/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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) Matrix Interpretation Processor: dim=3 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) = [0 1 0]x1, [a__plus#](x0, x1) = [0 1 0]x0 + [0 1 0]x1, [a__U21#](x0, x1, x2) = [0 1 0]x1 + [0 1 0]x2, [mark#](x0) = [0 1 0]x0, [a__U11#](x0, x1) = [0 1 0]x1, [0 0 0] [0 0 0] [and](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0 0 0] [0 0 0] [0 0 0] [U21](x0, x1, x2) = [0 1 1]x0 + [0 1 0]x1 + [0 1 0]x2 [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [0 0 0] [U11](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0] [isNat](x0) = [0] [0], [0 0 0] [0 0 0] [plus](x0, x1) = [1 1 0]x0 + [0 1 0]x1 [0 0 1] [0 0 0] , [0] [a__isNat](x0) = [0] [0], [0] [0] = [1] [0], [0 0 0] [0 0 0] [a__and](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0 0 0] [s](x0) = [1 1 1]x0 [0 0 0] , [0 0 0] [0 0 0] [a__plus](x0, x1) = [1 1 0]x0 + [0 1 0]x1 [0 0 1] [0 0 0] , [0 0 0] [0 0 0] [0 0 0] [a__U21](x0, x1, x2) = [0 1 1]x0 + [0 1 0]x1 + [0 1 0]x2 [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [0 0 0] [a__U11](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [0] [0] orientation: a__U11#(tt(),N) = [0 1 0]N >= [0 1 0]N = mark#(N) a__U21#(tt(),M,N) = [0 1 0]M + [0 1 0]N >= [0 1 0]M = mark#(M) a__U21#(tt(),M,N) = [0 1 0]M + [0 1 0]N >= [0 1 0]N = mark#(N) a__U21#(tt(),M,N) = [0 1 0]M + [0 1 0]N >= [0 1 0]M + [0 1 0]N = a__plus#(mark(N),mark(M)) a__and#(tt(),X) = [0 1 0]X >= [0 1 0]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()) = [0 1 0]N + [1] >= [0] = a__isNat#(N) a__plus#(N,0()) = [0 1 0]N + [1] >= [0 1 0]N = a__U11#(a__isNat(N),N) a__plus#(N,s(M)) = [1 1 1]M + [0 1 0]N >= [0] = a__isNat#(M) a__plus#(N,s(M)) = [1 1 1]M + [0 1 0]N >= [0] = a__and#(a__isNat(M),isNat(N)) a__plus#(N,s(M)) = [1 1 1]M + [0 1 0]N >= [0 1 0]M + [0 1 0]N = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) mark#(U11(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 = mark#(X1) mark#(U11(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X2 = a__U11#(mark(X1),X2) mark#(U21(X1,X2,X3)) = [0 1 1]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 = mark#(X1) mark#(U21(X1,X2,X3)) = [0 1 1]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X2 + [0 1 0]X3 = a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) = [1 1 0]X1 + [0 1 0]X2 >= [0 1 0]X2 = mark#(X2) mark#(plus(X1,X2)) = [1 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 = mark#(X1) mark#(plus(X1,X2)) = [1 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = a__plus#(mark(X1),mark(X2)) mark#(and(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 = mark#(X1) mark#(and(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X2 = a__and#(mark(X1),X2) mark#(isNat(X)) = [0] >= [0] = a__isNat#(X) mark#(s(X)) = [1 1 1]X >= [0 1 0]X = mark#(X) [0 0 0] [0 0 0] a__U11(tt(),N) = [0 1 0]N >= [0 1 0]N = mark(N) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U21(tt(),M,N) = [0 