MAYBE Problem: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Proof: DP Processor: DPs: active#(U11(tt(),V1,V2)) -> isNat#(V1) active#(U11(tt(),V1,V2)) -> U12#(isNat(V1),V2) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) active#(U12(tt(),V2)) -> isNat#(V2) active#(U12(tt(),V2)) -> U13#(isNat(V2)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),N)) -> mark#(N) active#(U41(tt(),M,N)) -> plus#(N,M) active#(U41(tt(),M,N)) -> s#(plus(N,M)) active#(U41(tt(),M,N)) -> mark#(s(plus(N,M))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(plus(V1,V2))) -> isNatKind#(V2) active#(isNat(plus(V1,V2))) -> isNatKind#(V1) active#(isNat(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNat(plus(V1,V2))) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V2) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1) active#(isNatKind(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(plus(N,0())) -> isNatKind#(N) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> and#(isNat(N),isNatKind(N)) active#(plus(N,0())) -> U31#(and(isNat(N),isNatKind(N)),N) active#(plus(N,0())) -> mark#(U31(and(isNat(N),isNatKind(N)),N)) active#(plus(N,s(M))) -> isNatKind#(N) active#(plus(N,s(M))) -> isNat#(N) active#(plus(N,s(M))) -> and#(isNat(N),isNatKind(N)) active#(plus(N,s(M))) -> isNatKind#(M) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> and#(isNat(M),isNatKind(M)) active#(plus(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(plus(N,s(M))) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) active#(plus(N,s(M))) -> mark#(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) mark#(tt()) -> active#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> U12#(mark(X1),X2) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> U13#(mark(X)) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> U21#(mark(X1),X2) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> U22#(mark(X)) mark#(U22(X)) -> active#(U22(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#(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#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(isNatKind(X)) -> active#(isNatKind(X)) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U12#(mark(X1),X2) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) U12#(active(X1),X2) -> U12#(X1,X2) U12#(X1,active(X2)) -> U12#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) U13#(mark(X)) -> U13#(X) U13#(active(X)) -> U13#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U22#(active(X)) -> U22#(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) 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) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) isNatKind#(mark(X)) -> isNatKind#(X) isNatKind#(active(X)) -> isNatKind#(X) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) TDG Processor: DPs: active#(U11(tt(),V1,V2)) -> isNat#(V1) active#(U11(tt(),V1,V2)) -> U12#(isNat(V1),V2) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) active#(U12(tt(),V2)) -> isNat#(V2) active#(U12(tt(),V2)) -> U13#(isNat(V2)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),N)) -> mark#(N) active#(U41(tt(),M,N)) -> plus#(N,M) active#(U41(tt(),M,N)) -> s#(plus(N,M)) active#(U41(tt(),M,N)) -> mark#(s(plus(N,M))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(plus(V1,V2))) -> isNatKind#(V2) active#(isNat(plus(V1,V2))) -> isNatKind#(V1) active#(isNat(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNat(plus(V1,V2))) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V2) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1) active#(isNatKind(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(plus(N,0())) -> isNatKind#(N) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> and#(isNat(N),isNatKind(N)) active#(plus(N,0())) -> U31#(and(isNat(N),isNatKind(N)),N) active#(plus(N,0())) -> mark#(U31(and(isNat(N),isNatKind(N)),N)) active#(plus(N,s(M))) -> isNatKind#(N) active#(plus(N,s(M))) -> isNat#(N) active#(plus(N,s(M))) -> and#(isNat(N),isNatKind(N)) active#(plus(N,s(M))) -> isNatKind#(M) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> and#(isNat(M),isNatKind(M)) active#(plus(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(plus(N,s(M))) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) active#(plus(N,s(M))) -> mark#(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) mark#(tt()) -> active#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> U12#(mark(X1),X2) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> U13#(mark(X)) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> U21#(mark(X1),X2) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> U22#(mark(X)) mark#(U22(X)) -> active#(U22(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#(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#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(isNatKind(X)) -> active#(isNatKind(X)) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U12#(mark(X1),X2) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) U12#(active(X1),X2) -> U12#(X1,X2) U12#(X1,active(X2)) -> U12#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) U13#(mark(X)) -> U13#(X) U13#(active(X)) -> U13#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U22#(active(X)) -> U22#(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) 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) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) isNatKind#(mark(X)) -> isNatKind#(X) isNatKind#(active(X)) -> isNatKind#(X) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) graph: ... SCC Processor: #sccs: 13 #rules: 83 #arcs: 1887/15129 DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) mark#(tt()) -> active#(tt()) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(U31(tt(),N)) -> mark#(N) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U41(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) 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(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) -> active#(0()) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(plus(N,0())) -> mark#(U31(and(isNat(N),isNatKind(N)),N)) active#(plus(N,s(M))) -> mark#(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 1, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 >= X1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 0 >= 0 = mark#(U13(isNat(V2))) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 >= X1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) active#(U31(tt(),N)) = N >= N = mark#(N) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U41(tt(),M,N)) = M + N >= M + N = mark#(s(plus(N,M))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U31(mark(X1),X2)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = X >= X = mark#(X) mark#(s(X)) = X >= X = active#(s(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) = 1 >= 1 = active#(0()) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(plus(N,0())) = N + 1 >= N = mark#(U31(and(isNat(N),isNatKind(N)),N)) active#(plus(N,s(M))) = M + N >= M + N = mark#(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = M + N >= M + N = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N >= M + N = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 >= X1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 >= X1 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 >= X1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 >= X1 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 >= X1 = U12(X1,X2) U12(X1,mark(X2)) = X1 >= X1 = U12(X1,X2) U12(active(X1),X2) = X1 >= X1 = U12(X1,X2) U12(X1,active(X2)) = X1 >= X1 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(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 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) mark#(tt()) -> active#(tt()) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(U31(tt(),N)) -> mark#(N) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U41(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) 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(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) -> active#(0()) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(plus(N,s(M))) -> mark#(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2 + 1, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 >= X1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 0 >= 0 = mark#(U13(isNat(V2))) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 >= X1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) active#(U31(tt(),N)) = N >= N = mark#(N) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark#(s(plus(N,M))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U31(mark(X1),X2)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = X + 1 >= X = mark#(X) mark#(s(X)) = X + 1 >= X + 1 = active#(s(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) = 0 >= 0 = active#(0()) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark#(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 >= X1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 >= X1 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 >= X1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 >= X1 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 >= X1 = U12(X1,X2) U12(X1,mark(X2)) = X1 >= X1 = U12(X1,X2) U12(active(X1),X2) = X1 >= X1 = U12(X1,X2) U12(X1,active(X2)) = X1 >= X1 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(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 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) mark#(tt()) -> active#(tt()) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(U31(tt(),N)) -> mark#(N) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U41(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) 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)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) -> active#(0()) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(plus(N,s(M))) -> mark#(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = 0, [plus](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = 0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 >= X1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 0 >= 0 = mark#(U13(isNat(V2))) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 >= X1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) active#(U31(tt(),N)) = N >= N = mark#(N) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U41(tt(),M,N)) = 0 >= 0 = mark#(s(plus(N,M))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U31(mark(X1),X2)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 0 >= 0 = active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = 0 >= 0 = active#(s(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) = 0 >= 0 = active#(0()) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(plus(N,s(M))) = N + 1 >= 0 = mark#(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 1 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 >= X1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 >= X1 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 >= X1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 >= X1 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 >= X1 = U12(X1,X2) U12(X1,mark(X2)) = X1 >= X1 = U12(X1,X2) U12(active(X1),X2) = X1 >= X1 = U12(X1,X2) U12(X1,active(X2)) = X1 >= X1 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(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) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) mark#(tt()) -> active#(tt()) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(U31(tt(),N)) -> mark#(N) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U41(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) 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)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) -> active#(0()) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 1, [and](x0, x1) = x0 + x1, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [U41](x0, x1, x2) = 1, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 >= X1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 0 >= 0 = mark#(U13(isNat(V2))) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 >= X1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) active#(U31(tt(),N)) = N >= N = mark#(N) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U41(tt(),M,N)) = 1 >= 0 = mark#(s(plus(N,M))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U31(mark(X1),X2)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = 0 >= 0 = active#(s(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) = X1 + 1 >= X1 + 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) = 1 >= 1 = active#(0()) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = 1 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 1 >= 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 >= X1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 >= X1 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + 1 >= X1 + 1 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 >= X1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 >= X1 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 >= X1 = U12(X1,X2) U12(X1,mark(X2)) = X1 >= X1 = U12(X1,X2) U12(active(X1),X2) = X1 >= X1 = U12(X1,X2) U12(X1,active(X2)) = X1 >= X1 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(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) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) mark#(tt()) -> active#(tt()) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(U31(tt(),N)) -> mark#(N) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) 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)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) -> active#(0()) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 1, [and](x0, x1) = x0 + x1, [s](x0) = 0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = 0, [U31](x0, x1) = x0 + x1 + 1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 0 >= 0 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 >= X1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 0 >= 0 = mark#(U13(isNat(V2))) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 >= X1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) active#(U31(tt(),N)) = N + 1 >= N = mark#(N) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U31(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U31(mark(X1),X2)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 0 >= 0 = active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = 0 >= 0 = active#(s(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) = 1 >= 1 = active#(0()) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 >= X1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 >= X1 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 >= X1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 >= X1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 >= X1 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 >= X1 = U12(X1,X2) U12(X1,mark(X2)) = X1 >= X1 = U12(X1,X2) U12(active(X1),X2) = X1 >= X1 = U12(X1,X2) U12(X1,active(X2)) = X1 >= X1 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U31(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U31(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U31(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) mark#(tt()) -> active#(tt()) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(isNat(0())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) -> active#(0()) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 1, [and](x0, x1) = 1, [s](x0) = 1, [plus](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U31](x0, x1) = 1, [U22](x0) = 1, [U21](x0, x1) = 1, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 1, [isNat](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 1 >= 1 = mark#(U13(isNat(V2))) mark#(tt()) = 1 >= 0 = active#(tt()) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U12(X1,X2)) = 1 >= 1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = 1 >= 1 = active#(s(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) = 1 >= 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) = 1 >= 1 = active#(0()) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(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) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(isNat(0())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) -> active#(0()) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 1, [plus](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U31](x0, x1) = 1, [U22](x0) = 1, [U21](x0, x1) = 1, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 1, [isNat](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 1 >= 1 = mark#(U13(isNat(V2))) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U12(X1,X2)) = 1 >= 1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = 1 >= 1 = active#(s(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) = 1 >= 1 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) mark#(0()) = 1 >= 0 = active#(0()) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(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) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(isNat(0())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 1, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x1 + x2, [U31](x0, x1) = x1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0 + x1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = V2 >= V2 = mark#(U13(isNat(V2))) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U31(X1,X2)) = X2 >= X2 = active#(U31(mark(X1),X2)) active#(isNat(0())) = 1 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X2 + X3 >= X2 + X3 = active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = X >= X = active#(s(mark(X))) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 >= V2 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = M + N >= M + N = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N >= M + N = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 + X2 >= X1 + X2 = active(U12(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X2 + X3 >= X2 + X3 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(active(X1),X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(X1,active(X2)) = X1 + X2 >= X1 + X2 = U12(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 1, [plus](x0, x1) = 0, [U41](x0, x1, x2) = 1, [U31](x0, x1) = 1, [U22](x0) = 1, [U21](x0, x1) = 1, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 1, [isNat](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 1 >= 1 = mark#(U13(isNat(V2))) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U12(X1,X2)) = 1 >= 1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = 1 >= 1 = active#(s(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(plus(X1,X2)) = 1 >= 0 = active#(plus(mark(X1),mark(X2))) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(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) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = 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) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 1, [plus](x0, x1) = 1, [U41](x0, x1, x2) = 0, [U31](x0, x1) = 1, [U22](x0) = 1, [U21](x0, x1) = 1, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 1, [isNat](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 1 >= 1 = mark#(U13(isNat(V2))) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U12(X1,X2)) = 1 >= 1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) mark#(U41(X1,X2,X3)) = 1 >= 0 = active#(U41(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = 1 >= 1 = active#(s(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(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) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 0, [plus](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U31](x0, x1) = 1, [U22](x0) = 1, [U21](x0, x1) = 1, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 1, [isNat](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 1 >= 1 = mark#(U13(isNat(V2))) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U12(X1,X2)) = 1 >= 1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(s(X)) = 1 >= 0 = active#(s(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(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) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = 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) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 0, [plus](x0, x1) = 0, [U41](x0, x1, x2) = 1, [U31](x0, x1) = 0, [U22](x0) = 1, [U21](x0, x1) = 1, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 1, [isNat](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 1 >= 1 = mark#(U13(isNat(V2))) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U12(X1,X2)) = 1 >= 1 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U31(X1,X2)) = 1 >= 0 = active#(U31(mark(X1),X2)) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = 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) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x1 + x2 + 1, [U31](x0, x1) = x1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0 + x1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = V2 >= V2 = mark#(U13(isNat(V2))) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNat(s(V1))) = V1 + 1 >= V1 = mark#(U21(isNatKind(V1),V1)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 >= V2 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 + X2 >= X1 + X2 = active(U12(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(active(X1),X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(X1,active(X2)) = X1 + X2 >= X1 + X2 = U12(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x1 + x2 + 1, [U31](x0, x1) = x1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1 + 1, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0 + x1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = V2 >= V2 = mark#(U13(isNat(V2))) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U12(mark(X1),X2)) active#(U21(tt(),V1)) = V1 + 1 >= V1 = mark#(U22(isNat(V1))) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) mark#(U21(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U21(mark(X1),X2)) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 >= V2 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= V1 + 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 + X2 >= X1 + X2 = active(U12(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(active(X1),X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(X1,active(X2)) = X1 + X2 >= X1 + X2 = U12(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 1, [plus](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U31](x0, x1) = 0, [U22](x0) = 1, [U21](x0, x1) = 0, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 1, [isNat](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 1 >= 1 = mark#(U13(isNat(V2))) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U12(X1,X2)) = 1 >= 1 = active#(U12(mark(X1),X2)) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) mark#(U21(X1,X2)) = 1 >= 0 = active#(U21(mark(X1),X2)) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = 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) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) active#(U22(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x1 + x2 + 1, [U31](x0, x1) = x1, [U22](x0) = x0 + 1, [U21](x0, x1) = x1 + 1, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0 + x1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = V2 >= V2 = mark#(U13(isNat(V2))) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U12(mark(X1),X2)) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U22(tt())) = 1 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) mark#(U22(X)) = X + 1 >= X = mark#(X) mark#(U22(X)) = X + 1 >= X + 1 = active#(U22(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 >= V2 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= V1 + 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= V1 + 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 + X2 >= X1 + X2 = active(U12(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = X2 + 1 >= X2 + 1 = active(U21(mark(X1),X2)) mark(U22(X)) = X + 1 >= X + 1 = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(active(X1),X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(X1,active(X2)) = X1 + X2 >= X1 + X2 = U12(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(active(X1),X2) = X2 + 1 >= X2 + 1 = U21(X1,X2) U21(X1,active(X2)) = X2 + 1 >= X2 + 1 = U21(X1,X2) U22(mark(X)) = X + 1 >= X + 1 = U22(X) U22(active(X)) = X + 1 >= X + 1 = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) mark#(U22(X)) -> active#(U22(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 0, [plus](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 1, [isNat](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = 1 >= 1 = mark#(U13(isNat(V2))) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U12(X1,X2)) = 1 >= 1 = active#(U12(mark(X1),X2)) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) mark#(U22(X)) = 1 >= 0 = active#(U22(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) active#(isNat(plus(V1,V2))) = 1 >= 1 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = 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) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = 1, [plus](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = 1, [U31](x0, x1) = x1, [U22](x0) = 1, [U21](x0, x1) = 1, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0 + x1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = V2 >= V2 = mark#(U13(isNat(V2))) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U12(mark(X1),X2)) mark#(isNat(X)) = X >= X = active#(isNat(X)) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 = mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 >= V2 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = 1 >= 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 2 >= 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 + X2 >= X1 + X2 = active(U12(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(active(X1),X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) U12(X1,active(X2)) = X1 + X2 >= X1 + X2 = U12(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = 0, [plus](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = 0, [U31](x0, x1) = x1, [U22](x0) = 0, [U21](x0, x1) = 0, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x0 + x1 + 1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x0 + x1 + x2 + 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U11(mark(X1),X2,X3)) active#(U12(tt(),V2)) = V2 + 1 >= V2 = mark#(U13(isNat(V2))) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U12(mark(X1),X2)) mark#(isNat(X)) = X >= X = active#(isNat(X)) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 + 1 >= V2 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 1 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U12(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) U12(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(X1,X2) U12(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(X1,X2) U12(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(X1,X2) U12(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 0, [plus](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 1, [isNat](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X1,X2)) = 1 >= 1 = active#(U12(mark(X1),X2)) mark#(isNat(X)) = 1 >= 0 = active#(isNat(X)) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = 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) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt())) -> mark#(tt()) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 1, [plus](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U31](x0, x1) = 1, [U22](x0) = 0, [U21](x0, x1) = 0, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 0, [isNat](x0) = 1, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = 1 >= 1 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = 1 >= 1 = active#(U11(mark(X1),X2,X3)) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X1,X2)) = 1 >= 0 = active#(U12(mark(X1),X2)) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,mark(X2)) = 0 >= 0 = U12(X1,X2) U12(active(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,active(X2)) = 0 >= 0 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(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) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = 0, [plus](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = 1, [U31](x0, x1) = x1, [U22](x0) = 0, [U21](x0, x1) = 0, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2 + 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U11(tt(),V1,V2)) = V2 + 1 >= V2 = mark#(U12(isNat(V1),V2)) mark#(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active#(U11(mark(X1),X2,X3)) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = V2 + 1 >= V2 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 >= V2 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = 1 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V2 + 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 1 >= 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X2 >= X2 = active(U12(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U12(mark(X1),X2) = X2 >= X2 = U12(X1,X2) U12(X1,mark(X2)) = X2 >= X2 = U12(X1,X2) U12(active(X1),X2) = X2 >= X2 = U12(X1,X2) U12(X1,active(X2)) = X2 >= X2 = U12(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x1 + x2 + 1, [U31](x0, x1) = x1, [U22](x0) = 0, [U21](x0, x1) = 0, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = 0, [isNat](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: mark#(isNatKind(X)) = X >= X = active#(isNatKind(X)) mark#(U11(X1,X2,X3)) = 0 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U13(tt())) = 0 >= 0 = mark#(tt()) mark#(U13(X)) = X >= X = mark#(X) mark#(U13(X)) = X >= X = active#(U13(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(and(isNatKind(V1),isNatKind(V2))) active#(isNatKind(s(V1))) = V1 + 1 >= V1 = mark#(isNatKind(V1)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 + 1 >= V1 = mark(isNatKind(V1)) active(plus(N,0())) = N >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,mark(X2)) = 0 >= 0 = U12(X1,X2) U12(active(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,active(X2)) = 0 >= 0 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U13(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 0, [plus](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 1, [U13](x0) = 1, [mark](x0) = 0, [U12](x0, x1) = 1, [isNat](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(U11(X1,X2,X3)) = 1 >= 0 = active#(U11(mark(X1),X2,X3)) active#(U13(tt())) = 1 >= 1 = mark#(tt()) mark#(U13(X)) = 1 >= 1 = mark#(X) mark#(U13(X)) = 1 >= 1 = active#(U13(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = 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) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U13(tt())) -> mark#(tt()) mark#(U13(X)) -> mark#(X) mark#(U13(X)) -> active#(U13(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = 0, [plus](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = 0, [U31](x0, x1) = x1, [U22](x0) = 0, [U21](x0, x1) = 