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#(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) 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) 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) SCC Processor: #sccs: 1 #rules: 24 #arcs: 146/576 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#(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__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#(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__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: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = x1, [a__plus#](x0, x1) = x0 + x1 + 1, [a__U21#](x0, x1, x2) = x1 + x2 + 1, [mark#](x0) = x0, [a__U11#](x0, x1) = x1, [and](x0, x1) = x0 + x1, [U21](x0, x1, x2) = x0 + x1 + x2 + 1, [U11](x0, x1) = x0 + x1, [isNat](x0) = 0, [plus](x0, x1) = x0 + x1 + 1, [a__isNat](x0) = 0, [0] = 0, [a__and](x0, x1) = x0 + x1, [s](x0) = x0, [a__plus](x0, x1) = x0 + x1 + 1, [a__U21](x0, x1, x2) = x0 + x1 + x2 + 1, [mark](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0 orientation: 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__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 + X2 >= X2 = a__U11#(mark(X1),X2) a__U11#(tt(),N) = N >= N = mark#(N) mark#(U21(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U21(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X2 + X3 + 1 = a__U21#(mark(X1),X2,X3) a__U21#(tt(),M,N) = M + N + 1 >= M = mark#(M) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) = N + 1 >= 0 = a__isNat#(N) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) a__plus#(N,0()) = N + 1 >= N = a__U11#(a__isNat(N),N) a__plus#(N,s(M)) = M + N + 1 >= 0 = a__isNat#(M) a__plus#(N,s(M)) = M + N + 1 >= 0 = a__and#(a__isNat(M),isNat(N)) a__plus#(N,s(M)) = M + N + 1 >= M + N + 1 = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) a__U21#(tt(),M,N) = M + N + 1 >= N = mark#(N) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(s(X)) = X >= X = mark#(X) a__U21#(tt(),M,N) = M + N + 1 >= M + N + 1 = a__plus#(mark(N),mark(M)) a__U11(tt(),N) = N >= N = mark(N) a__U21(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X >= 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()) = N + 1 >= N = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = 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()) = 0 >= 0 = 0() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U21(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNat(X) = 0 >= 0 = 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#(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)) -> a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__isNat#(s(V1)) -> a__isNat#(V1) 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) 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: dimension: 1 interpretation: [a__isNat#](x0) = 1, [a__and#](x0, x1) = 1, [a__plus#](x0, x1) = x0, [a__U21#](x0, x1, x2) = 0, [mark#](x0) = 1, [a__U11#](x0, x1) = 1, [and](x0, x1) = 0, [U21](x0, x1, x2) = 0, [U11](x0, x1) = 0, [isNat](x0) = 0, [plus](x0, x1) = x0, [a__isNat](x0) = 0, [0] = 0, [a__and](x0, x1) = 0, [s](x0) = 0, [a__plus](x0, x1) = x0, [a__U21](x0, x1, x2) = 0, [mark](x0) = 0, [a__U11](x0, x1) = x0, [tt] = 0 orientation: a__isNat#(plus(V1,V2)) = 1 >= 1 = a__isNat#(V1) a__isNat#(plus(V1,V2)) = 1 >= 1 = a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) = 1 >= 1 = mark#(X) mark#(U11(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2)) = 1 >= 1 = a__U11#(mark(X1),X2) a__U11#(tt(),N) = 1 >= 1 = mark#(N) mark#(U21(X1,X2,X3)) = 1 >= 0 = a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) = 1 >= 0 = a__plus#(mark(X1),mark(X2)) a__isNat#(s(V1)) = 1 >= 1 = a__isNat#(V1) a__plus#(N,s(M)) = N >= 0 = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),X2) mark#(isNat(X)) = 1 >= 1 = a__isNat#(X) mark#(s(X)) = 1 >= 1 = mark#(X) a__U21#(tt(),M,N) = 0 >= 0 = a__plus#(mark(N),mark(M)) a__U11(tt(),N) = 0 >= 0 = mark(N) a__U21(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = 0 >= 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 >= 0 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = N >= 0 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = 0 >= 0 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2) = X1 >= 0 = U11(X1,X2) a__U21(X1,X2,X3) = 0 >= 0 = U21(X1,X2,X3) a__plus(X1,X2) = X1 >= X1 = plus(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNat(X) = 0 >= 0 = 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#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),N) -> mark#(N) a__isNat#(s(V1)) -> a__isNat#(V1) 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) 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: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = 0, [a__plus#](x0, x1) = x1, [a__U21#](x0, x1, x2) = x1, [mark#](x0) = 0, [a__U11#](x0, x1) = 