1 0]M + [0 1 0]N >= [0 1 0]M + [0 1 0]N = s(a__plus(mark(N),mark(M))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__and(tt(),X) = [0 1 0]X >= [0 1 0]X = mark(X) [0 0 0] [0 0 0] [0] [0] a__isNat(0()) = [0] >= [0] = tt() [0] [0] [0] [0] a__isNat(plus(V1,V2)) = [0] >= [0] = a__and(a__isNat(V1),isNat(V2)) [0] [0] [0] [0] a__isNat(s(V1)) = [0] >= [0] = a__isNat(V1) [0] [0] [0 0 0] [0] [0 0 0] a__plus(N,0()) = [1 1 0]N + [1] >= [0 1 0]N = a__U11(a__isNat(N),N) [0 0 1] [0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__plus(N,s(M)) = [1 1 1]M + [1 1 0]N >= [0 1 0]M + [0 1 0]N = a__U21(a__and(a__isNat(M),isNat(N)),M,N) [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U11(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = a__U11(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U21(X1,X2,X3)) = [0 1 1]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = a__U21(mark(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(plus(X1,X2)) = [1 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = a__plus(mark(X1),mark(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(and(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = a__and(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(isNat(X)) = [0] >= [0] = a__isNat(X) [0] [0] [0] [0] mark(tt()) = [0] >= [0] = tt() [0] [0] [0 0 0] [0 0 0] mark(s(X)) = [1 1 1]X >= [0 1 0]X = s(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(0()) = [1] >= [1] = 0() [0] [0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U11(X1,X2) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = U11(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U21(X1,X2,X3) = [0 1 1]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 1]X1 + [0 1 0]X2 + [0 1 0]X3 = U21(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__plus(X1,X2) = [1 1 0]X1 + [0 1 0]X2 >= [1 1 0]X1 + [0 1 0]X2 = plus(X1,X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__and(X1,X2) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = and(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] a__isNat(X) = [0] >= [0] = isNat(X) [0] [0] 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#(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) 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#(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) Matrix Interpretation Processor: dim=2 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) = [2 1]x1, [a__plus#](x0, x1) = [1 1]x0 + [1 0]x1, [a__U21#](x0, x1, x2) = [1 0]x1 + [1 1]x2, [mark#](x0) = [1 0]x0, [a__U11#](x0, x1) = [1 0]x1 + [1], [1 0] [2 1] [and](x0, x1) = [0 0]x0 + [0 0]x1, [1 1] [1 0] [2 2] [U21](x0, x1, x2) = [0 0]x0 + [0 0]x1 + [0 0]x2, [1 2] [1 0] [1] [U11](x0, x1) = [0 0]x0 + [0 0]x1 + [0], [0] [isNat](x0) = [0], [2 2] [1 0] [plus](x0, x1) = [0 0]x0 + [0 0]x1, [0] [a__isNat](x0) = [0], [2] [0] = [0], [1 2] [2 1] [a__and](x0, x1) = [0 0]x0 + [0 0]x1, [1 2] [s](x0) = [0 0]x0, [2 2] [1 