1) = x0 + 2x1 + 0, [0] = 6, [s](x0) = x0 + 4, [U42](x0, x1, x2) = x0 + x1 + 2x2 + 0, [active#](x0) = x0, [U21](x0) = x0, [active](x0) = x0 orientation: mark#(s(X)) = X + 4 >= X = mark#(X) mark#(plus(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = M + 2N + 4 >= M + 2N + 4 = mark#(U41(isNat(M),M,N)) mark#(U41(X1,X2,X3)) = X1 + X2 + 2X3 + 4 >= X1 + X2 + 2X3 + 4 = active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) = M + 2N + 6 >= M + 2N + 4 = mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) = X1 + X2 + 2X3 + 0 >= X1 + X2 + 2X3 + 0 = active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) = M + 2N + 6 >= M + 2N + 4 = mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) = X1 + X2 + 2X3 + 0 >= X1 = mark#(X1) mark#(plus(X1,X2)) = 2X1 + X2 >= X1 = mark#(X1) mark#(plus(X1,X2)) = 2X1 + X2 >= X2 = mark#(X2) mark#(U41(X1,X2,X3)) = X1 + X2 + 2X3 + 4 >= X1 = mark#(X1) mark#(U21(X)) = X >= X = mark#(X) mark#(isNat(X)) = X + 4 >= X + 4 = active#(isNat(X)) active#(isNat(s(V1))) = V1 + 4 >= V1 + 4 = mark#(U21(isNat(V1))) active#(isNat(plus(V1,V2))) = 2V1 + V2 + 4 >= V1 + V2 + 4 = mark#(U11(isNat(V1),V2)) mark#(U11(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) = V2 + 6 >= V2 + 4 = mark#(U12(isNat(V2))) mark#(U12(X)) = X + 0 >= X = mark#(X) mark#(U11(X1,X2)) = X1 + X2 >= X1 = mark#(X1) active(U11(tt(),V2)) = V2 + 6 >= V2 + 4 = mark(U12(isNat(V2))) active(U12(tt())) = 6 >= 6 = mark(tt()) active(U21(tt())) = 6 >= 6 = mark(tt()) active(U31(tt(),N)) = 2N + 6 >= N = mark(N) active(U41(tt(),M,N)) = M + 2N + 6 >= M + 2N + 4 = mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) = M + 2N + 6 >= M + 2N + 4 = mark(s(plus(N,M))) active(isNat(0())) = 6 >= 6 = mark(tt()) active(isNat(plus(V1,V2))) = 2V1 + V2 + 4 >= V1 + V2 + 4 = mark(U11(isNat(V1),V2)) active(isNat(s(V1))) = V1 + 4 >= V1 + 4 = mark(U21(isNat(V1))) active(plus(N,0())) = 2N + 6 >= 2N + 4 = mark(U31(isNat(N),N)) active(plus(N,s(M))) = M + 2N + 4 >= M + 2N + 4 = mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = active(U11(mark(X1),X2)) mark(tt()) = 6 >= 6 = active(tt()) mark(U12(X)) = X + 0 >= X + 0 = active(U12(mark(X))) mark(isNat(X)) = X + 4 >= X + 4 = active(isNat(X)) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X1,X2)) = X1 + 2X2 + 0 >= X1 + 2X2 + 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X1 + X2 + 2X3 + 4 >= X1 + X2 + 2X3 + 4 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) = X1 + X2 + 2X3 + 0 >= X1 + X2 + 2X3 + 0 = active(U42(mark(X1),X2,X3)) mark(s(X)) = X + 4 >= X + 4 = active(s(mark(X))) mark(plus(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 6 >= 6 = active(0()) U11(mark(X1),X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) U11(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U11(X1,X2) U11(active(X1),X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) U11(X1,active(X2)) = X1 + X2 >= X1 + X2 = U11(X1,X2) U12(mark(X)) = X + 0 >= X + 0 = U12(X) U12(active(X)) = X + 0 >= X + 0 = U12(X) isNat(mark(X)) = X + 4 >= X + 4 = isNat(X) isNat(active(X)) = X + 4 >= X + 4 = isNat(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X1),X2) = X1 + 2X2 + 0 >= X1 + 2X2 + 0 = U31(X1,X2) U31(X1,mark(X2)) = X1 + 2X2 + 0 >= X1 + 2X2 + 0 = U31(X1,X2) U31(active(X1),X2) = X1 + 2X2 + 0 >= X1 + 2X2 + 0 = U31(X1,X2) U31(X1,active(X2)) = X1 + 2X2 + 0 >= X1 + 2X2 + 0 = U31(X1,X2) U41(mark(X1),X2,X3) = X1 + X2 + 2X3 + 4 >= X1 + X2 + 2X3 + 4 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 + X2 + 2X3 + 4 >= X1 + X2 + 2X3 + 4 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 + X2 + 2X3 + 4 >= X1 + X2 + 2X3 + 4 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 + X2 + 2X3 + 4 >= X1 + X2 + 2X3 + 4 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 + X2 + 2X3 + 4 >= X1 + X2 + 2X3 + 4 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 + X2 + 2X3 + 4 >= X1 + X2 + 2X3 + 4 = U41(X1,X2,X3) U42(mark(X1),X2,X3) = X1 + X2 + 2X3 + 0 >= X1 + X2 + 2X3 + 0 = U42(X1,X2,X3) U42(X1,mark(X2),X3) = X1 + X2 + 2X3 + 0 >= X1 + X2 + 2X3 + 0 = U42(X1,X2,X3) U42(X1,X2,mark(X3)) = X1 + X2 + 2X3 + 0 >= X1 + X2 + 2X3 + 0 = U42(X1,X2,X3) U42(active(X1),X2,X3) = X1 + X2 + 2X3 + 0 >= X1 + X2 + 2X3 + 0 = U42(X1,X2,X3) U42(X1,active(X2),X3) = X1 + X2 + 2X3 + 0 >= X1 + X2 + 2X3 + 0 = U42(X1,X2,X3) U42(X1,X2,active(X3)) = X1 + X2 + 2X3 + 0 >= X1 + X2 + 2X3 + 0 = U42(X1,X2,X3) s(mark(X)) = X + 4 >= X + 4 = s(X) s(active(X)) = X + 4 >= X + 4 = s(X) plus(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = 2X1 + X2 >= 2X1 + X2 = plus(X1,X2) plus(active(X1),X2) = 2X1 + X2 >= 2X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = 2X1 + X2 >= 2X1 + X2 = plus(X1,X2) problem: DPs: mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) 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#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(isNat(plus(V1,V2))) -> mark#(U11(isNat(V1),V2)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) 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) Arctic Interpretation Processor: dimension: 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: [U12](x0) = x0, [U41](x0, x1, x2) = 1x0 + 1x1 + 1x2 + 0, [U11](x0, x1) = x0 + x1 + 0, [isNat](x0) = x0 + 0, [mark#](x0) = x0 + 0, [mark](x0) = x0, [plus](x0, x1) = 1x0 + 1x1 + 1, [tt] = 0, [U31](x0, x1) = 1x1, [0] = 2, [s](x0) = x0, [U42](x0, x1, x2) = 1x0 + 1x1 + 1x2, [active#](x0) = x0 + 0, [U21](x0) = x0 + 0, [active](x0) = x0 orientation: mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(plus(X1,X2)) = 1X1 + 1X2 + 1 >= 1X1 + 1X2 + 1 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = 1M + 1N + 1 >= 1M + 1N + 1 = mark#(U41(isNat(M),M,N)) mark#(U41(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 0 >= 1X1 + 1X2 + 1X3 + 0 = active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) = 1M + 1N + 1 >= 1M + 1N + 1 = mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 0 >= 1X1 + 1X2 + 1X3 + 0 = active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) = 1M + 1N + 1 >= 1M + 1N + 1 = mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 0 >= X1 + 0 = mark#(X1) mark#(plus(X1,X2)) = 1X1 + 1X2 + 1 >= X2 + 0 = mark#(X2) mark#(U41(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 0 >= X1 + 0 = mark#(X1) mark#(U21(X)) = X + 0 >= X + 0 = mark#(X) mark#(isNat(X)) = X + 0 >= X + 0 = active#(isNat(X)) active#(isNat(s(V1))) = V1 + 0 >= V1 + 0 = mark#(U21(isNat(V1))) active#(isNat(plus(V1,V2))) = 1V1 + 1V2 + 1 >= V1 + V2 + 0 = mark#(U11(isNat(V1),V2)) mark#(U11(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) = V2 + 0 >= V2 + 0 = mark#(U12(isNat(V2))) mark#(U12(X)) = X + 0 >= X + 0 = mark#(X) mark#(U11(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) active(U11(tt(),V2)) = V2 + 0 >= V2 + 0 = mark(U12(isNat(V2))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 1N >= N = mark(N) active(U41(tt(),M,N)) = 1M + 1N + 1 >= 1M + 1N + 1 = mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) = 1M + 1N + 1 >= 1M + 1N + 1 = mark(s(plus(N,M))) active(isNat(0())) = 2 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 1V1 + 1V2 + 1 >= V1 + V2 + 0 = mark(U11(isNat(V1),V2)) active(isNat(s(V1))) = V1 + 0 >= V1 + 0 = mark(U21(isNat(V1))) active(plus(N,0())) = 1N + 3 >= 1N = mark(U31(isNat(N),N)) active(plus(N,s(M))) = 1M + 1N + 1 >= 1M + 1N + 1 = mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNat(X)) = X + 0 >= X + 0 = active(isNat(X)) mark(U21(X)) = X + 0 >= X + 0 = active(U21(mark(X))) mark(U31(X1,X2)) = 1X2 >= 1X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 0 >= 1X1 + 1X2 + 1X3 + 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) = 1X1 + 1X2 + 1X3 >= 1X1 + 1X2 + 1X3 = active(U42(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = 1X1 + 1X2 + 1 >= 1X1 + 1X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 2 >= 2 = active(0()) U11(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U11(X1,X2) U11(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = U11(X1,X2) U11(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U11(X1,X2) U11(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNat(mark(X)) = X + 0 >= X + 0 = isNat(X) isNat(active(X)) = X + 0 >= X + 0 = isNat(X) U21(mark(X)) = X + 0 >= X + 0 = U21(X) U21(active(X)) = X + 0 >= X + 0 = U21(X) U31(mark(X1),X2) = 1X2 >= 1X2 = U31(X1,X2) U31(X1,mark(X2)) = 1X2 >= 1X2 = U31(X1,X2) U31(active(X1),X2) = 1X2 >= 1X2 = U31(X1,X2) U31(X1,active(X2)) = 1X2 >= 1X2 = U31(X1,X2) U41(mark(X1),X2,X3) = 1X1 + 1X2 + 1X3 + 0 >= 1X1 + 1X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1X1 + 1X2 + 1X3 + 0 >= 1X1 + 1X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1X1 + 1X2 + 1X3 + 0 >= 1X1 + 1X2 + 1X3 + 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1X1 + 1X2 + 1X3 + 0 >= 1X1 + 1X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1X1 + 1X2 + 1X3 + 0 >= 1X1 + 1X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1X1 + 1X2 + 1X3 + 0 >= 1X1 + 1X2 + 1X3 + 0 = U41(X1,X2,X3) U42(mark(X1),X2,X3) = 1X1 + 1X2 + 1X3 >= 1X1 + 1X2 + 1X3 = U42(X1,X2,X3) U42(X1,mark(X2),X3) = 1X1 + 1X2 + 1X3 >= 1X1 + 1X2 + 1X3 = U42(X1,X2,X3) U42(X1,X2,mark(X3)) = 1X1 + 1X2 + 1X3 >= 1X1 + 1X2 + 1X3 = U42(X1,X2,X3) U42(active(X1),X2,X3) = 1X1 + 1X2 + 1X3 >= 1X1 + 1X2 + 1X3 = U42(X1,X2,X3) U42(X1,active(X2),X3) = 1X1 + 1X2 + 1X3 >= 1X1 + 1X2 + 1X3 = U42(X1,X2,X3) U42(X1,X2,active(X3)) = 1X1 + 1X2 + 1X3 >= 1X1 + 1X2 + 1X3 = U42(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = 1X1 + 1X2 + 1 >= 1X1 + 1X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = 1X1 + 1X2 + 1 >= 1X1 + 1X2 + 1 = plus(X1,X2) plus(active(X1),X2) = 1X1 + 1X2 + 1 >= 1X1 + 1X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = 1X1 + 1X2 + 1 >= 1X1 + 1X2 + 1 = plus(X1,X2) problem: DPs: mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) 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#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) 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: 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) 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: [U12](x0) = x0, [U41](x0, x1, x2) = 2x0, [U11](x0, x1) = x0 + 4x1 + 3, [isNat](x0) = 0, [mark#](x0) = 4x0, [mark](x0) = 2x0 + 1, [plus](x0, x1) = 0, [tt] = 3, [U31](x0, x1) = 1, [0] = 1, [s](x0) = 2x0, [U42](x0, x1, x2) = x0, [active#](x0) = 0, [U21](x0) = 2x0, [active](x0) = 4x0 + 5 orientation: mark#(s(X)) = 8X >= 4X = mark#(X) mark#(plus(X1,X2)) = 0 >= 0 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = 0 >= 0 = mark#(U41(isNat(M),M,N)) mark#(U41(X1,X2,X3)) = 8X1 >= 0 = active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) = 0 >= 0 = mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) = 4X1 >= 0 = active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) = 0 >= 0 = mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) = 4X1 >= 4X1 = mark#(X1) mark#(U41(X1,X2,X3)) = 8X1 >= 4X1 = mark#(X1) mark#(U21(X)) = 8X >= 4X = mark#(X) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNat(V1))) mark#(U11(X1,X2)) = 4X1 + 16X2 + 12 >= 0 = active#(U11(mark(X1),X2)) active#(U11(tt(),V2)) = 0 >= 0 = mark#(U12(isNat(V2))) mark#(U12(X)) = 4X >= 4X = mark#(X) mark#(U11(X1,X2)) = 4X1 + 16X2 + 12 >= 4X1 = mark#(X1) active(U11(tt(),V2)) = 16V2 + 29 >= 1 = mark(U12(isNat(V2))) active(U12(tt())) = 17 >= 7 = mark(tt()) active(U21(tt())) = 29 >= 7 = mark(tt()) active(U31(tt(),N)) = 9 >= 2N + 1 = mark(N) active(U41(tt(),M,N)) = 29 >= 1 = mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) = 17 >= 1 = mark(s(plus(N,M))) active(isNat(0())) = 5 >= 7 = mark(tt()) active(isNat(plus(V1,V2))) = 5 >= 8V2 + 7 = mark(U11(isNat(V1),V2)) active(isNat(s(V1))) = 5 >= 1 = mark(U21(isNat(V1))) active(plus(N,0())) = 5 >= 3 = mark(U31(isNat(N),N)) active(plus(N,s(M))) = 5 >= 1 = mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) = 2X1 + 8X2 + 7 >= 8X1 + 16X2 + 21 = active(U11(mark(X1),X2)) mark(tt()) = 7 >= 17 = active(tt()) mark(U12(X)) = 2X + 1 >= 8X + 9 = active(U12(mark(X))) mark(isNat(X)) = 1 >= 5 = active(isNat(X)) mark(U21(X)) = 4X + 1 >= 16X + 13 = active(U21(mark(X))) mark(U31(X1,X2)) = 3 >= 9 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 4X1 + 1 >= 16X1 + 13 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) = 2X1 + 1 >= 8X1 + 9 = active(U42(mark(X1),X2,X3)) mark(s(X)) = 4X + 1 >= 16X + 13 = active(s(mark(X))) mark(plus(X1,X2)) = 1 >= 5 = active(plus(mark(X1),mark(X2))) mark(0()) = 3 >= 9 = active(0()) U11(mark(X1),X2) = 2X1 + 4X2 + 4 >= X1 + 4X2 + 3 = U11(X1,X2) U11(X1,mark(X2)) = X1 + 8X2 + 7 >= X1 + 4X2 + 3 = U11(X1,X2) U11(active(X1),X2) = 4X1 + 4X2 + 8 >= X1 + 4X2 + 3 = U11(X1,X2) U11(X1,active(X2)) = X1 + 16X2 + 23 >= X1 + 4X2 + 3 = U11(X1,X2) U12(mark(X)) = 2X + 1 >= X = U12(X) U12(active(X)) = 4X + 5 >= X = U12(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U21(mark(X)) = 4X + 2 >= 2X = U21(X) U21(active(X)) = 8X + 10 >= 2X = U21(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U41(mark(X1),X2,X3) = 4X1 + 2 >= 2X1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 2X1 >= 2X1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 2X1 >= 2X1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 8X1 + 10 >= 2X1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 2X1 >= 2X1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 2X1 >= 2X1 = U41(X1,X2,X3) U42(mark(X1),X2,X3) = 2X1 + 1 >= X1 = U42(X1,X2,X3) U42(X1,mark(X2),X3) = X1 >= X1 = U42(X1,X2,X3) U42(X1,X2,mark(X3)) = X1 >= X1 = U42(X1,X2,X3) U42(active(X1),X2,X3) = 4X1 + 5 >= X1 = U42(X1,X2,X3) U42(X1,active(X2),X3) = X1 >= X1 = U42(X1,X2,X3) U42(X1,X2,active(X3)) = X1 >= X1 = U42(X1,X2,X3) s(mark(X)) = 4X + 2 >= 2X = s(X) s(active(X)) = 8X + 10 >= 2X = s(X) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) problem: DPs: mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(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#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(U11(tt(),V2)) -> mark#(U12(isNat(V2))) mark#(U12(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: 13 #arcs: 119/196 DPs: mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U12(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) Arctic Interpretation Processor: dimension: 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: [U12](x0) = x0 + 0, [U41](x0, x1, x2) = x0 + x1 + 1x2 + 0, [U11](x0, x1) = x0 + x1 + 0, [isNat](x0) = x0 + 0, [mark#](x0) = x0 + 0, [mark](x0) = x0, [plus](x0, x1) = 1x0 + x1, [tt] = 4, [U31](x0, x1) = x0 + x1, [0] = 4, [s](x0) = x0 + 0, [U42](x0, x1, x2) = 1x0 + x1 + 1x2 + 4, [active#](x0) = x0 + 0, [U21](x0) = x0 + 0, [active](x0) = x0 orientation: mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(plus(X1,X2)) = 1X1 + X2 + 0 >= 1X1 + X2 + 0 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = M + 1N + 0 >= M + 1N + 0 = mark#(U41(isNat(M),M,N)) mark#(U41(X1,X2,X3)) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) = M + 1N + 4 >= M + 1N + 4 = mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) = 1X1 + X2 + 1X3 + 4 >= 1X1 + X2 + 1X3 + 4 = active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) = M + 1N + 5 >= M + 1N + 0 = mark#(s(plus(N,M))) mark#(U42(X1,X2,X3)) = 1X1 + X2 + 1X3 + 4 >= X1 + 0 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + 1X3 + 0 >= X1 + 0 = mark#(X1) mark#(U21(X)) = X + 0 >= X + 0 = mark#(X) mark#(isNat(X)) = X + 0 >= X + 0 = active#(isNat(X)) active#(isNat(s(V1))) = V1 + 0 >= V1 + 0 = mark#(U21(isNat(V1))) mark#(U12(X)) = X + 0 >= X + 0 = mark#(X) active(U11(tt(),V2)) = V2 + 4 >= V2 + 0 = mark(U12(isNat(V2))) active(U12(tt())) = 4 >= 4 = mark(tt()) active(U21(tt())) = 4 >= 4 = mark(tt()) active(U31(tt(),N)) = N + 4 >= N = mark(N) active(U41(tt(),M,N)) = M + 1N + 4 >= M + 1N + 4 = mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) = M + 1N + 5 >= M + 1N + 0 = mark(s(plus(N,M))) active(isNat(0())) = 4 >= 4 = mark(tt()) active(isNat(plus(V1,V2))) = 1V1 + V2 + 0 >= V1 + V2 + 0 = mark(U11(isNat(V1),V2)) active(isNat(s(V1))) = V1 + 0 >= V1 + 0 = mark(U21(isNat(V1))) active(plus(N,0())) = 1N + 4 >= N + 0 = mark(U31(isNat(N),N)) active(plus(N,s(M))) = M + 1N + 0 >= M + 1N + 0 = mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(U11(mark(X1),X2)) mark(tt()) = 4 >= 4 = active(tt()) mark(U12(X)) = X + 0 >= X + 0 = active(U12(mark(X))) mark(isNat(X)) = X + 0 >= X + 0 = active(isNat(X)) mark(U21(X)) = X + 0 >= X + 0 = active(U21(mark(X))) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) = 1X1 + X2 + 1X3 + 4 >= 1X1 + X2 + 1X3 + 4 = active(U42(mark(X1),X2,X3)) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(plus(X1,X2)) = 1X1 + X2 >= 1X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 4 >= 4 = active(0()) U11(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U11(X1,X2) U11(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = U11(X1,X2) U11(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U11(X1,X2) U11(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = U11(X1,X2) U12(mark(X)) = X + 0 >= X + 0 = U12(X) U12(active(X)) = X + 0 >= X + 0 = U12(X) isNat(mark(X)) = X + 0 >= X + 0 = isNat(X) isNat(active(X)) = X + 0 >= X + 0 = isNat(X) U21(mark(X)) = X + 0 >= X + 0 = U21(X) U21(active(X)) = X + 0 >= X + 0 = 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) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = U41(X1,X2,X3) U42(mark(X1),X2,X3) = 1X1 + X2 + 1X3 + 4 >= 1X1 + X2 + 1X3 + 4 = U42(X1,X2,X3) U42(X1,mark(X2),X3) = 1X1 + X2 + 1X3 + 4 >= 1X1 + X2 + 1X3 + 4 = U42(X1,X2,X3) U42(X1,X2,mark(X3)) = 1X1 + X2 + 1X3 + 4 >= 1X1 + X2 + 1X3 + 4 = U42(X1,X2,X3) U42(active(X1),X2,X3) = 1X1 + X2 + 1X3 + 4 >= 1X1 + X2 + 1X3 + 4 = U42(X1,X2,X3) U42(X1,active(X2),X3) = 1X1 + X2 + 1X3 + 4 >= 1X1 + X2 + 1X3 + 4 = U42(X1,X2,X3) U42(X1,X2,active(X3)) = 1X1 + X2 + 1X3 + 4 >= 1X1 + X2 + 1X3 + 4 = U42(X1,X2,X3) s(mark(X)) = X + 0 >= X + 0 = s(X) s(active(X)) = X + 0 >= X + 0 = s(X) plus(mark(X1),X2) = 1X1 + X2 >= 1X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = 1X1 + X2 >= 1X1 + X2 = plus(X1,X2) plus(active(X1),X2) = 1X1 + X2 >= 1X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = 1X1 + X2 >= 1X1 + X2 = plus(X1,X2) problem: DPs: mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U12(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#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U12(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) Arctic Interpretation Processor: dimension: 1 usable rules: isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) 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: [U12](x0) = 5x0 + 5, [U41](x0, x1, x2) = x0 + 0, [U11](x0, x1) = x1, [isNat](x0) = 0, [mark#](x0) = 1x0 + 4, [mark](x0) = x0 + 0, [plus](x0, x1) = 3, [tt] = 0, [U31](x0, x1) = 1x0 + 4x1 + 0, [0] = 1, [s](x0) = x0, [U42](x0, x1, x2) = 