0, [and](x0, x1) = x1, [U21](x0, x1, x2) = x1 + x2 + 1, [U11](x0, x1) = x1, [isNat](x0) = 0, [plus](x0, x1) = x0 + x1, [a__isNat](x0) = 0, [0] = 0, [a__and](x0, x1) = x1, [s](x0) = x0 + 1, [a__plus](x0, x1) = x0 + x1, [a__U21](x0, x1, x2) = x1 + x2 + 1, [mark](x0) = x0, [a__U11](x0, x1) = x1, [tt] = 0 orientation: 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__and#(tt(),X) = 0 >= 0 = mark#(X) mark#(U11(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U11(X1,X2)) = 0 >= 0 = a__U11#(mark(X1),X2) a__U11#(tt(),N) = 0 >= 0 = mark#(N) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) a__plus#(N,s(M)) = M + 1 >= M = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__U21#(tt(),M,N) = M >= M = a__plus#(mark(N),mark(M)) a__U11(tt(),N) = N >= N = mark(N) a__U21(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X >= 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()) = N >= N = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X2 >= X2 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 >= X2 = 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) = X2 >= X2 = U11(X1,X2) a__U21(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__isNat(X) = 0 >= 0 = 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#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),N) -> mark#(N) a__isNat#(s(V1)) -> a__isNat#(V1) 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__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: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = 0, [a__plus#](x0, x1) = 0, [a__U21#](x0, x1, x2) = 1, [mark#](x0) = 0, [a__U11#](x0, x1) = 0, [and](x0, x1) = 0, [U21](x0, x1, x2) = 0, [U11](x0, x1) = 0, [isNat](x0) = 0, [plus](x0, x1) = x0, [a__isNat](x0) = 0, [0] = 0, [a__and](x0, x1) = 0, [s](x0) = 0, [a__plus](x0, x1) = x0, [a__U21](x0, x1, x2) = 0, [mark](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0 orientation: 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__and#(tt(),X) = 0 >= 0 = mark#(X) mark#(U11(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U11(X1,X2)) = 0 >= 0 = a__U11#(mark(X1),X2) a__U11#(tt(),N) = 0 >= 0 = mark#(N) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__U21#(tt(),M,N) = 1 >= 0 = a__plus#(mark(N),mark(M)) a__U11(tt(),N) = 0 >= 0 = mark(N) a__U21(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = 0 >= 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 >= 0 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = N >= 0 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = 0 >= 0 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U21(X1,X2,X3) = 0 >= 0 = U21(X1,X2,X3) a__plus(X1,X2) = X1 >= X1 = plus(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNat(X) = 0 >= 0 = 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#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),N) -> mark#(N) a__isNat#(s(V1)) -> a__isNat#(V1) 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: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = x1, [mark#](x0) = x0, [a__U11#](x0, x1) = x1, [and](x0, x1) = x0 + x1, [U21](x0, x1, x2) = x1 + x2, [U11](x0, x1) = x0 + x1 + 1, [isNat](x0) = 0, [plus](x0, x1) = x0 + x1, [a__isNat](x0) = 0, [0] = 1, [a__and](x0, x1) = x0 + x1, [s](x0) = x0, [a__plus](x0, x1) = x0 + x1, [a__U21](x0, x1, x2) = x1 + x2, [mark](x0) = x0, [a__U11](x0, x1) = x0 + x1 + 1, [tt] = 0 orientation: 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__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 + X2 + 1 >= X2 = a__U11#(mark(X1),X2) a__U11#(tt(),N) = N >= N = mark#(N) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(s(X)) = X >= X = mark#(X) a__U11(tt(),N) = N + 1 >= N = mark(N) a__U21(tt(),M,N) = M + N >= M + N = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X >= 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()) = N + 1 >= N + 1 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N >= M + N = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X2 + X3 >= X2 + X3 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = 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()) = 1 >= 1 = 0() a__U11(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(X1,X2) a__U21(X1,X2,X3) = X2 + X3 >= X2 + X3 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNat(X) = 0 >= 0 = 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) a__U11#(tt(),N) -> mark#(N) a__isNat#(s(V1)) -> a__isNat#(V1) 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: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = 0, [mark#](x0) = 0, [a__U11#](x0, x1) = 1, [and](x0, x1) = 0, [U21](x0, x1, x2) = 0, [U11](x0, x1) = 0, [isNat](x0) = 0, [plus](x0, x1) = 0, [a__isNat](x0) = 0, [0] = 0, [a__and](x0, x1) = 0, [s](x0) = 0, [a__plus](x0, x1) = 0, [a__U21](x0, x1, x2) = 0, [mark](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0 orientation: 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__and#(tt(),X) = 