0] [a__plus](x0, x1) = [0 0]x0 + [0 2]x1, [1 1] [1 0] [2 2] [a__U21](x0, x1, x2) = [0 2]x0 + [0 0]x1 + [0 0]x2, [1 0] [mark](x0) = [0 0]x0, [1 3] [1 0] [1] [a__U11](x0, x1) = [0 2]x0 + [0 0]x1 + [0], [0] [tt] = [0] orientation: a__U11#(tt(),N) = [1 0]N + [1] >= [1 0]N = mark#(N) a__U21#(tt(),M,N) = [1 0]M + [1 1]N >= [1 0]M = mark#(M) a__U21#(tt(),M,N) = [1 0]M + [1 1]N >= [1 0]N = mark#(N) a__U21#(tt(),M,N) = [1 0]M + [1 1]N >= [1 0]M + [1 0]N = a__plus#(mark(N),mark(M)) a__and#(tt(),X) = [2 1]X >= [1 0]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,s(M)) = [1 2]M + [1 1]N >= [0] = a__isNat#(M) a__plus#(N,s(M)) = [1 2]M + [1 1]N >= [0] = a__and#(a__isNat(M),isNat(N)) a__plus#(N,s(M)) = [1 2]M + [1 1]N >= [1 0]M + [1 1]N = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) mark#(U11(X1,X2)) = [1 2]X1 + [1 0]X2 + [1] >= [1 0]X1 = mark#(X1) mark#(U11(X1,X2)) = [1 2]X1 + [1 0]X2 + [1] >= [1 0]X2 + [1] = a__U11#(mark(X1),X2) mark#(U21(X1,X2,X3)) = [1 1]X1 + [1 0]X2 + [2 2]X3 >= [1 0]X1 = mark#(X1) mark#(U21(X1,X2,X3)) = [1 1]X1 + [1 0]X2 + [2 2]X3 >= [1 0]X2 + [1 1]X3 = a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) = [2 2]X1 + [1 0]X2 >= [1 0]X2 = mark#(X2) mark#(plus(X1,X2)) = [2 2]X1 + [1 0]X2 >= [1 0]X1 = mark#(X1) mark#(plus(X1,X2)) = [2 2]X1 + [1 0]X2 >= [1 0]X1 + [1 0]X2 = a__plus#(mark(X1),mark(X2)) mark#(and(X1,X2)) = [1 0]X1 + [2 1]X2 >= [1 0]X1 = mark#(X1) mark#(and(X1,X2)) = [1 0]X1 + [2 1]X2 >= [2 1]X2 = a__and#(mark(X1),X2) mark#(isNat(X)) = [0] >= [0] = a__isNat#(X) mark#(s(X)) = [1 2]X >= [1 0]X = mark#(X) [1 0] [1] [1 0] a__U11(tt(),N) = [0 0]N + [0] >= [0 0]N = mark(N) [1 0] [2 2] [1 0] [2 0] a__U21(tt(),M,N) = [0 0]M + [0 0]N >= [0 0]M + [0 0]N = s(a__plus(mark(N),mark(M))) [2 1] [1 0] a__and(tt(),X) = [0 0]X >= [0 0]X = mark(X) [0] [0] a__isNat(0()) = [0] >= [0] = tt() [0] [0] a__isNat(plus(V1,V2)) = [0] >= [0] = a__and(a__isNat(V1),isNat(V2)) [0] [0] a__isNat(s(V1)) = [0] >= [0] = a__isNat(V1) [2 2] [2] [1 0] [1] a__plus(N,0()) = [0 0]N + [0] >= [0 0]N + [0] = a__U11(a__isNat(N),N) [1 2] [2 2] [1 0] [2 2] a__plus(N,s(M)) = [0 0]M + [0 0]N >= [0 0]M + [0 0]N = a__U21(a__and(a__isNat(M),isNat(N)),M,N) [1 2] [1 0] [1] [1 0] [1 0] [1] mark(U11(X1,X2)) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = a__U11(mark(X1),X2) [1 1] [1 0] [2 2] [1 0] [1 0] [2 2] mark(U21(X1,X2,X3)) = [0 0]X1 + [0 0]X2 + [0 0]X3 >= [0 0]X1 + [0 0]X2 + [0 0]X3 = a__U21(mark(X1),X2,X3) [2 2] [1 0] [2 0] [1 0] mark(plus(X1,X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = a__plus(mark(X1),mark(X2)) [1 0] [2 1] [1 0] [2 1] mark(and(X1,X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = a__and(mark(X1),X2) [0] [0] mark(isNat(X)) = [0] >= [0] = a__isNat(X) [0] [0] mark(tt()) = [0] >= [0] = tt() [1 2] [1 0] mark(s(X)) = [0 0]X >= [0 0]X = s(mark(X)) [2] [2] mark(0()) = [0] >= [0] = 0() [1 3] [1 0] [1] [1 2] [1 0] [1] a__U11(X1,X2) = [0 2]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = U11(X1,X2) [1 