0, [U13](x0) = x0 + 1, [mark](x0) = x0, [U12](x0, x1) = x1 + 1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2 + 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(U13(tt())) = 1 >= 0 = mark#(tt()) mark#(U13(X)) = X + 1 >= X = mark#(X) mark#(U13(X)) = X + 1 >= X + 1 = active#(U13(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) active(U11(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 + 1 >= V2 + 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= V2 + 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 1 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = X2 + 1 >= X2 + 1 = active(U12(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U13(X)) = X + 1 >= X + 1 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3) U12(mark(X1),X2) = X2 + 1 >= X2 + 1 = U12(X1,X2) U12(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U12(X1,X2) U12(active(X1),X2) = X2 + 1 >= X2 + 1 = U12(X1,X2) U12(X1,active(X2)) = X2 + 1 >= X2 + 1 = U12(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U13(mark(X)) = X + 1 >= X + 1 = U13(X) U13(active(X)) = X + 1 >= X + 1 = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X)) -> active#(U13(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = x0, [0] = 1, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x1 + x2, [U31](x0, x1) = x1, [U22](x0) = 1, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0, [U12](x0, x1) = x1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 1 orientation: mark#(isNatKind(X)) = X >= X = active#(isNatKind(X)) mark#(U13(X)) = X >= X = active#(U13(mark(X))) active#(and(tt(),X)) = X + 1 >= X = mark#(X) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(and(isNatKind(V1),isNatKind(V2))) active(U11(tt(),V1,V2)) = V2 >= V2 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 >= V2 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = M + N >= M + N = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 >= V2 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 >= V1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N >= M + N = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = X3 >= X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X1,X2)) = X2 >= X2 = active(U12(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U13(X)) = X >= X = active(U13(mark(X))) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X2 + X3 >= X2 + X3 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) U11(mark(X1),X2,X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X3 >= X3 = U11(X1,X2,X3) U11(active(X1),X2,X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,active(X2),X3) = X3 >= X3 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = X3 >= X3 = U11(X1,X2,X3) U12(mark(X1),X2) = X2 >= X2 = U12(X1,X2) U12(X1,mark(X2)) = X2 >= X2 = U12(X1,X2) U12(active(X1),X2) = X2 >= X2 = U12(X1,X2) U12(X1,active(X2)) = X2 >= X2 = U12(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U13(mark(X)) = X >= X = U13(X) U13(active(X)) = X >= X = U13(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X2 + X3 >= X2 + X3 = U41(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(U13(X)) -> active#(U13(mark(X))) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = 0, [plus](x0, x1) = x0, [U41](x0, x1, x2) = 0, [U31](x0, x1) = x1, [U22](x0) = x0, [U21](x0, x1) = 1, [U13](x0) = 1, [mark](x0) = x0, [U12](x0, x1) = 1, [isNat](x0) = 1, [active](x0) = x0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(U13(X)) = 1 >= 0 = active#(U13(mark(X))) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= 0 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatKind(V2))) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 1 >= 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 1 >= 1 = active(U12(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U13(X)) = 1 >= 1 = active(U13(mark(X))) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = X1 >= X1 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) plus(mark(X1),X2) = X1 >= X1 = plus(X1,X2) plus(X1,mark(X2)) = X1 >= X1 = plus(X1,X2) plus(active(X1),X2) = X1 >= X1 = plus(X1,X2) plus(X1,active(X2)) = X1 >= X1 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))), M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U31](x0, x1) = 0, [U22](x0) = 1, [U21](x0, x1) = 0, [U13](x0) = 0, [mark](x0) = 0, [U12](x0, x1) = 0, [isNat](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 0 = active#(and(mark(X1),X2)) active#(isNatKind(plus(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatKind(V2))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,mark(X2)) = 0 >= 0 = U12(X1,X2) U12(active(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,active(X2)) = 0 >= 0 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = 0 >= 0 = U13(X) U13(active(X)) = 0 >= 0 = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = 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) and(mark(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,mark(X2)) = 0 >= 0 = and(X1,X2) and(active(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,active(X2)) = 0 >= 0 = and(X1,X2) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(isNatKind(0())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N))), M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = x0, [0] = 1, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x1 + x2 + 1, [U31](x0, x1) = x1 + 1, [U22](x0) = 1, [U21](x0, x1) = 1, [U13](x0) = 0, [mark](x0) = x0, [U12](x0, x1) = 0, [isNat](x0) = 1, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: mark#(isNatKind(X)) = X >= X = active#(isNatKind(X)) active#(isNatKind(0())) = 1 >= 0 = mark#(tt()) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) active#(isNatKind(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(and(isNatKind(V1),isNatKind(V2))) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 >= V1 + V2 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 + 1 >= V1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(U31(X1,X2)) = X2 + 1 >= X2 + 1 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,mark(X2)) = 0 >= 0 = U12(X1,X2) U12(active(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,active(X2)) = 0 >= 0 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 0 >= 0 = U13(X) U13(active(X)) = 0 >= 0 = U13(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) U31(mark(X1),X2) = X2 + 1 >= X2 + 1 = U31(X1,X2) U31(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U31(X1,X2) U31(active(X1),X2) = X2 + 1 >= X2 + 1 = U31(X1,X2) U31(X1,active(X2)) = X2 + 1 >= X2 + 1 = U31(X1,X2) U41(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(and(X1,X2)) -> mark#(X1) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N))), M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = x0, [0] = 1, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = x1 + x2 + 1, [U31](x0, x1) = x1, [U22](x0) = 0, [U21](x0, x1) = 0, [U13](x0) = 1, [mark](x0) = x0, [U12](x0, x1) = 1, [isNat](x0) = 1, [active](x0) = x0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = X >= X = active#(isNatKind(X)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) active#(isNatKind(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 = mark#(and(isNatKind(V1),isNatKind(V2))) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 >= V1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 2 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 1 >= 1 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 1 >= 1 = active(U12(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U13(X)) = 1 >= 1 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U13(mark(X)) = 1 >= 1 = U13(X) U13(active(X)) = 1 >= 1 = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(and(X1,X2)) -> mark#(X1) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N))), M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x0 + x1 + 1, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = x1 + x2 + 1, [U31](x0, x1) = x1, [U22](x0) = 0, [U21](x0, x1) = 0, [U13](x0) = 0, [mark](x0) = x0, [U12](x0, x1) = 0, [isNat](x0) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: mark#(isNatKind(X)) = X >= 0 = active#(isNatKind(X)) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= N = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 >= V1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U41(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) U11(mark(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 0 >= 0 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 0 >= 0 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 0 >= 0 = U11(X1,X2,X3) U12(mark(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,mark(X2)) = 0 >= 0 = U12(X1,X2) U12(active(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,active(X2)) = 0 >= 0 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = 0 >= 0 = U13(X) U13(active(X)) = 0 >= 0 = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U41(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U41(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) and(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) and(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N))), M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = 0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = 0, [s](x0) = 0, [plus](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [U13](x0) = 0, [mark](x0) = 0, [U12](x0, x1) = 0, [isNat](x0) = 0, [active](x0) = 0, [U11](x0, x1, x2) = 1, [tt] = 0 orientation: mark#(isNatKind(X)) = 1 >= 0 = active#(isNatKind(X)) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) = 0 >= 0 = active(U11(mark(X1),X2,X3)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X1,X2)) = 0 >= 0 = active(U12(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U13(X)) = 0 >= 0 = active(U13(mark(X))) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) U11(mark(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 1 >= 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 1 >= 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 1 >= 1 = U11(X1,X2,X3) U12(mark(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,mark(X2)) = 0 >= 0 = U12(X1,X2) U12(active(X1),X2) = 0 >= 0 = U12(X1,X2) U12(X1,active(X2)) = 0 >= 0 = U12(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U13(mark(X)) = 0 >= 0 = U13(X) U13(active(X)) = 0 >= 0 = U13(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = 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) and(mark(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,mark(X2)) = 0 >= 0 = and(X1,X2) and(active(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,active(X2)) = 0 >= 0 = and(X1,X2) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N))), M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U12#(mark(X1),X2) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) U12#(active(X1),X2) -> U12#(X1,X2) U12#(X1,active(X2)) -> U12#(X1,X2) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open DPs: isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open DPs: U13#(mark(X)) -> U13#(X) U13#(active(X)) -> U13#(X) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open DPs: U22#(mark(X)) -> U22#(X) U22#(active(X)) -> U22#(X) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open 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(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open DPs: s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open DPs: isNatKind#(mark(X)) -> isNatKind#(X) isNatKind#(active(X)) -> isNatKind#(X) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open DPs: U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open DPs: U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) TRS: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open 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(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open 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(),V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) -> mark(U13(isNat(V2))) active(U13(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),N)) -> mark(N) active(U41(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(0()) -> active(0()) mark(isNatKind(X)) -> active(isNatKind(X)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U13(mark(X)) -> U13(X) U13(active(X)) -> U13(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(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) 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) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open