/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: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Proof: DP Processor: DPs: active#(U11(tt(),N)) -> mark#(N) active#(U21(tt(),M,N)) -> plus#(N,M) active#(U21(tt(),M,N)) -> s#(plus(N,M)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(plus(V1,V2))) -> isNat#(V2) active#(isNat(plus(V1,V2))) -> isNat#(V1) active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> U11#(isNat(N),N) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) active#(plus(N,s(M))) -> isNat#(N) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(0()) -> active#(0()) U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) TDG Processor: DPs: active#(U11(tt(),N)) -> mark#(N) active#(U21(tt(),M,N)) -> plus#(N,M) active#(U21(tt(),M,N)) -> s#(plus(N,M)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(plus(V1,V2))) -> isNat#(V2) active#(isNat(plus(V1,V2))) -> isNat#(V1) active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> U11#(isNat(N),N) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) active#(plus(N,s(M))) -> isNat#(N) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(0()) -> active#(0()) U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) graph: U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) -> U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) -> U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) -> U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) -> U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) -> U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) -> U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) -> U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) -> U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) -> U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) -> U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) -> U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) -> U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) -> U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) -> U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) -> U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) -> U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) -> U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) -> U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) -> U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) -> U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) -> U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) -> U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) -> U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) -> U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) -> U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U11#(mark(X1),X2) -> U11#(X1,X2) -> U11#(X1,active(X2)) -> U11#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) -> U11#(active(X1),X2) -> U11#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) -> U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) -> U11#(mark(X1),X2) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) -> U11#(X1,active(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) -> U11#(active(X1),X2) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) -> U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) -> U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) -> U11#(X1,active(X2)) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) -> U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) -> U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) -> U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) -> U11#(X1,active(X2)) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) -> U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) -> U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) -> U11#(mark(X1),X2) -> U11#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) -> and#(X1,active(X2)) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) -> and#(active(X1),X2) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) -> and#(X1,mark(X2)) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) -> and#(mark(X1),X2) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) -> and#(X1,active(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) -> and#(active(X1),X2) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) -> and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) -> and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) -> and#(X1,active(X2)) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) -> and#(active(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) -> and#(X1,mark(X2)) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) -> and#(mark(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) -> and#(X1,active(X2)) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) -> and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) -> and#(X1,mark(X2)) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) -> and#(mark(X1),X2) -> and#(X1,X2) isNat#(mark(X)) -> isNat#(X) -> isNat#(active(X)) -> isNat#(X) isNat#(mark(X)) -> isNat#(X) -> isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) -> isNat#(active(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) -> isNat#(mark(X)) -> isNat#(X) s#(mark(X)) -> s#(X) -> s#(active(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) -> s#(active(X)) -> s#(X) s#(active(X)) -> s#(X) -> s#(mark(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) -> plus#(X1,active(X2)) -> plus#(X1,X2) plus#(mark(X1),X2) -> plus#(X1,X2) -> plus#(active(X1),X2) -> plus#(X1,X2) plus#(mark(X1),X2) -> plus#(X1,X2) -> plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(mark(X1),X2) -> plus#(X1,X2) -> plus#(mark(X1),X2) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) -> plus#(X