MAYBE Problem: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__U31(tt()) -> 0() a__U41(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__plus(N,0()) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__x(N,0()) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U31(X) -> U31(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Proof: DP Processor: DPs: a__U11#(tt(),N) -> mark#(N) a__U21#(tt(),M,N) -> mark#(M) a__U21#(tt(),M,N) -> mark#(N) a__U21#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__U41#(tt(),M,N) -> mark#(M) a__U41#(tt(),M,N) -> mark#(N) a__U41#(tt(),M,N) -> a__x#(mark(N),mark(M)) a__U41#(tt(),M,N) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) a__and#(tt(),X) -> mark#(X) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(x(V1,V2)) -> a__isNat#(V1) a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__U11#(a__isNat(N),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) a__x#(N,0()) -> a__isNat#(N) a__x#(N,0()) -> a__U31#(a__isNat(N)) a__x#(N,s(M)) -> a__isNat#(M) a__x#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) a__x#(N,s(M)) -> a__U41#(a__and(a__isNat(M),isNat(N)),M,N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U31(X)) -> mark#(X) mark#(U31(X)) -> a__U31#(mark(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__U31(tt()) -> 0() a__U41(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__plus(N,0()) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__x(N,0()) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U31(X) -> U31(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) TDG Processor: DPs: a__U11#(tt(),N) -> mark#(N) a__U21#(tt(),M,N) -> mark#(M) a__U21#(tt(),M,N) -> mark#(N) a__U21#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__U41#(tt(),M,N) -> mark#(M) a__U41#(tt(),M,N) -> mark#(N) a__U41#(tt(),M,N) -> a__x#(mark(N),mark(M)) a__U41#(tt(),M,N) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) a__and#(tt(),X) -> mark#(X) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(x(V1,V2)) -> a__isNat#(V1) a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__plus#(N,0()) -> a__isNat#(N) a__plus#(N,0()) -> a__U11#(a__isNat(N),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) a__x#(N,0()) -> a__isNat#(N) a__x#(N,0()) -> a__U31#(a__isNat(N)) a__x#(N,s(M)) -> a__isNat#(M) a__x#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) a__x#(N,s(M)) -> a__U41#(a__and(a__isNat(M),isNat(N)),M,N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U31(X)) -> mark#(X) mark#(U31(X)) -> a__U31#(mark(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__U31(tt()) -> 0() a__U41(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__plus(N,0()) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__x(N,0()) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U31(X) -> U31(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) graph: a__isNat#(x(V1,V2)) -> a__isNat#(V1) -> a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__isNat#(x(V1,V2)) -> a__isNat#(V1) -> a__isNat#(x(V1,V2)) -> a__isNat#(V1) a__isNat#(x(V1,V2)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(x(V1,V2)) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__isNat#(x(V1,V2)) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) -> a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) -> a__isNat#(x(V1,V2)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(x(V1,V2)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__and#(tt(),X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__and#(tt(),X) -> mark#(X) -> mark#(x(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(x(X1,X2)) -> mark#(X2) a__and#(tt(),X) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__and#(tt(),X) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U31(X)) -> a__U31#(mark(X)) a__and#(tt(),X) -> mark#(X) -> mark#(U31(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) a__and#(tt(),X) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) a__and#(tt(),X) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) a__x#(N,0()) -> a__isNat#(N) -> a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__x#(N,0()) -> a__isNat#(N) -> a__isNat#(x(V1,V2)) -> a__isNat#(V1) a__x#(N,0()) -> a__isNat#(N) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__x#(N,0()) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__x#(N,0()) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__x#(N,s(M)) -> a__isNat#(M) -> a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__x#(N,s(M)) -> a__isNat#(M) -> a__isNat#(x(V1,V2)) -> a__isNat#(V1) a__x#(N,s(M)) -> a__isNat#(M) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__x#(N,s(M)) -> a__isNat#(M) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__x#(N,s(M)) -> a__isNat#(M) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__x#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) -> a__and#(tt(),X) -> mark#(X) a__x#(N,s(M)) -> a__U41#(a__and(a__isNat(M),isNat(N)),M,N) -> a__U41#(tt(),M,N) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) a__x#(N,s(M)) -> a__U41#(a__and(a__isNat(M),isNat(N)),M,N) -> a__U41#(tt(),M,N) -> a__x#(mark(N),mark(M)) a__x#(N,s(M)) -> a__U41#(a__and(a__isNat(M),isNat(N)),M,N) -> a__U41#(tt(),M,N) -> mark#(N) a__x#(N,s(M)) -> a__U41#(a__and(a__isNat(M),isNat(N)),M,N) -> a__U41#(tt(),M,N) -> mark#(M) a__U41#(tt(),M,N) -> a__x#(mark(N),mark(M)) -> a__x#(N,s(M)) -> a__U41#(a__and(a__isNat(M),isNat(N)),M,N) a__U41#(tt(),M,N) -> a__x#(mark(N),mark(M)) -> a__x#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) a__U41#(tt(),M,N) -> a__x#(mark(N),mark(M)) -> a__x#(N,s(M)) -> a__isNat#(M) a__U41#(tt(),M,N) -> a__x#(mark(N),mark(M)) -> a__x#(N,0()) -> a__U31#(a__isNat(N)) a__U41#(tt(),M,N) -> a__x#(mark(N),mark(M)) -> a__x#(N,0()) -> a__isNat#(N) a__U41#(tt(),M,N) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) -> a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) a__U41#(tt(),M,N) