YES Problem: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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__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) 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(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__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__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__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) 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#(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__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__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) 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(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__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__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__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) 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#(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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) graph: 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#(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#(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__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#(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#(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#(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#(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#(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#(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#(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#(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#(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#(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#(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#(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) Arctic Interpretation Processor: dimension: 1 usable rules: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = x0 + x1 + 0, [a__plus#](x0, x1) = x0 + x1 + 0, [a__U21#](x0, x1, x2) = x0 + x1 + x2 + 0, [mark#](x0) = x0 + 0, [a__U11#](x0, x1) = x0 + x1 + 0, [and](x0, x1) = x0 + x1 + 0, [U21](x0, x1, x2) = x0 + x1 + x2 + 0, [U11](x0, x1) = 4x0 + x1 + 4, [isNat](x0) = 0, [plus](x0, x1) = x0 + x1 + 0, [a__isNat](x0) = 0, [0] = 6, [a__and](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 0, [a__plus](x0, x1) = x0 + x1 + 0, [a__U21](x0, x1, x2) = x0 + x1 + x2 + 0, [mark](x0) = x0 + 0, [a__U11](x0, x1) = 4x0 + x1 + 4, [tt] = 0 orientation: a__U11#(tt(),N) = N + 0 >= N + 0 = mark#(N) a__U21#(tt(),M,N) = M + N + 0 >= M + 0 = mark#(M) a__U21#(tt(),M,N) = M + N + 0 >= N + 0 = mark#(N) a__U21#(tt(),M,N) = M + N + 0 >= M + N + 0 = a__plus#(mark(N),mark(M)) a__and#(tt(),X) = X + 0 >= X + 0 = mark#(X) a__isNat#(plus(V1,V2)) = 0 >= 0 = a__isNat#(V1) a__isNat#(plus(V1,V2)) = 0 >= 0 = a__and#(a__isNat(V1),isNat(V2)) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) a__plus#(N,0()) = N + 6 >= 0 = a__isNat#(N) a__plus#(N,0()) = N + 6 >= N + 0 = a__U11#(a__isNat(N),N) a__plus#(N,s(M)) = M + N + 0 >= 0 = a__isNat#(M) a__plus#(N,s(M)) = M + N + 0 >= 0 = a__and#(a__isNat(M),isNat(N)) a__plus#(N,s(M)) = M + N + 0 >= M + N + 0 = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) mark#(U11(X1,X2)) = 4X1 + X2 + 4 >= X1 + 0 = mark#(X1) mark#(U11(X1,X2)) = 4X1 + X2 + 4 >= X1 + X2 + 0 = a__U11#(mark(X1),X2) mark#(U21(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + 0 = mark#(X1) mark#(U21(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__plus#(mark(X1),mark(X2)) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and#(mark(X1),X2) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__U11(tt(),N) = N + 4 >= N + 0 = mark(N) a__U21(tt(),M,N) = M + N + 0 >= M + N + 0 