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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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(),V1,V2)) -> isNat#(V1) active#(U31(tt(),V1,V2)) -> U32#(isNat(V1),V2) active#(U31(tt(),V1,V2)) -> mark#(U32(isNat(V1),V2)) active#(U32(tt(),V2)) -> isNat#(V2) active#(U32(tt(),V2)) -> U33#(isNat(V2)) active#(U32(tt(),V2)) -> mark#(U33(isNat(V2))) active#(U33(tt())) -> mark#(tt()) active#(U41(tt(),N)) -> mark#(N) active#(U51(tt(),M,N)) -> plus#(N,M) active#(U51(tt(),M,N)) -> s#(plus(N,M)) active#(U51(tt(),M,N)) -> mark#(s(plus(N,M))) active#(U61(tt())) -> mark#(0()) active#(U71(tt(),M,N)) -> x#(N,M) active#(U71(tt(),M,N)) -> plus#(x(N,M),N) active#(U71(tt(),M,N)) -> mark#(plus(x(N,M),N)) 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#(isNat(x(V1,V2))) -> isNatKind#(V2) active#(isNat(x(V1,V2))) -> isNatKind#(V1) active#(isNat(x(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNat(x(V1,V2))) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2) active#(isNat(x(V1,V2))) -> mark#(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) 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#(isNatKind(x(V1,V2))) -> isNatKind#(V2) active#(isNatKind(x(V1,V2))) -> isNatKind#(V1) active#(isNatKind(x(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNatKind(x(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) 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())) -> U41#(and(isNat(N),isNatKind(N)),N) active#(plus(N,0())) -> mark#(U41(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))) -> U51#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) active#(plus(N,s(M))) -> mark#(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active#(x(N,0())) -> isNatKind#(N) active#(x(N,0())) -> isNat#(N) active#(x(N,0())) -> and#(isNat(N),isNatKind(N)) active#(x(N,0())) -> U61#(and(isNat(N),isNatKind(N))) active#(x(N,0())) -> mark#(U61(and(isNat(N),isNatKind(N)))) active#(x(N,s(M))) -> isNatKind#(N) active#(x(N,s(M))) -> isNat#(N) active#(x(N,s(M))) -> and#(isNat(N),isNatKind(N)) active#(x(N,s(M))) -> isNatKind#(M) active#(x(N,s(M))) -> isNat#(M) active#(x(N,s(M))) -> and#(isNat(M),isNatKind(M)) active#(x(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(x(N,s(M))) -> U71#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) active#(x(N,s(M))) -> mark#(U71(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,X3)) -> mark#(X1) mark#(U31(X1,X2,X3)) -> U31#(mark(X1),X2,X3) mark#(U31(X1,X2,X3)) -> active#(U31(mark(X1),X2,X3)) mark#(U32(X1,X2)) -> mark#(X1) mark#(U32(X1,X2)) -> U32#(mark(X1),X2) mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2)) mark#(U33(X)) -> mark#(X) mark#(U33(X)) -> U33#(mark(X)) mark#(U33(X)) -> active#(U33(mark(X))) mark#(U41(X1,X2)) -> mark#(X1) mark#(U41(X1,X2)) -> U41#(mark(X1),X2) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> active#(U51(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#(U61(X)) -> mark#(X) mark#(U61(X)) -> U61#(mark(X)) mark#(U61(X)) -> active#(U61(mark(X))) mark#(0()) -> active#(0()) mark#(U71(X1,X2,X3)) -> mark#(X1) mark#(U71(X1,X2,X3)) -> U71#(mark(X1),X2,X3) mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3)) mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> x#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> active#(x(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#(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,X3) -> U31#(X1,X2,X3) U31#(X1,mark(X2),X3) -> U31#(X1,X2,X3) U31#(X1,X2,mark(X3)) -> U31#(X1,X2,X3) U31#(active(X1),X2,X3) -> U31#(X1,X2,X3) U31#(X1,active(X2),X3) -> U31#(X1,X2,X3) U31#(X1,X2,active(X3)) -> U31#(X1,X2,X3) U32#(mark(X1),X2) -> U32#(X1,X2) U32#(X1,mark(X2)) -> U32#(X1,X2) U32#(active(X1),X2) -> U32#(X1,X2) U32#(X1,active(X2)) -> U32#(X1,X2) U33#(mark(X)) -> U33#(X) U33#(active(X)) -> U33#(X) U41#(mark(X1),X2) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) U41#(active(X1),X2) -> U41#(X1,X2) U41#(X1,active(X2)) -> U41#(X1,X2) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,mark(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