1] [1 0] [2 2] [1 1] [1 0] [2 2] a__U21(X1,X2,X3) = [0 2]X1 + [0 0]X2 + [0 0]X3 >= [0 0]X1 + [0 0]X2 + [0 0]X3 = U21(X1,X2,X3) [2 2] [1 0] [2 2] [1 0] a__plus(X1,X2) = [0 0]X1 + [0 2]X2 >= [0 0]X1 + [0 0]X2 = plus(X1,X2) [1 2] [2 1] [1 0] [2 1] a__and(X1,X2) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = and(X1,X2) [0] [0] a__isNat(X) = [0] >= [0] = isNat(X) problem: DPs: 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#(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__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#(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) SCC Processor: #sccs: 1 #rules: 19 #arcs: 146/400 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#(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,s(M)) -> a__isNat#(M) a__isNat#(s(V1)) -> a__isNat#(V1) 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) 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) = x0 + 6, [a__and#](x0, x1) = x1, [a__plus#](x0, x1) = 3x0 + 4x1, [a__U21#](x0, x1, x2) = 4x1 + 3x2, [mark#](x0) = x0, [and](x0, x1) = x0 + x1 + 0, [U21](x0, x1, x2) = x0 + 4x1 + 4x2, [U11](x0, x1) = x0 + 3x1 + 0, [isNat](x0) = x0 + 6, [plus](x0, x1) = 4x0 + 4x1, [a__isNat](x0) = x0 + 6, [0] = 3, [a__and](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 2, [a__plus](x0, x1) = 4x0 + 4x1, [a__U21](x0, x1, x2) = x0 + 4x1 + 4x2, [mark](x0) = x0, [a__U11](x0, x1) = x0 + 3x1 + 0, [tt] = 4 orientation: a__isNat#(plus(V1,V2)) = 4V1 + 4V2 + 6 >= V1 + 6 = a__isNat#(V1) a__isNat#(plus(V1,V2)) = 4V1 + 4V2 + 6 >= V2 + 6 = a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) = X >= X = mark#(X) mark#(U21(X1,X2,X3)) = X1 + 4X2 + 4X3 >= X1 = mark#(X1) mark#(U21(X1,X2,X3)) = X1 + 4X2 + 4X3 >= 4X2 + 3X3 = a__U21#(mark(X1),X2,X3) a__U21#(tt(),M,N) = 4M + 3N >= M = mark#(M) mark#(plus(X1,X2)) = 4X1 + 4X2 >= X2 = mark#(X2) mark#(plus(X1,X2)) = 4X1 + 4X2 >= X1 = mark#(X1) mark#(plus(X1,X2)) = 4X1 + 4X2 >= 3X1 + 4X2 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = 4M + 3N + 6 >= M + 6 = a__isNat#(M) a__isNat#(s(V1)) = V1 + 6 >= V1 + 6 = a__isNat#(V1) a__plus#(N,s(M)) = 4M + 3N + 6 >= N + 6 = a__and#(a__isNat(M),isNat(N)) a__plus#(N,s(M)) = 4M + 3N + 6 >= 4M + 3N = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) a__U21#(tt(),M,N) = 4M + 3N >= N = mark#(N) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 = a__and#(mark(X1),X2) mark#(isNat(X)) = X + 6 >= X + 6 = a__isNat#(X) mark#(s(X)) = X + 2 >= X = mark#(X) a__U21#(tt(),M,N) = 4M + 3N >= 4M + 3N = a__plus#(mark(N),mark(M)) a__U11(tt(),N) = 3N + 4 >= N = mark(N) a__U21(tt(),M,N) = 4M + 4N + 4 >= 4M + 4N + 2 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X + 4 >= X = mark(X) a__isNat(0()) = 6 >= 4 = tt() a__isNat(plus(V1,V2)) = 4V1 + 4V2 + 6 >= V1 + V2 + 6 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = V1 + 6 >= V1 + 6 = a__isNat(V1) a__plus(N,0()) = 4N + 7 >= 3N + 6 = a__U11(a__isNat(N),N) a__plus(N,s(M)) = 4M + 4N + 6 >= 