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X + 0 >= X + 0 = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(plus(V1,V2)) = 0 >= 0 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = 0 >= 0 = a__isNat(V1) a__plus(N,0()) = N + 6 >= N + 4 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 0 >= M + N + 0 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 6 >= 6 = 0() a__U11(X1,X2) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = U11(X1,X2) a__U21(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = plus(X1,X2) a__and(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) problem: 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__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__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) 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#(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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Restore Modifier: 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__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__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) 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#(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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Matrix Interpretation Processor: dim=1 usable rules: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = 4x1, [a__plus#](x0, x1) = 2x0 + x1, [a__U21#](x0, x1, x2) = x1 + 2x2, [mark#](x0) = x0, [a__U11#](x0, x1) = 2x1, [and](x0, x1) = 2x0 + 6x1, [U21](x0, x1, x2) = x0 + 2x1 + 4x2 + 2, [U11](x0, x1) = 2x1 + 1, [isNat](x0) = 0, [plus](x0, x1) = 4x0 + 2x1 + 2, [a__isNat](x0) = 0, [0] = 7, [a__and](x0, x1) = 2x0 + 6x1, [s](x0) = x0, [a__plus](x0, x1) = 4x0 + 2x1 + 2, [a__U21](x0, x1, x2) = x0 + 2x1 + 4x2 + 2, [mark](x0) = x0, [a__U11](x0, x1) = 2x1 + 1, [tt] = 0 orientation: a__U11#(tt(),N) = 2N >= N = mark#(N) a__U21#(tt(),M,N) = M + 2N >= M = mark#(M) a__U21#(tt(),M,N) = M + 2N >= N = mark#(N) a__U21#(tt(),M,N) = M + 2N >= M + 2N = a__plus#(mark(N),mark(M)) a__and#(tt(),X) = 4X >= X = mark#(X) a__isNat#(plus(V1,V2)) = 0 >= 0 = a__isNat#(V1) a__isNat#(plus(V1,V2)) = 0 >= 0 = a__and#(a__isNat(V1),isNat(V2)) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) a__plus#(N,0()) = 2N + 7 >= 2N = a__U11#(a__isNat(N),N) a__plus#(N,s(M)) = M + 2N >= 0 = a__isNat#(M) a__plus#(N,s(M)) = M + 2N >= 0 = a__and#(a__isNat(M),isNat(N)) a__plus#(N,s(M)) = M + 2N >= M + 2N = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) mark#(U11(X1,X2)) = 2X2 + 1 >= 2X2 = a__U11#(mark(X1),X2) mark#(U21(X1,X2,X3)) = X1 + 2X2 + 4X3 + 2 >= X1 = mark#(X1) mark#(U21(X1,X2,X3)) = X1 + 2X2 + 4X3 + 2 >= X2 + 2X3 = a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) = 4X1 + 2X2 + 2 >= X2 = mark#(X2) mark#(plus(X1,X2)) = 4X1 + 2X2 + 2 >= X1 = mark#(X1) mark#(plus(X1,X2)) = 4X1 + 2X2 + 2 >= 2X1 + X2 = a__plus#(mark(X1),mark(X2)) mark#(and(X1,X2)) = 2X1 + 6X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = 2X1 + 6X2 >= 4X2 = a__and#(mark(X1),X2) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(s(X)) = X >= X = mark#(X) a__U11(tt(),N) = 2N + 1 >= N = mark(N) a__U21(tt(),M,N) = 2M + 4N + 2 >= 2M + 4N + 2 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = 6X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(plus(V1,V2)) = 0 >= 0 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = 0 >= 0 = a__isNat(V1) a__plus(N,0()) = 4N + 16 >= 2N + 1 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = 2M + 4N + 2 >= 2M + 4N + 2 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = 2X2 + 1 >= 2X2 + 1 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X1 + 2X2 + 4X3 + 2 >= X1 + 2X2 + 4X3 + 2 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 4X1 + 2X2 + 2 >= 4X1 + 2X2 + 2 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = 2X1 + 6X2 >= 2X1 + 6X2 = a__and(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 7 >= 7 = 0() a__U11(X1,X2) = 