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) -> a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) a__U41#(tt(),M,N) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) -> a__plus#(N,s(M)) -> a__isNat#(M) a__U41#(tt(),M,N) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) -> a__plus#(N,0()) -> a__U11#(a__isNat(N),N) a__U41#(tt(),M,N) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) -> a__plus#(N,0()) -> a__isNat#(N) a__U41#(tt(),M,N) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> mark#(N) -> mark#(isNat(X)) -> a__isNat#(X) a__U41#(tt(),M,N) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__U41#(tt(),M,N) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X2) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U31(X)) -> a__U31#(mark(X)) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U31(X)) -> mark#(X) a__U41#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U41#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> mark#(M) -> mark#(isNat(X)) -> a__isNat#(X) a__U41#(tt(),M,N) -> mark#(M) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(M) -> mark#(and(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__U41#(tt(),M,N) -> mark#(M) -> mark#(x(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(x(X1,X2)) -> mark#(X2) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U31(X)) -> a__U31#(mark(X)) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U31(X)) -> mark#(X) a__U41#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U41#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X2) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U21(X1,X2,X3)) -> mark#(X1) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U41#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2)) -> mark#(X1) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(x(V1,V2)) -> a__isNat#(V1) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__plus#(N,0()) -> a__isNat#(N) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__plus#(N,0()) -> a__U11#(a__isNat(N),N) -> a__U11#(tt(),N) -> mark#(N) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(x(V1,V2)) -> a__isNat#(V1) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__plus#(N,s(M)) -> a__isNat#(M) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) -> a__and#(tt(),X) -> mark#(X) a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) -> a__U21#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) -> a__U21#(tt(),M,N) -> mark#(N) a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) -> a__U21#(tt(),M,N) -> mark#(M) a__U21#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) a__U21#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) a__U21#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__isNat#(M) a__U21#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,0()) -> a__U11#(a__isNat(N),N) a__U21#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,0()) -> a__isNat#(N) a__U21#(tt(),M,N) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__U21#(tt(),M,N) -> mark#(N) -> mark#(isNat(X)) -> a__isNat#(X) a__U21#(tt(),M,N) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U21#(tt(),M,N) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(N) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__U21#(tt(),M,N) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X2) a__U21#(tt(),M,N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U21#(tt(),M,N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(N) -> mark#(U31(X)) -> a__U31#(mark(X)) a__U21#(tt(),M,N) -> mark#(N) -> mark#(U31(X)) -> mark#(X) a__U21#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U21#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__U21#(tt(),M,N) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) a__U21#(tt(),M,N) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U21#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(M) -> mark#(s(X)) -> mark#(X) a__U21#(tt(),M,N) -> mark#(M) -> mark#(isNat(X)) -> a__isNat#(X) a__U21#(tt(),M,N) -> mark#(M) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U21#(tt(),M,N) -> mark#(M) -> mark#(and(X1,X2)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(M) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__U21#(tt(),M,N) -> mark#(M) -> mark#(x(X1,X2)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(M) -> mark#(x(X1,X2)) -> mark#(X2) a__U21#(tt(),M,N) -> mark#(M) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U21#(tt(),M,N) -> mark#(M) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(M) -> mark#(U31(X)) -> a__U31#(mark(X)) a__U21#(tt(),M,N) -> mark#(M) -> mark#(U31(X)) -> mark#(X) a__U21#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U21#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X2) a__U21#(tt(),M,N) -> mark#(M) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) a__U21#(tt(),M,N) -> mark#(M) -> mark#(U21(X1,X2,X3)) -> mark#(X1) a__U21#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U21#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) -> a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) -> a__U41#(tt(),M,N) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) -> a__U41#(tt(),M,N) -> a__x#(mark(N),mark(M)) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) -> a__U41#(tt(),M,N) -> mark#(N) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) -> a__U41#(tt(),M,N) -> mark#(M) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X2) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U41(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U31(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(U31(X)) -> mark#(X) -> mark#(x(X1,X2)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(x(X1,X2)) -> mark#(X2) mark#(U31(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U31(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U31(X)) -> mark#(X) -> mark#(U31(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U31(