0 >= 0 = mark#(X) a__U11#(tt(),N) = 1 >= 0 = mark#(N) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__U11(tt(),N) = 0 >= 0 = mark(N) a__U21(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = 0 >= 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()) = 0 >= 0 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = 0 >= 0 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = 0 >= 0 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U21(X1,X2,X3) = 0 >= 0 = U21(X1,X2,X3) a__plus(X1,X2) = 0 >= 0 = plus(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNat(X) = 0 >= 0 = 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) a__isNat#(s(V1)) -> a__isNat#(V1) 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: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = x1, [mark#](x0) = x0, [and](x0, x1) = x0 + x1, [U21](x0, x1, x2) = x1 + x2 + 1, [U11](x0, x1) = x1, [isNat](x0) = 0, [plus](x0, x1) = x0 + x1, [a__isNat](x0) = 0, [0] = 0, [a__and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [a__plus](x0, x1) = x0 + x1, [a__U21](x0, x1, x2) = x1 + x2 + 1, [mark](x0) = x0, [a__U11](x0, x1) = x1, [tt] = 0 orientation: 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__and#(tt(),X) = X >= X = mark#(X) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(s(X)) = X + 1 >= X = mark#(X) a__U11(tt(),N) = N >= N = mark(N) a__U21(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X >= 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()) = N >= N = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X2 >= X2 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = 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) = X2 >= X2 = U11(X1,X2) a__U21(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNat(X) = 0 >= 0 = 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) a__isNat#(s(V1)) -> a__isNat#(V1) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(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: dimension: 1 interpretation: [a__isNat#](x0) = x0, [a__and#](x0, x1) = x1, [mark#](x0) = x0, [and](x0, x1) = x0 + x1, [U21](x0, x1, x2) = x1 + x2 + 1, [U11](x0, x1) = x1, [isNat](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [a__isNat](x0) = x0, [0] = 0, [a__and](x0, x1) = x0 + x1, [s](x0) = x0, [a__plus](x0, x1) = x0 + x1 + 1, [a__U21](x0, x1, x2) = x1 + x2 + 1, [mark](x0) = x0, [a__U11](x0, x1) = x1, [tt] = 0 orientation: a__isNat#(plus(V1,V2)) = V1 + V2 + 1 >= V1 = a__isNat#(V1) a__isNat#(plus(V1,V2)) = V1 + V2 + 1 >= V2 = a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) = X >= X = mark#(X) a__isNat#(s(V1)) = V1 >= V1 = a__isNat#(V1) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNat(X)) = X >= X = a__isNat#(X) a__U11(tt(),N) = N >= N = mark(N) a__U21(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(plus(V1,V2)) = V1 + V2 + 1 >= V1 + V2 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = V1 >= V1 = a__isNat(V1) a__plus(N,0()) = N + 1 >= N = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X2 >= X2 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNat(X)) = X >= X = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2) = X2 >= X2 = U11(X1,X2) a__U21(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNat(X) = X >= X = isNat(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNat#(s(V1)) -> a__isNat#(V1) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(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: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = 1, [mark#](x0) = 1, [and](x0, x1) = x1, [U21](x0, x1, x2) = 1, [U11](x0, x1) = x1 + 1, [isNat](x0) = 1, [plus](x0, x1) = x0 + x1 + 1, [a__isNat](x0) = 1, [0] = 1, [a__and](x0, x1) = x1, [s](x0) = 1, [a__plus](x0, x1) = x0 + x1 + 1, [a__U21](x0, x1, x2) = 1, [mark](x0) = x0, [a__U11](x0, x1) = x1 + 1, [tt] = 0 orientation: a__and#(tt(),X) = 1 >= 1 = mark#(X) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),X2) mark#(isNat(X)) = 1 >= 0 = a__isNat#(X) a__U11(tt(),N) = N + 1 >= N = mark(N) a__U21(tt(),M,N) = 1 >= 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 1 >= 0 = tt() a__isNat(plus(V1,V2)) = 1 >= 1 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = 1 >= 1 = a__isNat(V1) a__plus(N,0()) = N + 2 >= N + 1 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = N + 2 >= 1 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X2 + 1 >= X2 + 1 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = 1 >= 1 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 >= X2 = a__and(mark(X1),X2) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 1 >= 1 = s(mark(X)) mark(0()) = 1 >= 1 = 0() a__U11(X1,X2) = X2 + 1 >= X2 + 1 = U11(X1,X2) a__U21(X1,X2,X3) = 1 >= 1 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__isNat(X) = 1 >= 1 = isNat(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNat#(s(V1)) -> a__isNat#(V1) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) 