1,active(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) -> plus#(active(X1),X2) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) -> plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) -> plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) -> plus#(X1,active(X2)) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) -> plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) -> plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) -> plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) -> plus#(X1,active(X2)) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) -> plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) -> plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) -> plus#(mark(X1),X2) -> plus#(X1,X2) mark#(isNat(X)) -> active#(isNat(X)) -> active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) mark#(isNat(X)) -> active#(isNat(X)) -> active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(plus(N,s(M))) -> isNat#(M) mark#(isNat(X)) -> active#(isNat(X)) -> active#(plus(N,s(M))) -> isNat#(N) mark#(isNat(X)) -> active#(isNat(X)) -> active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(plus(N,0())) -> U11#(isNat(N),N) mark#(isNat(X)) -> active#(isNat(X)) -> active#(plus(N,0())) -> isNat#(N) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(s(V1))) -> isNat#(V1) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(plus(V1,V2))) -> isNat#(V1) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(plus(V1,V2))) -> isNat#(V2) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(0())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) -> active#(and(tt(),X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) -> active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(isNat(X)) -> active#(isNat(X)) -> active#(U21(tt(),M,N)) -> s#(plus(N,M)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(U21(tt(),M,N)) -> plus#(N,M) mark#(isNat(X)) -> active#(isNat(X)) -> active#(U11(tt(),N)) -> mark#(N) mark#(0()) -> active#(0()) -> active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(0()) -> active#(0()) -> active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) mark#(0()) -> active#(0()) -> active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) mark#(0()) -> active#(0()) -> active#(plus(N,s(M))) -> isNat#(M) mark#(0()) -> active#(0()) -> active#(plus(N,s(M))) -> isNat#(N) mark#(0()) -> active#(0()) -> active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(0()) -> active#(0()) -> active#(plus(N,0())) -> U11#(isNat(N),N) mark#(0()) -> active#(0()) -> active#(plus(N,0())) -> isNat#(N) mark#(0()) -> active#(0()) -> active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(0()) -> active#(0()) -> active#(isNat(s(V1))) -> isNat#(V1) mark#(0()) -> active#(0()) -> active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(0()) -> active#(0()) -> active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) mark#(0()) -> active#(0()) -> active#(isNat(plus(V1,V2))) -> isNat#(V1) mark#(0()) -> active#(0()) -> active#(isNat(plus(V1,V2))) -> isNat#(V2) mark#(0()) -> active#(0()) -> active#(isNat(0())) -> mark#(tt()) mark#(0()) -> active#(0()) -> active#(and(tt(),X)) -> mark#(X) mark#(0()) -> active#(0()) -> active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(0()) -> active#(0()) -> active#(U21(tt(),M,N)) -> s#(plus(N,M)) mark#(0()) -> active#(0()) -> active#(U21(tt(),M,N)) -> plus#(N,M) mark#(0()) -> active#(0()) -> active#(U11(tt(),N)) -> mark#(N) mark#(and(X1,X2)) -> and#(mark(X1),X2) -> and#(X1,active(X2)) -> and#(X1,X2) mark#(and(X1,X2)) -> and#(mark(X1),X2) -> and#(active(X1),X2) -> and#(X1,X2) mark#(and(X1,X2)) -> and#(mark(X1),X2) -> and#(X1,mark(X2)) -> and#(X1,X2) mark#(and(X1,X2)) -> and#(mark(X1),X2) -> and#(mark(X1),X2) -> and#(X1,X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> 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)) -> active#(plus(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> 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#(s(X)) -> active#(s(mark(X))) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> s#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> 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#(tt()) -> active#(tt()) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,s(M))) -> isNat#(M) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,s(M))) -> isNat#(N) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,0())) -> U11#(isNat(N),N) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,0())) -> isNat#(N) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(isNat(s(V1))) -> isNat#(V1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(isNat(plus(V1,V2))) -> isNat#(V1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(isNat(plus(V1,V2))) -> isNat#(V2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(isNat(0())) -> mark#(tt()) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(and(tt(),X)) -> mark#(X) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(U21(tt(),M,N)) -> s#(plus(N,M)) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(U21(tt(),M,N)) -> plus#(N,M) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(U11(tt(),N)) -> mark#(N) mark#(s(X)) -> s#(mark(X)) -> s#(active(X)) -> s#(X) mark#(s(X)) -> s#(mark(X)) -> s#(mark(X)) -> s#(X) mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> 