,mark(X3)) -> U51#(X1,X2,X3) U51#(active(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,active(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,active(X3)) -> U51#(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) U61#(mark(X)) -> U61#(X) U61#(active(X)) -> U61#(X) U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3) U71#(X1,mark(X2),X3) -> U71#(X1,X2,X3) U71#(X1,X2,mark(X3)) -> U71#(X1,X2,X3) U71#(active(X1),X2,X3) -> U71#(X1,X2,X3) U71#(X1,active(X2),X3) -> U71#(X1,X2,X3) U71#(X1,X2,active(X3)) -> U71#(X1,X2,X3) x#(mark(X1),X2) -> x#(X1,X2) x#(X1,mark(X2)) -> x#(X1,X2) x#(active(X1),X2) -> x#(X1,X2) x#(X1,active(X2)) -> x#(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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) EDG 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(),V1,V2)) -> isNat#(V1) active#(U31(tt(),V1,V2)) -> U32#(isNat(V1),V2) active#(U31(tt(),V1,V2)) -> mark#(U32(isNat(V1),V2)) active#(U32(tt(),V2)) -> isNat#(V2) active#(U32(tt(),V2)) -> U33#(isNat(V2)) active#(U32(tt(),V2)) -> mark#(U33(isNat(V2))) active#(U33(tt())) -> mark#(tt()) active#(U41(tt(),N)) -> mark#(N) active#(U51(tt(),M,N)) -> plus#(N,M) active#(U51(tt(),M,N)) -> s#(plus(N,M)) active#(U51(tt(),M,N)) -> mark#(s(plus(N,M))) active#(U61(tt())) -> mark#(0()) active#(U71(tt(),M,N)) -> x#(N,M) active#(U71(tt(),M,N)) -> plus#(x(N,M),N) active#(U71(tt(),M,N)) -> mark#(plus(x(N,M),N)) 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#(isNat(x(V1,V2))) -> isNatKind#(V2) active#(isNat(x(V1,V2))) -> isNatKind#(V1) active#(isNat(x(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNat(x(V1,V2))) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2) active#(isNat(x(V1,V2))) -> mark#(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) 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#(isNatKind(x(V1,V2))) -> isNatKind#(V2) active#(isNatKind(x(V1,V2))) -> isNatKind#(V1) active#(isNatKind(x(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNatKind(x(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) 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())) -> U41#(and(isNat(N),isNatKind(N)),N) active#(plus(N,0())) -> mark#(U41(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))) -> U51#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) active#(plus(N,s(M))) -> mark#(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active#(x(N,0())) -> isNatKind#(N) active#(x(N,0())) -> isNat#(N) active#(x(N,0())) -> and#(isNat(N),isNatKind(N)) active#(x(N,0())) -> U61#(and(isNat(N),isNatKind(N))) active#(x(N,0())) -> mark#(U61(and(isNat(N),isNatKind(N)))) active#(x(N,s(M))) -> isNatKind#(N) active#(x(N,s(M))) -> isNat#(N) active#(x(N,s(M))) -> and#(isNat(N),isNatKind(N)) active#(x(N,s(M))) -> isNatKind#(M) active#(x(N,s(M))) -> isNat#(M) active#(x(N,s(M))) -> and#(isNat(M),isNatKind(M)) active#(x(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(x(N,s(M))) -> U71#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) active#(x(N,s(M))) -> mark#(U71(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,X3)) -> mark#(X1) mark#(U31(X1,X2,X3)) -> U31#(mark(X1),X2,X3) mark#(U31(X1,X2,X3)) -> active#(U31(mark(X1),X2,X3)) mark#(U32(X1,X2)) -> mark#(X1) mark#(U32(X1,X2)) -> U32#(mark(X1),X2) mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2)) mark#(U33(X)) -> mark#(X) mark#(U33(X)) -> U33#(mark(X)) mark#(U33(X)) -> active#(U33(mark(X))) mark#(U41(X1,X2)) -> mark#(X1) mark#(U41(X1,X2)) -> U41#(mark(X1),X2) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> active#(U51(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#(U61(X)) -> mark#(X) mark#(U61(X)) -> U61#(mark(X)) mark#(U61(X)) -> active#(U61(mark(X))) mark#(0()) -> active#(0()) mark#(U71(X1,X2,X3)) -> mark#(X1) mark#(U71(X1,X2,X3)) -> U71#(mark(X1),X2,X3) mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3)) mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> x#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> active#(x(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#(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