4M + 4N + 6 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = X1 + 3X2 + 0 >= X1 + 3X2 + 0 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = X1 + 4X2 + 4X3 >= X1 + 4X2 + 4X3 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and(mark(X1),X2) mark(isNat(X)) = X + 6 >= X + 6 = a__isNat(X) mark(tt()) = 4 >= 4 = tt() mark(s(X)) = X + 2 >= X + 2 = s(mark(X)) mark(0()) = 3 >= 3 = 0() a__U11(X1,X2) = X1 + 3X2 + 0 >= X1 + 3X2 + 0 = U11(X1,X2) a__U21(X1,X2,X3) = X1 + 4X2 + 4X3 >= X1 + 4X2 + 4X3 = U21(X1,X2,X3) a__plus(X1,X2) = 4X1 + 4X2 >= 4X1 + 4X2 = plus(X1,X2) a__and(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) a__isNat(X) = X + 6 >= X + 6 = 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#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> a__isNat#(M) a__isNat#(s(V1)) -> a__isNat#(V1) 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) 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) 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#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> a__isNat#(M) a__isNat#(s(V1)) -> a__isNat#(V1) 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) 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) Arctic Interpretation Processor: dimension: 1 usable rules: interpretation: [a__isNat#](x0) = 3, [a__and#](x0, x1) = x1 + 0, [a__plus#](x0, x1) = 4, [a__U21#](x0, x1, x2) = 4, [mark#](x0) = x0, [and](x0, x1) = x0 + x1 + 0, [U21](x0, x1, x2) = 4x0 + 5x1 + x2 + 4, [U11](x0, x1) = x0 + x1 + 4, [isNat](x0) = 3, [plus](x0, x1) = x1 + 5, [a__isNat](x0) = x0 + 0, [0] = 3, [a__and](x0, x1) = x1, [s](x0) = x0 + 4, [a__plus](x0, x1) = x0 + x1 + 4, [a__U21](x0, x1, x2) = x0 + 2x1 + x2 + 0, [mark](x0) = 0, [a__U11](x0, x1) = 15x1, [tt] = 0 orientation: a__isNat#(plus(V1,V2)) = 3 >= 3 = a__isNat#(V1) a__isNat#(plus(V1,V2)) = 3 >= 3 = a__and#(a__isNat(V1),isNat(V2)) a__and#(tt(),X) = X + 0 >= X = mark#(X) mark#(U21(X1,X2,X3)) = 4X1 + 5X2 + X3 + 4 >= X1 = mark#(X1) mark#(U21(X1,X2,X3)) = 4X1 + 5X2 + X3 + 4 >= 4 = a__U21#(mark(X1),X2,X3) mark#(plus(X1,X2)) = X2 + 5 >= 4 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = 4 >= 3 = a__isNat#(M) a__isNat#(s(V1)) = 3 >= 3 = a__isNat#(V1) a__plus#(N,s(M)) = 4 >= 3 = a__and#(a__isNat(M),isNat(N)) a__plus#(N,s(M)) = 4 >= 4 = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = a__and#(mark(X1),X2) mark#(isNat(X)) = 3 >= 3 = a__isNat#(X) mark#(s(X)) = X + 4 >= X = mark#(X) a__U21#(tt(),M,N) = 4 >= 4 = a__plus#(mark(N),mark(M)) a__U11(tt(),N) = 15N >= 0 = mark(N) a__U21(tt(),M,N) = 2M + N + 0 >= 4 = s(a__plus(mark(N),mark(M))) a__and(tt(),X) = X >= 0 = mark(X) a__isNat(0()) = 3 >= 0 = tt() a__isNat(plus(V1,V2)) = V2 + 5 >= 3 = a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) = V1 + 4 >= V1 + 0 = a__isNat(V1) a__plus(N,0()) = N + 4 >= 15N = a__U11(a__isNat(N),N) a__plus(N,s(M)) = M + N + 4 >= 2M + N + 3 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = 0 >= 15X2 