3x0 + 3, [active#](x0) = 4, [U21](x0) = x0 + 3, [active](x0) = x0 + 0 orientation: mark#(s(X)) = 1X + 4 >= 1X + 4 = mark#(X) mark#(plus(X1,X2)) = 4 >= 4 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = 4 >= 4 = mark#(U41(isNat(M),M,N)) mark#(U41(X1,X2,X3)) = 1X1 + 4 >= 4 = active#(U41(mark(X1),X2,X3)) active#(U41(tt(),M,N)) = 4 >= 4 = mark#(U42(isNat(N),M,N)) mark#(U42(X1,X2,X3)) = 4X1 + 4 >= 4 = active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) = 4 >= 4 = mark#(s(plus(N,M))) mark#(U41(X1,X2,X3)) = 1X1 + 4 >= 1X1 + 4 = mark#(X1) mark#(U21(X)) = 1X + 4 >= 1X + 4 = mark#(X) mark#(isNat(X)) = 4 >= 4 = active#(isNat(X)) active#(isNat(s(V1))) = 4 >= 4 = mark#(U21(isNat(V1))) mark#(U12(X)) = 6X + 6 >= 1X + 4 = mark#(X) active(U11(tt(),V2)) = V2 + 0 >= 5 = mark(U12(isNat(V2))) active(U12(tt())) = 5 >= 0 = mark(tt()) active(U21(tt())) = 3 >= 0 = mark(tt()) active(U31(tt(),N)) = 4N + 1 >= N + 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 3 = mark(U42(isNat(N),M,N)) active(U42(tt(),M,N)) = 3 >= 3 = mark(s(plus(N,M))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= V2 + 0 = mark(U11(isNat(V1),V2)) active(isNat(s(V1))) = 0 >= 3 = mark(U21(isNat(V1))) active(plus(N,0())) = 3 >= 4N + 1 = mark(U31(isNat(N),N)) active(plus(N,s(M))) = 3 >= 0 = mark(U41(isNat(M),M,N)) mark(U11(X1,X2)) = X2 + 0 >= X2 + 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 5X + 5 >= 5X + 5 = active(U12(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U21(X)) = X + 3 >= X + 3 = active(U21(mark(X))) mark(U31(X1,X2)) = 1X1 + 4X2 + 0 >= 1X1 + 4X2 + 1 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X1 + 0 >= X1 + 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2,X3)) = 3X1 + 3 >= 3X1 + 3 = active(U42(mark(X1),X2,X3)) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(plus(X1,X2)) = 3 >= 3 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) U11(mark(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,mark(X2)) = X2 + 0 >= X2 = U11(X1,X2) U11(active(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,active(X2)) = X2 + 0 >= X2 = U11(X1,X2) U12(mark(X)) = 5X + 5 >= 5X + 5 = U12(X) U12(active(X)) = 5X + 5 >= 5X + 5 = U12(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U21(mark(X)) = X + 3 >= X + 3 = U21(X) U21(active(X)) = X + 3 >= X + 3 = U21(X) U31(mark(X1),X2) = 1X1 + 4X2 + 1 >= 1X1 + 4X2 + 0 = U31(X1,X2) U31(X1,mark(X2)) = 1X1 + 4X2 + 4 >= 1X1 + 4X2 + 0 = U31(X1,X2) U31(active(X1),X2) = 1X1 + 4X2 + 1 >= 1X1 + 4X2 + 0 = U31(X1,X2) U31(X1,active(X2)) = 1X1 + 4X2 + 4 >= 1X1 + 4X2 + 0 = U31(X1,X2) U41(mark(X1),X2,X3) = X1 + 0 >= X1 + 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 + 0 >= X1 + 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 + 0 >= X1 + 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 + 0 >= X1 + 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 + 0 >= X1 + 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 + 0 >= X1 + 0 = U41(X1,X2,X3) U42(mark(X1),X2,X3) = 3X1 + 3 >= 3X1 + 3 = U42(X1,X2,X3) U42(X1,mark(X2),X3) = 3X1 + 3 >= 3X1 + 3 = U42(X1,X2,X3) U42(X1,X2,mark(X3)) = 3X1 + 3 >= 3X1 + 3 = U42(X1,X2,X3) U42(active(X1),X2,X3) = 3X1 + 3 >= 3X1 + 3 = U42(X1,X2,X3) U42(X1,active(X2),X3) = 3X1 + 3 >= 3X1 + 3 = U42(X1,X2,X3) U42(X1,X2,active(X3)) = 3X1 + 3 >= 3X1 + 3 = U42(X1,X2,X3) s(mark(X)) = X + 0 >= X = s(X) s(active(X)) = X + 0 >= X = s(X) plus(mark(X1),X2) = 3 >= 3 = plus(X1,X2) plus(X1,mark(X2)) = 3 >= 3 = plus(X1,X2) plus(active(X1),X2) = 3 >= 3 = plus(X1,X2) plus(X1,active(X2)) = 3 >= 3 = plus(X1,X2) problem: DPs: mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) 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#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U41(isNat(M),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)) -> active#(U42(mark(X1),X2,X3)) active#(U42(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) 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) KBO Processor: argument filtering: pi(tt) = [] pi(U11) = [] pi(active) = 0 pi(isNat) = [] pi(U12) = [] pi(mark) = 0 pi(U21) = 0 pi(U31) = [0,1] pi(U41) = [0,1,2] pi(U42) = [0,1,2] pi(plus) = [0,1] pi(s) = [0] pi(0) = [] pi(active#) = 0 pi(mark#) = 0 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) weight function: w0 = 1 w(0) = w(s) = w(plus) = w(U42) = w(U41) = w(U12) = w( isNat) = w(U11) = w(tt) = 1 w(mark#) = w(active#) = w(U31) = w(U21) = w(mark) = w( active) = 0 precedence: active# > U31 > plus > U41 > mark > U42 > U21 > active > s > mark# > 0 > isNat > U11 > U12 > tt problem: DPs: mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) 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#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3)) mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) 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: 55/36 DPs: mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) 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 pi(U21) = 0 pi(active#) = 0 pi(mark#) = 0 problem: DPs: mark#(U21(X)) -> mark#(X) mark#(isNat(X)) -> active#(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) SCC Processor: #sccs: 1 #rules: 1 #arcs: 4/4 DPs: 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) Size-Change Termination Processor: 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) The DP: mark#(U21(X)) -> mark#(X) has the edges: 0 > 0 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#) = 1 problem: DPs: U42#(mark(X1),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,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#(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#) = 0 problem: DPs: plus#(X1,mark(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: 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#) = 0 problem: DPs: U11#(X1,mark(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: 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#) = 0 problem: DPs: U31#(X1,mark(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: 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#) = 1 problem: DPs: U41#(mark(X1),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,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#(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