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#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,s(M))) -> isNat#(M) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,s(M))) -> isNat#(N) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,0())) -> U11#(isNat(N),N) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,0())) -> isNat#(N) mark#(s(X)) -> active#(s(mark(X))) -> active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(s(X)) -> active#(s(mark(X))) -> active#(isNat(s(V1))) -> isNat#(V1) mark#(s(X)) -> active#(s(mark(X))) -> active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(s(X)) -> active#(s(mark(X))) -> active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) mark#(s(X)) -> active#(s(mark(X))) -> active#(isNat(plus(V1,V2))) -> isNat#(V1) mark#(s(X)) -> active#(s(mark(X))) -> active#(isNat(plus(V1,V2))) -> isNat#(V2) mark#(s(X)) -> active#(s(mark(X))) -> active#(isNat(0())) -> mark#(tt()) mark#(s(X)) -> active#(s(mark(X))) -> active#(and(tt(),X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) -> active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(s(X)) -> active#(s(mark(X))) -> active#(U21(tt(),M,N)) -> s#(plus(N,M)) mark#(s(X)) -> active#(s(mark(X))) -> active#(U21(tt(),M,N)) -> plus#(N,M) mark#(s(X)) -> active#(s(mark(X))) -> active#(U11(tt(),N)) -> mark#(N) mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) -> plus#(X1,active(X2)) -> plus#(X1,X2) mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) -> plus#(active(X1),X2) -> plus#(X1,X2) mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) -> plus#(X1,mark(X2)) -> plus#(X1,X2) mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) -> plus#(mark(X1),X2) -> plus#(X1,X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(0()) -> active#(0()) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> 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)) -> active#(plus(mark(X1),mark(X2))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> 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#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> s#(mark(X)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2,X3)) -> 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#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> 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)) -> active#(plus(mark(X1),mark(X2))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> 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#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> s#(mark(X)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> 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#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> isNat#(M) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> isNat#(N) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,0())) -> U11#(isNat(N),N) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,0())) -> isNat#(N) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(isNat(s(V1))) -> isNat#(V1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(isNat(plus(V1,V2))) -> isNat#(V1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(isNat(plus(V1,V2))) -> isNat#(V2) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(isNat(0())) -> mark#(tt()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(and(tt(),X)) -> mark#(X) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(U21(tt(),M,N)) -> s#(plus(N,M)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(U21(tt(),M,N)) -> plus#(N,M) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(U11(tt(),N)) -> mark#(N) mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) -> U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) -> U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) -> U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) -> U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) -> U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) -> U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> 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)) -> active#(plus(mark(X1),mark(X2))) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> 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#(s(X)) -> active#(s(mark(X))) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> s#(mark(X)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> 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#(tt()) -> active#(tt()) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(plus(N,s(M))) -> isNat#(M) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(plus(N,s(M))) -> isNat#(N) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(plus(N,0())) -> U11#(isNat(N),N) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(plus(N,0())) -> isNat#(N) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(isNat(s(V1))) -> isNat#(V1) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(isNat(plus(V1,V2))) -> isNat#(V1) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(isNat(plus(V1,V2))) -> isNat#(V2) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(isNat(0())) -> mark#(tt()) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(and(tt(),X)) -> mark#(X) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(U21(tt(),M,N)) -> s#(plus(N,M)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(U21(tt(),M,N)) -> plus#(N,M) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(U11(tt(),N)) -> mark#(N) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) -> U11#(X1,active(X2)) -> U11#(X1,X2) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) -> U11#(active(X1),X2) -> U11#(X1,X2) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) -> U11#(X1,mark(X2)) -> U11#(X1,X2) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) -> U11#(mark(X1),X2) -> U11#(X1,X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> 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)) -> active#(plus(mark(X1),mark(X2))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> 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#(s(X)) -> active#(s(mark(X))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> s#(mark(X)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> 