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: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = x0 + x1, [mark#](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [U21](x0, x1, x2) = x1 + x2 + 1, [U11](x0, x1) = x1, [isNat](x0) = x0 + 1, [plus](x0, x1) = x0 + x1 + 1, [a__isNat](x0) = x0 + 1, [0] = 0, [a__and](x0, x1) = x0 + x1, [s](x0) = x0, [a__plus](x0, x1) = x0 + x1 + 1, [a__U21](x0, x1, x2) = x1 + x2 + 1, [mark](x0) = x0, [a__U11](x0, x1) = x1, [tt] = 1 orientation: a__and#(tt(),X) = X + 1 >= X + 1 = mark#(X) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 = a__and#(mark(X1),X2) a__U11(tt(),N) = N >= N = mark(N) a__U21(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X + 1 >= X = mark(X) a__isNat(0()) = 1 >= 1 = tt() a__isNat(plus(V1,V2)) = V1 + V2 + 2 >= V1 + V2 + 2 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = V1 + 1 >= V1 + 1 = a__isNat(V1) a__plus(N,0()) = N + 1 >= N = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X2 >= X2 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNat(X)) = X + 1 >= X + 1 = a__isNat(X) mark(tt()) = 1 >= 1 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2) = X2 >= X2 = U11(X1,X2) a__U21(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNat(X) = X + 1 >= X + 1 = isNat(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNat#(s(V1)) -> a__isNat#(V1) mark#(and(X1,X2)) -> mark#(X1) 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: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__and#](x0, x1) = x1 + 1, [mark#](x0) = 0, [and](x0, x1) = 0, [U21](x0, x1, x2) = 0, [U11](x0, x1) = 0, [isNat](x0) = 0, [plus](x0, x1) = x0, [a__isNat](x0) = 0, [0] = 0, [a__and](x0, x1) = 0, [s](x0) = 0, [a__plus](x0, x1) = x0, [a__U21](x0, x1, x2) = 0, [mark](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0 orientation: a__and#(tt(),X) = X + 1 >= 0 = mark#(X) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) a__U11(tt(),N) = 0 >= 0 = mark(N) a__U21(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = 0 >= 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 >= 0 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = N >= 0 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = 0 >= 0 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U21(X1,X2,X3) = 0 >= 0 = U21(X1,X2,X3) a__plus(X1,X2) = X1 >= X1 = plus(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) problem: DPs: a__isNat#(s(V1)) -> a__isNat#(V1) mark#(and(X1,X2)) -> mark#(X1) 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: dimension: 1 interpretation: [a__isNat#](x0) = x0 + 1, [mark#](x0) = 1, [and](x0, x1) = x1, [U21](x0, x1, x2) = x1 + x2 + 1, [U11](x0, x1) = x1 + 1, [isNat](x0) = 0, [plus](x0, x1) = x0 + x1, [a__isNat](x0) = 0, [0] = 1, [a__and](x0, x1) = x1, [s](x0) = x0 + 1, [a__plus](x0, x1) = x0 + x1, [a__U21](x0, x1, x2) = x1 + x2 + 1, [mark](x0) = x0, [a__U11](x0, x1) = x1 + 1, [tt] = 0 orientation: a__isNat#(s(V1)) = V1 + 2 >= V1 + 1 = a__isNat#(V1) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) a__U11(tt(),N) = N + 1 >= N = mark(N) a__U21(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X >= 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()) = N + 1 >= N + 1 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X2 + 1 >= X2 + 1 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 >= X2 = 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()) = 1 >= 1 = 0() a__U11(X1,X2) = X2 + 1 >= X2 + 1 = U11(X1,X2) a__U21(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) problem: DPs: mark#(and(X1,X2)) -> mark#(X1) 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: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [and](x0, x1) = x0 + x1 + 1, [U21](x0, x1, x2) = x1 + x2 + 1, [U11](x0, x1) = x1, [isNat](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [a__isNat](x0) = x0, [0] = 0, [a__and](x0, x1) = x0 + x1 + 1, [s](x0) = x0, [a__plus](x0, x1) = x0 + x1 + 1, [a__U21](x0, x1, x2) = x1 + x2 + 1, [mark](x0) = x0, [a__U11](x0, x1) = x1, [tt] = 0 orientation: mark#(and(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = mark#(X1) a__U11(tt(),N) = N >= N = mark(N) a__U21(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X + 1 >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(plus(V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = V1 >= V1 = a__isNat(V1) a__plus(N,0()) = N + 1 >= N = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X2 >= X2 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__and(mark(X1),X2) mark(isNat(X)) = X >= X = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2) = X2 >= X2 = U11(X1,X2) a__U21(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) a__and(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) a__isNat(X) = X >= X = isNat(X) 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