,X3) -> U31#(X1,X2,X3) U31#(X1,mark(X2),X3) -> U31#(X1,X2,X3) U31#(X1,X2,mark(X3)) -> U31#(X1,X2,X3) U31#(active(X1),X2,X3) -> U31#(X1,X2,X3) U31#(X1,active(X2),X3) -> U31#(X1,X2,X3) U31#(X1,X2,active(X3)) -> U31#(X1,X2,X3) U32#(mark(X1),X2) -> U32#(X1,X2) U32#(X1,mark(X2)) -> U32#(X1,X2) U32#(active(X1),X2) -> U32#(X1,X2) U32#(X1,active(X2)) -> U32#(X1,X2) U33#(mark(X)) -> U33#(X) U33#(active(X)) -> U33#(X) U41#(mark(X1),X2) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) U41#(active(X1),X2) -> U41#(X1,X2) U41#(X1,active(X2)) -> U41#(X1,X2) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,mark(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,mark(X3)) -> U51#(X1,X2,X3) U51#(active(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,active(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,active(X3)) -> U51#(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) U61#(mark(X)) -> U61#(X) U61#(active(X)) -> U61#(X) U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3) U71#(X1,mark(X2),X3) -> U71#(X1,X2,X3) U71#(X1,X2,mark(X3)) -> U71#(X1,X2,X3) U71#(active(X1),X2,X3) -> U71#(X1,X2,X3) U71#(X1,active(X2),X3) -> U71#(X1,X2,X3) U71#(X1,X2,active(X3)) -> U71#(X1,X2,X3) x#(mark(X1),X2) -> x#(X1,X2) x#(X1,mark(X2)) -> x#(X1,X2) x#(active(X1),X2) -> x#(X1,X2) x#(X1,active(X2)) -> x#(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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: 19 #rules: 121 #arcs: 3671/40000 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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(isNatKind(x(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(x(N,s(M))) -> mark#(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark#(and(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> active#(x(mark(X1),mark(X2))) active#(x(N,0())) -> mark#(U61(and(isNat(N),isNatKind(N)))) mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X2) mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3)) active#(plus(N,s(M))) -> mark#(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) mark#(U71(X1,X2,X3)) -> mark#(X1) mark#(U61(X)) -> active#(U61(mark(X))) active#(plus(N,0())) -> mark#(U41(and(isNat(N),isNatKind(N)),N)) mark#(U61(X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(s(X)) -> active#(s(mark(X))) active#(isNatKind(plus(V1,V2))) -> mark#(and(isNatKind(V1),isNatKind(V2))) mark#(s(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) active#(isNat(x(V1,V2))) -> mark#(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U41(X1,X2)) -> mark#(X1) mark#(U33(X)) -> active#(U33(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) mark#(U33(X)) -> mark#(X) mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(U32(X1,X2)) -> mark#(X1) mark#(U31(X1,X2,X3)) -> active#(U31(mark(X1),X2,X3)) active#(U71(tt(),M,N)) -> mark#(plus(x(N,M),N)) mark#(U31(X1,X2,X3)) -> mark#(X1) mark#(U22(X)) -> active#(U22(mark(X))) active#(U51(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U41(tt(),N)) -> mark#(N) mark#(U21(X1,X2)) -> mark#(X1) mark#(U13(X)) -> active#(U13(mark(X))) active#(U32(tt(),V2)) -> mark#(U33(isNat(V2))) mark#(U13(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(U31(tt(),V1,V2)) -> mark#(U32(isNat(V1),V2)) mark#(U12(X1,X2)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> active#(U11(mark(X1),X2,X3)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2,X3)) -> mark#(X1) active#(U12(tt(),V2)) -> mark#(U13(isNat(V2))) active#(U11(tt(),V1,V2)) -> mark#(U12(isNat(V1),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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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: U12#(active(X1),X2) -> U12#(X1,X2) U12#(X1,active(X2)) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) U12#(mark(X1),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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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: U32#(mark(X1),X2) -> U32#(X1,X2) U32#(X1,active(X2)) -> U32#(X1,X2) U32#(active(X1),X2) -> U32#(X1,X2) U32#(X1,mark(X2)) -> U32#(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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: U33#(mark(X)) -> U33#(X) U33#(active(X)) -> U33#(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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,active(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,mark(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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: x#(mark(X1),X2) -> x#(X1,X2) x#(X1,active(X2)) -> x#(X1,X2) x#(active(X1),X2) -> x#(X1,X2) x#(X1,mark(X2)) -> x#(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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,active(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,mark(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U11#(X1,mark(X2),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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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,active(X2)) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) U21#(X1,mark(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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,X3) -> U31#(X1,X2,X3) U31#(X1,X2,active(X3)) -> U31#(X1,X2,X3) U31#(X1,active(X2),X3) -> U31#(X1,X2,X3) U31#(active(X1),X2,X3) -> U31#(X1,X2,X3) U31#(X1,X2,mark(X3)) -> U31#(X1,X2,X3) U31#(X1,mark(X2),X3) -> U31#(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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) -> U41#(X1,X2) U41#(X1,active(X2)) -> U41#(X1,X2) U41#(active(X1),X2) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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: U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,X2,active(X3)) -> U51#(X1,X2,X3) U51#(X1,active(X2),X3) -> U51#(X1,X2,X3) U51#(active(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,X2,mark(X3)) -> U51#(X1,X2,X3) U51#(X1,mark(X2),X3) -> U51#(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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: U61#(mark(X)) -> U61#(X) U61#(active(X)) -> U61#(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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: U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3) U71#(X1,X2,active(X3)) -> U71#(X1,X2,X3) U71#(X1,active(X2),X3) -> U71#(X1,X2,X3) U71#(active(X1),X2,X3) -> U71#(X1,X2,X3) U71#(X1,X2,mark(X3)) -> U71#(X1,X2,X3) U71#(X1,mark(X2),X3) -> U71#(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(),V1,V2)) -> mark(U32(isNat(V1),V2)) active(U32(tt(),V2)) -> mark(U33(isNat(V2))) active(U33(tt())) -> mark(tt()) active(U41(tt(),N)) -> mark(N) active(U51(tt(),M,N)) -> mark(s(plus(N,M))) active(U61(tt())) -> mark(0()) active(U71(tt(),M,N)) -> mark(plus(x(N,M),N)) 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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N)))) active(x(N,s(M))) -> mark(U71(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,X3)) -> active(U31(mark(X1),X2,X3)) mark(U32(X1,X2)) -> active(U32(mark(X1),X2)) mark(U33(X)) -> active(U33(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0()) -> active(0()) mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3)) mark(x(X1,X2)) -> active(x(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) 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,X3) -> U31(X1,X2,X3) U31(X1,mark(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,mark(X3)) -> U31(X1,X2,X3) U31(active(X1),X2,X3) -> U31(X1,X2,X3) U31(X1,active(X2),X3) -> U31(X1,X2,X3) U31(X1,X2,active(X3)) -> U31(X1,X2,X3) U32(mark(X1),X2) -> U32(X1,X2) U32(X1,mark(X2)) -> U32(X1,X2) U32(active(X1),X2) -> U32(X1,X2) U32(X1,active(X2)) -> U32(X1,X2) U33(mark(X)) -> U33(X) U33(active(X)) -> U33(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,mark(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,mark(X3)) -> U71(X1,X2,X3) U71(active(X1),X2,X3) -> U71(X1,X2,X3) U71(X1,active(X2),X3) -> U71(X1,X2,X3) U71(X1,X2,active(X3)) -> U71(X1,X2,X3) x(mark(X1),X2) -> x(X1,X2) x(X1,mark(X2)) -> x(X1,X2) x(active(X1),X2) -> x(X1,X2) x(X1,active(X2)) -> x(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