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#(tt()) -> active#(tt()) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(plus(N,s(M))) -> isNat#(M) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(plus(N,s(M))) -> isNat#(N) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(plus(N,0())) -> U11#(isNat(N),N) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(plus(N,0())) -> isNat#(N) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(isNat(s(V1))) -> isNat#(V1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(isNat(plus(V1,V2))) -> isNat#(V1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(isNat(plus(V1,V2))) -> isNat#(V2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(isNat(0())) -> mark#(tt()) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(and(tt(),X)) -> mark#(X) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U21(tt(),M,N)) -> s#(plus(N,M)) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U21(tt(),M,N)) -> plus#(N,M) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U11(tt(),N)) -> mark#(N) mark#(tt()) -> active#(tt()) -> active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(tt()) -> active#(tt()) -> active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) mark#(tt()) -> active#(tt()) -> active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) mark#(tt()) -> active#(tt()) -> active#(plus(N,s(M))) -> isNat#(M) mark#(tt()) -> active#(tt()) -> active#(plus(N,s(M))) -> isNat#(N) mark#(tt()) -> active#(tt()) -> active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(tt()) -> active#(tt()) -> active#(plus(N,0())) -> U11#(isNat(N),N) mark#(tt()) -> active#(tt()) -> active#(plus(N,0())) -> isNat#(N) mark#(tt()) -> active#(tt()) -> active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(tt()) -> active#(tt()) -> active#(isNat(s(V1))) -> isNat#(V1) mark#(tt()) -> active#(tt()) -> active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(tt()) -> active#(tt()) -> active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) mark#(tt()) -> active#(tt()) -> active#(isNat(plus(V1,V2))) -> isNat#(V1) mark#(tt()) -> active#(tt()) -> active#(isNat(plus(V1,V2))) -> isNat#(V2) mark#(tt()) -> active#(tt()) -> active#(isNat(0())) -> mark#(tt()) mark#(tt()) -> active#(tt()) -> active#(and(tt(),X)) -> mark#(X) mark#(tt()) -> active#(tt()) -> active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(tt()) -> active#(tt()) -> active#(U21(tt(),M,N)) -> s#(plus(N,M)) mark#(tt()) -> active#(tt()) -> active#(U21(tt(),M,N)) -> plus#(N,M) mark#(tt()) -> active#(tt()) -> active#(U11(tt(),N)) -> mark#(N) active#(isNat(0())) -> mark#(tt()) -> mark#(0()) -> active#(0()) active#(isNat(0())) -> mark#(tt()) -> mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(0())) -> mark#(tt()) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNat(0())) -> mark#(tt()) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) active#(isNat(0())) -> mark#(tt()) -> mark#(and(X1,X2)) -> mark#(X1) active#(isNat(0())) -> mark#(tt()) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNat(0())) -> mark#(tt()) -> mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) active#(isNat(0())) -> mark#(tt()) -> mark#(plus(X1,X2)) -> mark#(X1) active#(isNat(0())) -> mark#(tt()) -> mark#(plus(X1,X2)) -> mark#(X2) active#(isNat(0())) -> mark#(tt()) -> mark#(s(X)) -> active#(s(mark(X))) active#(isNat(0())) -> mark#(tt()) -> mark#(s(X)) -> s#(mark(X)) active#(isNat(0())) -> mark#(tt()) -> mark#(s(X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) -> mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) active#(isNat(0())) -> mark#(tt()) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(isNat(0())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNat(0())) -> mark#(tt()) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(isNat(0())) -> mark#(tt()) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) active#(isNat(0())) -> mark#(tt()) -> mark#(U11(X1,X2)) -> mark#(X1) active#(isNat(s(V1))) -> isNat#(V1) -> isNat#(active(X)) -> isNat#(X) active#(isNat(s(V1))) -> isNat#(V1) -> isNat#(mark(X)) -> isNat#(X) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(0()) -> active#(0()) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(and(X1,X2)) -> mark#(X1) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(plus(X1,X2)) -> mark#(X1) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(plus(X1,X2)) -> mark#(X2) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(s(X)) -> active#(s(mark(X))) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(s(X)) -> s#(mark(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(s(X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(tt()) -> active#(tt()) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(U11(X1,X2)) -> mark#(X1) active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) -> and#(X1,active(X2)) -> and#(X1,X2) active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) -> and#(active(X1),X2) -> and#(X1,X2) active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) -> and#(X1,mark(X2)) -> and#(X1,X2) active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) -> and#(mark(X1),X2) -> and#(X1,X2) active#(isNat(plus(V1,V2))) -> isNat#(V2) -> isNat#(active(X)) -> isNat#(X) active#(isNat(plus(V1,V2))) -> isNat#(V2) -> isNat#(mark(X)) -> isNat#(X) active#(isNat(plus(V1,V2))) -> isNat#(V1) -> isNat#(active(X)) -> isNat#(X) active#(isNat(plus(V1,V2))) -> isNat#(V1) -> isNat#(mark(X)) -> isNat#(X) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(0()) -> active#(0()) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(and(X1,X2)) -> mark#(X1) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(plus(X1,X2)) -> mark#(X1) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(plus(X1,X2)) -> mark#(X2) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(s(X)) -> active