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U31(X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(U31(X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U31(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) -> a__U21#(tt(),M,N) -> a__plus#(mark(N),mark(M)) mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) -> a__U21#(tt(),M,N) -> mark#(N) mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) -> a__U21#(tt(),M,N) -> mark#(M) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X2) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) -> a__U11#(tt(),N) -> mark#(N) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) -> a__x#(N,s(M)) -> a__U41#(a__and(a__isNat(M),isNat(N)),M,N) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) -> a__x#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) -> a__x#(N,s(M)) -> a__isNat#(M) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) -> a__x#(N,0()) -> a__U31#(a__isNat(N)) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) -> a__x#(N,0()) -> a__isNat#(N) mark#(x(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(x(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> a__isNat#(X) mark#(x(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(x(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X2) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(x(X1,X2)) -> mark#(X2) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X2) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(x(X1,X2)) -> mark#(X2) -> mark#(U31(X)) -> mark#(X) mark#(x(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(x(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(x(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(x(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(x(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(x(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(x(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(x(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(x(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(x(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(x(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(x(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(x(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(x(V1,V2)) -> a__isNat#(V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__isNat#(V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(plus(V1,V2)) -> a__isNat#(V1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__isNat#(M) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,0()) -> a__U11#(a__isNat(N),N) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,0()) -> a__isNat#(N) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> a__isNat#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(x(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U31(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(x(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(x(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(x(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U31(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) a__U11#(tt(),N) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__U11#(tt(),N) -> mark#(N) -> mark#(isNat(X)) -> a__isNat#(X) a__U11#(tt(),N) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U11#(tt(),N) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__U11#(tt(),N) -> mark#(N) -> mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__U11#(tt(),N) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X1) a__U11#(tt(),N) -> mark#(N) -> mark#(x(X1,X2)) -> mark#(X2) a__U11#(tt(),N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U11#(tt(),N) -> mark#(N) -> mark#(U41(X1,X2,X3)) -> mark#(X1) a__U11#(tt(),N) -> mark#(N) -> mark#(U31(X)) -> a__U31#(mark(X)) a__U11#(tt(),N) -> mark#(N) -> mark#(U31(X)) -> mark#(X) a__U11#(tt(),N) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U11#(tt(),N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__U11#(tt(),N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__U11#(tt(),N) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) a__U11#(tt(),N) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> mark#(X1) a__U11#(tt(),N) -> mark#(N) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),N) -> mark#(N) -> mark#(U11(X1,X2)) -> mark#(X1) SCC Processor: #sccs: 1 #rules: 40 #arcs: 374/1764 DPs: a__isNat#(x(V1,V2)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__isNat#(V1) a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),N) -> mark#(N) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) a__U21#(tt(),M,N) -> mark#(M) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) -> a__isNat#(N) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(x(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__plus#(N,0()) -> a__U11#(a__isNat(N),N) a__plus#(N,s(M)) -> a__isNat#(M) a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) a__U21#(tt(),M,N) -> mark#(N) mark#(U31(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) a__U41#(tt(),M,N) -> mark#(M) mark#(x(X1,X2)) -> mark#(X2) mark#(x(X1,X2)) -> mark#(X1) mark#(x(X1,X2)) -> a__x#(mark(X1),mark(X2)) a__x#(N,0()) -> a__isNat#(N) a__x#(N,s(M)) -> a__isNat#(M) a__x#(N,s(M)) -> a__and#(a__isNat(M),isNat(N)) a__x#(N,s(M)) -> a__U41#(a__and(a__isNat(M),isNat(N)),M,N) a__U41#(tt(),M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) a__U41#(tt(),M,N) -> a__x#(mark(N),mark(M)) a__U41#(tt(),M,N) -> a__plus#(a__x(mark(N),mark(M)),mark(N)) a__U21#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__U31(tt()) -> 0() a__U41(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__plus(N,0()) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__x(N,0()) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2) -> U11(X1,X2) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U31(X) -> U31(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Open