2X2 + 1 >= 2X2 + 1 = U11(X1,X2) a__U21(X1,X2,X3) = X1 + 2X2 + 4X3 + 2 >= X1 + 2X2 + 4X3 + 2 = U21(X1,X2,X3) a__plus(X1,X2) = 4X1 + 2X2 + 2 >= 4X1 + 2X2 + 2 = plus(X1,X2) a__and(X1,X2) = 2X1 + 6X2 >= 2X1 + 6X2 = and(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) problem: 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__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__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) 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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Restore Modifier: 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__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__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) 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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) SCC Processor: #sccs: 2 #rules: 10 #arcs: 146/225 DPs: 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)) TRS: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Matrix Interpretation Processor: dim=1 usable rules: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) interpretation: [a__plus#](x0, x1) = x1, [a__U21#](x0, x1, x2) = x1 + 1, [and](x0, x1) = 2x1, [U21](x0, x1, x2) = 4x0 + 4x1 + 4x2 + 4, [U11](x0, x1) = 4x1, [isNat](x0) = 0, [plus](x0, x1) = 4x0 + 4x1, [a__isNat](x0) = 0, [0] = 0, [a__and](x0, x1) = 2x1, [s](x0) = x0 + 1, [a__plus](x0, x1) = 4x0 + 4x1, [a__U21](x0, x1, x2) = 4x0 + 4x1 + 4x2 + 4, [mark](x0) = x0, [a__U11](x0, x1) = 4x1, [tt] = 0 orientation: a__plus#(N,s(M)) = M + 1 >= M + 1 = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) a__U21#(tt(),M,N) = M + 1 >= M = a__plus#(mark(N),mark(M)) a__U11(tt(),N) = 4N >= N = mark(N) a__U21(tt(),M,N) = 4M + 4N + 4 >= 4M + 4N + 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = 2X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(plus(V1,V2)) = 0 >= 0 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = 0 >= 0 = a__isNat(V1) a__plus(N,0()) = 4N >= 4N = a__U11(a__isNat(N),N) a__plus(N,s(M)) = 4M + 4N + 4 >= 4M + 4N + 4 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = 4X2 >= 4X2 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = 2X2 >= 2X2 = a__and(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2) = 4X2 >= 4X2 = U11(X1,X2) a__U21(X1,X2,X3) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = U21(X1,X2,X3) a__plus(X1,X2) = 4X1 + 4X2 >= 4X1 + 4X2 = plus(X1,X2) a__and(X1,X2) = 2X2 >= 2X2 = and(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) problem: DPs: a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) TRS: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Restore Modifier: DPs: a__plus#(N,s(M)) -> a__U21#(a__and(a__isNat(M),isNat(N)),M,N) TRS: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/1 DPs: 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#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) a__isNat#(s(V1)) -> a__isNat#(V1) 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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Arctic Interpretation Processor: dimension: 1 usable rules: interpretation: [a__isNat#](x0) = x0, [a__and#](x0, x1) = x1 + 0, [mark#](x0) = x0, [and](x0, x1) = 7x0 + x1 + 0, [U21](x0, x1, x2) = x0 + x1 + x2 + 6, [U11](x0, x1) = x0 + x1 + 6, [isNat](x0) = x0 + 4, [plus](x0, x1) = x0 + x1 + 4, [a__isNat](x0) = x0, [0] = 7, [a__and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [a__plus](x0, x1) = x0 + x1 + 0, [a__U21](x0, x1, x2) = x0 + x1 + x2 + 0, [mark](x0) = x0 + 0, [a__U11](x0, x1) = 2x0 + x1 + 3, [tt] = 0 orientation: a__isNat#(plus(V1,V2)) = V1 + V2 + 4 >= V1 = a__isNat#(V1) a__isNat#(plus(V1,V2)) = V1 + V2 + 4 >= V2 + 4 = a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) = X + 0 >= X = mark#(X) mark#(and(X1,X2)) = 7X1 + X2 + 0 >= X1 = mark#(X1) mark#(and(X1,X2)) = 7X1 + X2 + 0 >= X2 + 0 = a__and#(mark(X1),X2) mark#(isNat(X)) = X + 4 >= X = a__isNat#(X) a__isNat#(s(V1)) = V1 + 1 >= V1 = a__isNat#(V1) mark#(s(X)) = X + 1 >= X = mark#(X) a__U11(tt(),N) = N + 3 >= N + 0 = mark(N) a__U21(tt(),M,N) = M + N + 0 >= M + N + 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X + 0 >= X + 0 = mark(X) a__isNat(0()) = 7 >= 0 = tt() a__isNat(plus(V1,V2)) = V1 + V2 + 4 >= V1 + V2 + 4 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = V1 + 1 >= V1 = a__isNat(V1) a__plus(N,0()) = N + 7 >= 2N + 3 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 1 >= M + N + 4 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X1 + X2 + 6 >= 2X1 + X2 + 3 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X1 + X2 + X3 + 6 >= X1 + X2 + X3 + 0 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 0 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = 7X1 + X2 + 0 >= X1 + X2 + 0 = a__and(mark(X1),X2) mark(isNat(X)) = X + 4 >= X = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(0()) = 7 >= 7 = 0() a__U11(X1,X2) = 2X1 + X2 + 3 >= X1 + X2 + 6 = U11(X1,X2) a__U21(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 6 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 4 = plus(X1,X2) a__and(X1,X2) = X1 + X2 >= 7X1 + X2 + 0 = and(X1,X2) a__isNat(X) = X >= X + 4 = isNat(X) problem: DPs: 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#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) a__isNat#(s(V1)) -> a__isNat#(V1) 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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Restore Modifier: DPs: 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#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) a__isNat#(s(V1)) -> a__isNat#(V1) 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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Arctic Interpretation Processor: dimension: 1 usable rules: interpretation: [a__isNat#](x0) = x0, [a__and#](x0, x1) = x1 + 1, [mark#](x0) = x0, [and](x0, x1) = x0 + x1 + 1, [U21](x0, x1, x2) = x0 + x1 + x2 + 6, [U11](x0, x1) = x0 + x1 + 2, [isNat](x0) = x0 + 7, [plus](x0, x1) = 7x0 + x1 + 7, [a__isNat](x0) = x0 + 1, [0] = 0, [a__and](x0, x1) = x1 + 0, [s](x0) = x0 + 6, [a__plus](x0, x1) = x0 + x1 + 0, [a__U21](x0, x1, x2) = x0 + x1 + x2 + 0, [mark](x0) = x0 + 2, [a__U11](x0, x1) = x1 + 12, [tt] = 0 orientation: a__isNat#(plus(V1,V2)) = 7V1 + V2 + 7 >= V1 = a__isNat#(V1) a__isNat#(plus(V1,V2)) = 7V1 + V2 + 7 >= V2 + 7 = a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) = X + 1 >= X = mark#(X) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = a__and#(mark(X1),X2) mark#(isNat(X)) = X + 7 >= X = a__isNat#(X) a__isNat#(s(V1)) = V1 + 6 >= V1 = a__isNat#(V1) mark#(s(X)) = X + 6 >= X = mark#(X) a__U11(tt(),N) = N + 12 >= N + 2 = mark(N) a__U21(tt(),M,N) = M + N + 0 >= M + N + 6 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X + 0 >= X + 2 = mark(X) a__isNat(0()) = 1 >= 0 = tt() a__isNat(plus(V1,V2)) = 7V1 + V2 + 7 >= V2 + 7 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = V1 + 6 >= V1 + 1 = a__isNat(V1) a__plus(N,0()) = N + 0 >= N + 12 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 6 >= M + N + 7 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X1 + X2 + 2 >= X2 + 12 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X1 + X2 + X3 + 6 >= X1 + X2 + X3 + 2 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 7X1 + X2 + 7 >= X1 + X2 + 2 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 + 2 >= X2 + 0 = a__and(mark(X1),X2) mark(isNat(X)) = X + 7 >= X + 1 = a__isNat(X) mark(tt()) = 2 >= 0 = tt() mark(s(X)) = X + 6 >= X + 6 = s(mark(X)) mark(0()) = 2 >= 0 = 0() a__U11(X1,X2) = X2 + 12 >= X1 + X2 + 2 = U11(X1,X2) a__U21(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 6 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 0 >= 7X1 + X2 + 7 = plus(X1,X2) a__and(X1,X2) = X2 + 0 >= X1 + X2 + 1 = and(X1,X2) a__isNat(X) = X + 1 >= X + 7 = isNat(X) problem: DPs: a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) a__isNat#(s(V1)) -> a__isNat#(V1) 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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Restore Modifier: DPs: a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) a__isNat#(s(V1)) -> a__isNat#(V1) 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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Arctic Interpretation Processor: dimension: 1 usable rules: interpretation: [a__isNat#](x0) = x0, [a__and#](x0, x1) = x1 + 0, [mark#](x0) = x0, [and](x0, x1) = 4x1 + 0, [U21](x0, x1, x2) = 2x0 + x1 + x2 + 4, [U11](x0, x1) = 3, [isNat](x0) = 2x0 + 0, [plus](x0, x1) = 1x0 + 2x1 + 0, [a__isNat](x0) = x0 + 3, [0] = 1, [a__and](x0, x1) = 2x1 + 3, [s](x0) = x0 + 0, [a__plus](x0, x1) = x0 + x1 + 1, [a__U21](x0, x1, x2) = x0 + x1 + 10x2 + 0, [mark](x0) = x0, [a__U11](x0, x1) = 2x0 + 13x1 + 4, [tt] = 4 orientation: a__isNat#(plus(V1,V2)) = 1V1 + 2V2 + 0 >= 2V2 + 0 = a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) = X + 0 >= X = mark#(X) mark#(and(X1,X2)) = 4X2 + 0 >= X2 + 0 = a__and#(mark(X1),X2) mark#(isNat(X)) = 2X + 0 >= X = a__isNat#(X) a__isNat#(s(V1)) = V1 + 0 >= V1 = a__isNat#(V1) mark#(s(X)) = X + 0 >= X = mark#(X) a__U11(tt(),N) = 13N + 6 >= N = mark(N) a__U21(tt(),M,N) = M + 10N + 4 >= M + N + 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = 2X + 3 >= X = mark(X) a__isNat(0()) = 3 >= 4 = tt() a__isNat(plus(V1,V2)) = 1V1 + 2V2 + 3 >= 4V2 + 3 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = V1 + 3 >= V1 + 3 = a__isNat(V1) a__plus(N,0()) = N + 1 >= 13N + 5 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 1 >= M + 10N + 3 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = 3 >= 2X1 + 13X2 + 4 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = 2X1 + X2 + X3 + 4 >= X1 + X2 + 10X3 + 0 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 1X1 + 2X2 + 0 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = 4X2 + 0 >= 2X2 + 3 = a__and(mark(X1),X2) mark(isNat(X)) = 2X + 0 >= X + 3 = a__isNat(X) mark(tt()) = 4 >= 4 = tt() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 1 >= 1 = 0() a__U11(X1,X2) = 2X1 + 13X2 + 4 >= 3 = U11(X1,X2) a__U21(X1,X2,X3) = X1 + X2 + 10X3 + 0 >= 2X1 + X2 + X3 + 4 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 1 >= 1X1 + 2X2 + 0 = plus(X1,X2) a__and(X1,X2) = 2X2 + 3 >= 4X2 + 0 = and(X1,X2) a__isNat(X) = X + 3 >= 2X + 0 = isNat(X) problem: DPs: a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__isNat#(s(V1)) -> a__isNat#(V1) 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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Restore Modifier: DPs: a__isNat#(plus(V1,V2)) -> a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__isNat#(s(V1)) -> a__isNat#(V1) 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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) SCC Processor: #sccs: 2 #rules: 4 #arcs: 23/25 DPs: a__isNat#(s(V1)) -> a__isNat#(V1) TRS: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Subterm Criterion Processor: simple projection: pi(a__isNat#) = 0 problem: DPs: TRS: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Qed DPs: a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) 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__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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) Subterm Criterion Processor: simple projection: pi(mark#) = 0 pi(a__and#) = 1 problem: DPs: a__and#(tt(),X) -> mark#(X) TRS: a__U11(tt(),N) -> mark(N) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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__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) 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(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__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) SCC Processor: #sccs: 0 #rules: 0 #arcs: 5/1