#(s(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(s(X)) -> s#(mark(X)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(s(X)) -> mark#(X) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(tt()) -> active#(tt()) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(U11(X1,X2)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(0()) -> active#(0()) active#(and(tt(),X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) active#(and(tt(),X)) -> mark#(X) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) active#(and(tt(),X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(and(tt(),X)) -> mark#(X) -> mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) active#(and(tt(),X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) active#(and(tt(),X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) active#(and(tt(),X)) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) active#(and(tt(),X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) active#(and(tt(),X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(and(tt(),X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) active#(and(tt(),X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) active#(and(tt(),X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) active#(and(tt(),X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> U11#(isNat(N),N) -> U11#(X1,active(X2)) -> U11#(X1,X2) active#(plus(N,0())) -> U11#(isNat(N),N) -> U11#(active(X1),X2) -> U11#(X1,X2) active#(plus(N,0())) -> U11#(isNat(N),N) -> U11#(X1,mark(X2)) -> U11#(X1,X2) active#(plus(N,0())) -> U11#(isNat(N),N) -> U11#(mark(X1),X2) -> U11#(X1,X2) active#(plus(N,0())) -> isNat#(N) -> isNat#(active(X)) -> isNat#(X) active#(plus(N,0())) -> isNat#(N) -> isNat#(mark(X)) -> isNat#(X) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(0()) -> active#(0()) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(isNat(X)) -> active#(isNat(X)) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(and(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(plus(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(plus(X1,X2)) -> mark#(X2) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(s(X)) -> active#(s(mark(X))) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(s(X)) -> s#(mark(X)) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(s(X)) -> mark#(X) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(tt()) -> active#(tt()) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(U11(X1,X2)) -> mark#(X1) active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) -> U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) -> U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) -> U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) -> U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) -> U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) -> U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) -> and#(X1,active(X2)) -> and#(X1,X2) active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) -> and#(active(X1),X2) -> and#(X1,X2) active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) -> and#(X1,mark(X2)) -> and#(X1,X2) active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) -> and#(mark(X1),X2) -> and#(X1,X2) active#(plus(N,s(M))) -> isNat#(M) -> isNat#(active(X)) -> isNat#(X) active#(plus(N,s(M))) -> isNat#(M) -> isNat#(mark(X)) -> isNat#(X) active#(plus(N,s(M))) -> isNat#(N) -> isNat#(active(X)) -> isNat#(X) active#(plus(N,s(M))) -> isNat#(N) -> isNat#(mark(X)) -> isNat#(X) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(0()) -> active#(0()) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(isNat(X)) -> active#(isNat(X)) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(and(X1,X2)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(plus(X1,X2)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(plus(X1,X2)) -> mark#(X2) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(s(X)) -> active#(s(mark(X))) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(s(X)) -> s#(mark(X)) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(s(X)) -> mark#(X) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(tt()) -> active#(tt()) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(U11(X1,X2)) -> mark#(X1) active#(U21(tt(),M,N)) -> s#(plus(N,M)) -> s#(active(X)) -> s#(X) active#(U21(tt(),M,N)) -> s#(plus(N,M)) -> s#(mark(X)) -> s#(X) active#(U21(tt(),M,N)) -> plus#(N,M) -> plus#(X1,active(X2)) -> plus#(X1,X2) active#(U21(tt(),M,N)) -> plus#(N,M) -> plus#(active(X1),X2) -> plus#(X1,X2) active#(U21(tt(),M,N)) -> plus#(N,M) -> plus#(X1,mark(X2)) -> plus#(X1,X2) active#(U21(tt(),M,N)) -> plus#(N,M) -> plus#(mark(X1),X2) -> plus#(X1,X2) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(0()) -> active#(0()) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(isNat(X)) -> active#(isNat(X)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(and(X1,X2)) -> mark#(X1) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> mark#(X1) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> mark#(X2) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> active#(s(mark(X))) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> s#(mark(X)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> mark#(X) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(tt()) -> active#(tt()) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(U11(X1,X2)) -> mark#(X1) active#(U11(tt(),N)) -> mark#(N) -> mark#(0()) -> active#(0()) active#(U11(tt(),N)) -> mark#(N) -> mark#(isNat(X)) -> active#(isNat(X)) active#(U11(tt(),N)) -> mark#(N) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(U11(tt(),N)) -> mark#(N) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) active#(U11(tt(),N)) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) active