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = 0 >= 2X2 + X3 + 0 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 0 >= 4 = a__plus(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= X2 = a__and(mark(X1),X2) mark(isNat(X)) = 0 >= X + 0 = a__isNat(X) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 4 = s(mark(X)) mark(0()) = 0 >= 3 = 0() a__U11(X1,X2) = 15X2 >= X1 + X2 + 4 = U11(X1,X2) a__U21(X1,X2,X3) = X1 + 2X2 + X3 + 0 >= 4X1 + 5X2 + X3 + 4 = U21(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 4 >= X2 + 5 = plus(X1,X2) a__and(X1,X2) = X2 >= X1 + X2 + 0 = and(X1,X2) a__isNat(X) = X + 0 >= 3 = 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#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) 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) 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#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3) 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) SCC Processor: #sccs: 2 #rules: 10 #arcs: 99/121 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) Subterm Criterion Processor: simple projection: pi(isNat) = 0 pi(mark#) = 0 pi(a__and#) = 1 pi(a__isNat#) = 0 problem: DPs: a__and#(tt(),X) -> mark#(X) 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) SCC Processor: #sccs: 0 #rules: 0 #arcs: 23/4 DPs: 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) 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) = 2x1, [a__U21#](x0, x1, x2) = 2x1, [and](x0, x1) = x1, [U21](x0, x1, x2) = 3/2x0 + 3/2x1 + 2x2 + 3, [U11](x0, x1) = 2x1, [isNat](x0) = 0, [plus](x0, x1) = 2x0 + 3/2x1, [a__isNat](x0) = 0, [0] = 0, [a__and](x0, x1) = x1, [s](x0) = x0 + 2, [a__plus](x0, x1) = 2x0 + 3/2x1, [a__U21](x0, x1, x2) = 3/2x0 + 3/2x1 + 2x2 + 3, [mark](x0) = x0, [a__U11](x0, x1) = 2x1, [tt] = 0 orientation: a__U21#(tt(),M,N) = 2M >= 2M = a__plus#(mark(N),mark(M)) a__plus#(N,s(M)) = 2M + 4 >= 2M = a__U21#(a__and(a__isNat(M),isNat(N)),M,N) a__U11(tt(),N) = 2N >= N = mark(N) a__U21(tt(),M,N) = 3/2M + 2N + 3 >= 3/2M + 2N + 2 = 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()) = 2N >= 2N = a__U11(a__isNat(N),N) a__plus(N,s(M)) = 3/2M + 2N + 3 >= 3/2M + 2N + 3 = a__U21(a__and(a__isNat(M),isNat(N)),M,N) mark(U11(X1,X2)) = 2X2 >= 2X2 = a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) = 3/2X1 + 3/2X2 + 2X3 + 3 >= 3/2X1 + 3/2X2 + 2X3 + 3 = a__U21(mark(X1),X2,X3) mark(plus(X1,X2)) = 2X1 + 3/2X2 >= 2X1 + 3/2X2 = 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 + 2 >= X + 2 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2) = 2X2 >= 2X2 = U11(X1,X2) a__U21(X1,X2,X3) = 3/2X1 + 3/2X2 + 2X3 + 3 >= 3/2X1 + 3/2X2 + 2X3 + 3 = U21(X1,X2,X3) a__plus(X1,X2) = 2X1 + 3/2X2 >= 2X1 + 3/2X2 = plus(X1,X2) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) problem: DPs: 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) Restore Modifier: DPs: 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) SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/1