#(U11(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(U11(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) active#(U11(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) active#(U11(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) active#(U11(tt(),N)) -> mark#(N) -> mark#(s(X)) -> active#(s(mark(X))) active#(U11(tt(),N)) -> mark#(N) -> mark#(s(X)) -> s#(mark(X)) active#(U11(tt(),N)) -> mark#(N) -> mark#(s(X)) -> mark#(X) active#(U11(tt(),N)) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U11(tt(),N)) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3) active#(U11(tt(),N)) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(U11(tt(),N)) -> mark#(N) -> mark#(tt()) -> active#(tt()) active#(U11(tt(),N)) -> mark#(N) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),N)) -> mark#(N) -> mark#(U11(X1,X2)) -> U11#(mark(X1),X2) active#(U11(tt(),N)) -> mark#(N) -> mark#(U11(X1,X2)) -> mark#(X1) SCC Processor: #sccs: 7 #rules: 44 #arcs: 574/3721 DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(U11(tt(),N)) -> mark#(N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(tt()) -> active#(tt()) active#(and(tt(),X)) -> mark#(X) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(0()) -> active#(0()) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) EDG Processor: DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(U11(tt(),N)) -> mark#(N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(tt()) -> active#(tt()) active#(and(tt(),X)) -> mark#(X) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(0()) -> active#(0()) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) graph: mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(0())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(and(tt(),X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(0()) -> active#(0()) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,0())) -> mark#(U11(isNat(N),N)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> 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)) -> active#(plus(mark(X1),mark(X2))) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U21(X1,X2,X3)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) -> active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U11(tt(),N)) -> mark#(N) active#(isNat(0())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNat(s(V1))) -> mark#(isNat(V1)) -> mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(and(X1,X2)) -> mark#(X1) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) active#(and(tt(),X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(and(tt(),X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) active#(and(tt(),X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) active#(and(tt(),X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) active#(and(tt(),X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(and(tt(),X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) active#(and(tt(),X)) -> mark#(X) -> mark#(0()) -> active#(0()) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(U11(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> mark#(X) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> active#(s(mark(X))) active#(U11(tt(),N)) -> mark#(N) -> mark#(U11(X1,X2)) -> mark#(X1) active#(U11(tt(),N)) -> mark#(N) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),N)) -> mark#(N) -> mark#(tt()) -> active#(tt()) active#(U11(tt(),N)) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> mark#(X1) active#(U11(tt(),N)) -> mark#(N) -> mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U11(tt(),N)) -> mark#(N) -> mark#(s(X)) -> mark#(X) active#(U11(tt(),N)) -> mark#(N) -> mark#(s(X)) -> active#(s(mark(X))) active#(U11(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) active#(U11(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) active#(U11(tt(),N)) -> mark#(N) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(U11(tt(),N)) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) active#(U11(tt(),N)) -> mark#(N) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(U11(tt(),N)) -> mark#(N) -> mark#(isNat(X)) -> active#(isNat(X)) active#(U11(tt(),N)) -> mark#(N) -> mark#(0()) -> active#(0()) SCC Processor: #sccs: 1 #rules: 18 #arcs: 130/484 DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),N)) -> mark#(N) mark#(U11(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(U11(isNat(N),N)) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=1 usable rules: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) interpretation: [mark#](x0) = 1/2x0, [active#](x0) = 1/2x0, [isNat](x0) = 0, [0] = 2, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = 2x0 + x1, [U21](x0, x1, x2) = 2x0 + x1 + 2x2, [mark](x0) = x0, [active](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0 orientation: mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(isNat(V1)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) = 1/2X1 + 1/2X2 >= 1/2X1 + 1/2X2 = active#(and(mark(X1),X2)) active#(and(tt(),X)) = 1/2X >= 1/2X = mark#(X) mark#(and(X1,X2)) = 1/2X1 + 1/2X2 >= 1/2X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + 1/2X2 >= X1 + 1/2X2 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = 1/2M + N >= 1/2M + N = mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) = X1 + 1/2X2 + X3 >= X1 + 1/2X2 + X3 = active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) = 1/2M + N >= 1/2M + N = mark#(s(plus(N,M))) mark#(s(X)) = 1/2X >= 1/2X = mark#(X) mark#(plus(X1,X2)) = X1 + 1/2X2 >= 1/2X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + 1/2X2 >= 1/2X2 = mark#(X2) mark#(U21(X1,X2,X3)) = X1 + 1/2X2 + X3 >= 1/2X1 = mark#(X1) mark#(U11(X1,X2)) = 1/2X1 + 1/2X2 >= 1/2X1 + 1/2X2 = active#(U11(mark(X1),X2)) active#(U11(tt(),N)) = 1/2N >= 1/2N = mark#(N) mark#(U11(X1,X2)) = 1/2X1 + 1/2X2 >= 1/2X1 = mark#(X1) active#(plus(N,0())) = N + 1 >= 1/2N = mark#(U11(isNat(N),N)) active(U11(tt(),N)) = N >= N = mark(N) active(U21(tt(),M,N)) = M + 2N >= M + 2N = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) = 0 >= 0 = mark(isNat(V1)) active(plus(N,0())) = 2N + 2 >= N = mark(U11(isNat(N),N)) active(plus(N,s(M))) = M + 2N >= M + 2N = mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X1,X2,X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = active(U21(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(0()) = 2 >= 2 = active(0()) U11(mark(X1),X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) U11(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U11(X1,X2) U11(active(X1),X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) U11(X1,active(X2)) = X1 + X2 >= X1 + X2 = U11(X1,X2) U21(mark(X1),X2,X3) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = U21(X1,X2,X3) U21(X1,mark(X2),X3) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = U21(X1,X2,X3) U21(X1,X2,mark(X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = U21(X1,X2,X3) U21(active(X1),X2,X3) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = U21(X1,X2,X3) U21(X1,active(X2),X3) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = U21(X1,X2,X3) U21(X1,X2,active(X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = U21(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = 2X1 + X2 >= 2X1 + X2 = plus(X1,X2) plus(active(X1),X2) = 2X1 + X2 >= 2X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = 2X1 + X2 >= 2X1 + X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) problem: DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),N)) -> mark#(N) mark#(U11(X1,X2)) -> mark#(X1) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Restore Modifier: DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),N)) -> mark#(N) mark#(U11(X1,X2)) -> mark#(X1) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=1 usable rules: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) interpretation: [mark#](x0) = 2x0, [active#](x0) = 2x0, [isNat](x0) = 0, [0] = 1, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + 3/2x1, [U21](x0, x1, x2) = x0 + 3/2x1 + x2, [mark](x0) = x0, [active](x0) = x0, [U11](x0, x1) = 3/2x0 + x1 + 3/2, [tt] = 0 orientation: mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(isNat(V1)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = active#(and(mark(X1),X2)) active#(and(tt(),X)) = 2X >= 2X = mark#(X) mark#(and(X1,X2)) = 2X1 + 2X2 >= 2X1 = mark#(X1) mark#(plus(X1,X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = 3M + 2N >= 3M + 2N = mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) = 2X1 + 3X2 + 2X3 >= 2X1 + 3X2 + 2X3 = active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) = 3M + 2N >= 3M + 2N = mark#(s(plus(N,M))) mark#(s(X)) = 2X >= 2X = mark#(X) mark#(plus(X1,X2)) = 2X1 + 3X2 >= 2X1 = mark#(X1) mark#(plus(X1,X2)) = 2X1 + 3X2 >= 2X2 = mark#(X2) mark#(U21(X1,X2,X3)) = 2X1 + 3X2 + 2X3 >= 2X1 = mark#(X1) mark#(U11(X1,X2)) = 3X1 + 2X2 + 3 >= 3X1 + 2X2 + 3 = active#(U11(mark(X1),X2)) active#(U11(tt(),N)) = 2N + 3 >= 2N = mark#(N) mark#(U11(X1,X2)) = 3X1 + 2X2 + 3 >= 2X1 = mark#(X1) active(U11(tt(),N)) = N + 3/2 >= N = mark(N) active(U21(tt(),M,N)) = 3/2M + N >= 3/2M + N = mark(s(plus(N,M))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) = 0 >= 0 = mark(isNat(V1)) active(plus(N,0())) = N + 3/2 >= N + 3/2 = mark(U11(isNat(N),N)) active(plus(N,s(M))) = 3/2M + N >= 3/2M + N = mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) = 3/2X1 + X2 + 3/2 >= 3/2X1 + X2 + 3/2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X1,X2,X3)) = X1 + 3/2X2 + X3 >= X1 + 3/2X2 + X3 = active(U21(mark(X1),X2,X3)) mark(s(X)) = X >= X = active(s(mark(X))) mark(plus(X1,X2)) = X1 + 3/2X2 >= X1 + 3/2X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(0()) = 1 >= 1 = active(0()) U11(mark(X1),X2) = 3/2X1 + X2 + 3/2 >= 3/2X1 + X2 + 3/2 = U11(X1,X2) U11(X1,mark(X2)) = 3/2X1 + X2 + 3/2 >= 3/2X1 + X2 + 3/2 = U11(X1,X2) U11(active(X1),X2) = 3/2X1 + X2 + 3/2 >= 3/2X1 + X2 + 3/2 = U11(X1,X2) U11(X1,active(X2)) = 3/2X1 + X2 + 3/2 >= 3/2X1 + X2 + 3/2 = U11(X1,X2) U21(mark(X1),X2,X3) = X1 + 3/2X2 + X3 >= X1 + 3/2X2 + X3 = U21(X1,X2,X3) U21(X1,mark(X2),X3) = X1 + 3/2X2 + X3 >= X1 + 3/2X2 + X3 = U21(X1,X2,X3) U21(X1,X2,mark(X3)) = X1 + 3/2X2 + X3 >= X1 + 3/2X2 + X3 = U21(X1,X2,X3) U21(active(X1),X2,X3) = X1 + 3/2X2 + X3 >= X1 + 3/2X2 + X3 = U21(X1,X2,X3) U21(X1,active(X2),X3) = X1 + 3/2X2 + X3 >= X1 + 3/2X2 + X3 = U21(X1,X2,X3) U21(X1,X2,active(X3)) = X1 + 3/2X2 + X3 >= X1 + 3/2X2 + X3 = U21(X1,X2,X3) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) plus(mark(X1),X2) = X1 + 3/2X2 >= X1 + 3/2X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 3/2X2 >= X1 + 3/2X2 = plus(X1,X2) plus(active(X1),X2) = X1 + 3/2X2 >= X1 + 3/2X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + 3/2X2 >= X1 + 3/2X2 = plus(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) problem: DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Restore Modifier: DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U21(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) SCC Processor: #sccs: 1 #rules: 14 #arcs: 103/225 DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(U21(X1,X2,X3)) -> mark#(X1) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=1 usable rules: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) interpretation: [mark#](x0) = 4x0, [active#](x0) = 4x0, [isNat](x0) = 0, [0] = 2, [and](x0, x1) = 7x0 + 2x1, [s](x0) = x0 + 2, [plus](x0, x1) = 2x0 + 2x1, [U21](x0, x1, x2) = 2x0 + 2x1 + 2x2 + 2, [mark](x0) = x0, [active](x0) = x0, [U11](x0, x1) = 2x1, [tt] = 0 orientation: mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(isNat(V1)) active#(isNat(plus(V1,V2))) = 0 >= 0 = mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) = 28X1 + 8X2 >= 28X1 + 8X2 = active#(and(mark(X1),X2)) active#(and(tt(),X)) = 8X >= 4X = mark#(X) mark#(and(X1,X2)) = 28X1 + 8X2 >= 4X1 = mark#(X1) mark#(plus(X1,X2)) = 8X1 + 8X2 >= 8X1 + 8X2 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = 8M + 8N + 16 >= 8M + 8N + 8 = mark#(U21(and(isNat(M),isNat(N)),M,N)) mark#(U21(X1,X2,X3)) = 8X1 + 8X2 + 8X3 + 8 >= 8X1 + 8X2 + 8X3 + 8 = active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) = 8M + 8N + 8 >= 8M + 8N + 8 = mark#(s(plus(N,M))) mark#(s(X)) = 4X + 8 >= 4X = mark#(X) mark#(plus(X1,X2)) = 8X1 + 8X2 >= 4X1 = mark#(X1) mark#(plus(X1,X2)) = 8X1 + 8X2 >= 4X2 = mark#(X2) mark#(U21(X1,X2,X3)) = 8X1 + 8X2 + 8X3 + 8 >= 4X1 = mark#(X1) active(U11(tt(),N)) = 2N >= N = mark(N) active(U21(tt(),M,N)) = 2M + 2N + 2 >= 2M + 2N + 2 = mark(s(plus(N,M))) active(and(tt(),X)) = 2X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) = 0 >= 0 = mark(isNat(V1)) active(plus(N,0())) = 2N + 4 >= 2N = mark(U11(isNat(N),N)) active(plus(N,s(M))) = 2M + 2N + 4 >= 2M + 2N + 2 = mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) = 2X2 >= 2X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X1,X2,X3)) = 2X1 + 2X2 + 2X3 + 2 >= 2X1 + 2X2 + 2X3 + 2 = active(U21(mark(X1),X2,X3)) mark(s(X)) = X + 2 >= X + 2 = active(s(mark(X))) mark(plus(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) = 7X1 + 2X2 >= 7X1 + 2X2 = active(and(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(0()) = 2 >= 2 = active(0()) U11(mark(X1),X2) = 2X2 >= 2X2 = U11(X1,X2) U11(X1,mark(X2)) = 2X2 >= 2X2 = U11(X1,X2) U11(active(X1),X2) = 2X2 >= 2X2 = U11(X1,X2) U11(X1,active(X2)) = 2X2 >= 2X2 = U11(X1,X2) U21(mark(X1),X2,X3) = 2X1 + 2X2 + 2X3 + 2 >= 2X1 + 2X2 + 2X3 + 2 = U21(X1,X2,X3) U21(X1,mark(X2),X3) = 2X1 + 2X2 + 2X3 + 2 >= 2X1 + 2X2 + 2X3 + 2 = U21(X1,X2,X3) U21(X1,X2,mark(X3)) = 2X1 + 2X2 + 2X3 + 2 >= 2X1 + 2X2 + 2X3 + 2 = U21(X1,X2,X3) U21(active(X1),X2,X3) = 2X1 + 2X2 + 2X3 + 2 >= 2X1 + 2X2 + 2X3 + 2 = U21(X1,X2,X3) U21(X1,active(X2),X3) = 2X1 + 2X2 + 2X3 + 2 >= 2X1 + 2X2 + 2X3 + 2 = U21(X1,X2,X3) U21(X1,X2,active(X3)) = 2X1 + 2X2 + 2X3 + 2 >= 2X1 + 2X2 + 2X3 + 2 = U21(X1,X2,X3) s(mark(X)) = X + 2 >= X + 2 = s(X) s(active(X)) = X + 2 >= X + 2 = s(X) plus(mark(X1),X2) = 2X1 + 2X2 >= 2X1 + 2X2 = plus(X1,X2) plus(X1,mark(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = plus(X1,X2) plus(active(X1),X2) = 2X1 + 2X2 >= 2X1 + 2X2 = plus(X1,X2) plus(X1,active(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = plus(X1,X2) and(mark(X1),X2) = 7X1 + 2X2 >= 7X1 + 2X2 = and(X1,X2) and(X1,mark(X2)) = 7X1 + 2X2 >= 7X1 + 2X2 = and(X1,X2) and(active(X1),X2) = 7X1 + 2X2 >= 7X1 + 2X2 = and(X1,X2) and(X1,active(X2)) = 7X1 + 2X2 >= 7X1 + 2X2 = and(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) problem: DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Restore Modifier: DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) mark#(U21(X1,X2,X3)) -> active#(U21(mark(X1),X2,X3)) active#(U21(tt(),M,N)) -> mark#(s(plus(N,M))) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) SCC Processor: #sccs: 1 #rules: 8 #arcs: 65/121 DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNat(plus(V1,V2))) -> mark#(and(isNat(V1),isNat(V2))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) KBO Processor: argument filtering: pi(tt) = [] pi(U11) = 1 pi(active) = 0 pi(mark) = 0 pi(U21) = [1,2] pi(plus) = [0,1] pi(s) = [0] pi(and) = [0,1] pi(0) = [] pi(isNat) = 0 pi(active#) = 0 pi(mark#) = 0 usable rules: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) weight function: w0 = 1 w(mark#) = w(0) = w(s) = w(U21) = w(U11) = w(tt) = 1 w(active#) = w(isNat) = w(and) = w(plus) = w(mark) = w( active) = 0 precedence: plus > U21 > 0 > mark# ~ active# ~ isNat ~ and ~ s ~ mark ~ active ~ U11 ~ tt problem: DPs: mark#(isNat(X)) -> active#(isNat(X)) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Restore Modifier: DPs: mark#(isNat(X)) -> active#(isNat(X)) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) SCC Processor: #sccs: 0 #rules: 0 #arcs: 26/4 DPs: plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(plus#) = 0 problem: DPs: plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(plus#) = 1 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Qed DPs: s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Qed DPs: isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(isNat#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Qed DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(and#) = 0 problem: DPs: and#(X1,mark(X2)) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(and#) = 1 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Qed DPs: U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(U11#) = 0 problem: DPs: U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(U11#) = 1 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Qed DPs: U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3) U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,active(X2),X3) -> U21#(X1,X2,X3) U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(U21#) = 1 problem: DPs: U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(U21#) = 2 problem: DPs: U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(active(X1),X2,X3) -> U21#(X1,X2,X3) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Subterm Criterion Processor: simple projection: pi(U21#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U21(mark(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,mark(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,mark(X3)) -> U21(X1,X2,X3) U21(active(X1),X2,X3) -> U21(X1,X2,X3) U21(X1,active(X2),X3) -> U21(X1,X2,X3) U21(X1,X2,active(X3)) -> U21(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Qed