YES Problem: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Proof: DP Processor: DPs: active#(and(tt(),X)) -> mark#(X) active#(plus(N,0())) -> mark#(N) active#(plus(N,s(M))) -> plus#(N,M) active#(plus(N,s(M))) -> s#(plus(N,M)) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(tt()) -> active#(tt()) 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#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) 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) 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) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) TDG Processor: DPs: active#(and(tt(),X)) -> mark#(X) active#(plus(N,0())) -> mark#(N) active#(plus(N,s(M))) -> plus#(N,M) active#(plus(N,s(M))) -> s#(plus(N,M)) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(tt()) -> active#(tt()) 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#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) 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) 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) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) graph: 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) 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#(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#(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#(0()) -> active#(0()) 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#(tt()) -> active#(tt()) 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)) -> active#(s(mark(X))) -> active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,s(M))) -> s#(plus(N,M)) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,s(M))) -> plus#(N,M) mark#(s(X)) -> active#(s(mark(X))) -> active#(plus(N,0())) -> mark#(N) mark#(s(X)) -> active#(s(mark(X))) -> active#(and(tt(),X)) -> mark#(X) 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#(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#(0()) -> active#(0()) 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#(tt()) -> active#(tt()) 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#(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#(0()) -> active#(0()) 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#(tt()) -> active#(tt()) 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)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> s#(plus(N,M)) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,s(M))) -> plus#(N,M) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(plus(N,0())) -> mark#(N) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) -> active#(and(tt(),X)) -> mark#(X) mark#(0()) -> active#(0()) -> active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(0()) -> active#(0()) -> active#(plus(N,s(M))) -> s#(plus(N,M)) mark#(0()) -> active#(0()) -> active#(plus(N,s(M))) -> plus#(N,M) mark#(0()) -> active#(0()) -> active#(plus(N,0())) -> mark#(N) mark#(0()) -> active#(0()) -> active#(and(tt(),X)) -> mark#(X) 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#(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#(0()) -> active#(0()) 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#(tt()) -> active#(tt()) 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)) -> active#(and(mark(X1),X2)) -> active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,s(M))) -> s#(plus(N,M)) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,s(M))) -> plus#(N,M) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(plus(N,0())) -> mark#(N) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(and(tt(),X)) -> mark#(X) mark#(tt()) -> active#(tt()) -> active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(tt()) -> active#(tt()) -> active#(plus(N,s(M))) -> s#(plus(N,M)) mark#(tt()) -> active#(tt()) -> active#(plus(N,s(M))) -> plus#(N,M) mark#(tt()) -> active#(tt()) -> active#(plus(N,0())) -> mark#(N) mark#(tt()) -> active#(tt()) -> active#(and(tt(),X)) -> mark#(X) active#(plus(N,s(M))) -> s#(plus(N,M)) -> s#(active(X)) -> s#(X) active#(plus(N,s(M))) -> s#(plus(N,M)) -> s#(mark(X)) -> s#(X) active#(plus(N,s(M))) -> plus#(N,M) -> plus#(X1,active(X2)) -> plus#(X1,X2) active#(plus(N,s(M))) -> plus#(N,M) -> plus#(active(X1),X2) -> plus#(X1,X2) active#(plus(N,s(M))) -> plus#(N,M) -> plus#(X1,mark(X2)) -> plus#(X1,X2) active#(plus(N,s(M))) -> plus#(N,M) -> plus#(mark(X1),X2) -> plus#(X1,X2) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> active#(s(mark(X))) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> s#(mark(X)) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(s(X)) -> mark#(X) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(0()) -> active#(0()) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> mark#(X1) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(plus(X1,X2)) -> mark#(X2) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(tt()) -> active#(tt()) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) -> mark#(and(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(N) -> mark#(s(X)) -> active#(s(mark(X))) active#(plus(N,0())) -> mark#(N) -> mark#(s(X)) -> s#(mark(X)) active#(plus(N,0())) -> mark#(N) -> mark#(s(X)) -> mark#(X) active#(plus(N,0())) -> mark#(N) -> mark#(0()) -> active#(0()) active#(plus(N,0())) -> mark#(N) -> mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,0())) -> mark#(N) -> mark#(plus(X1,X2)) -> plus#(mark(X1),mark(X2)) active#(plus(N,0())) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) active#(plus(N,0())) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) active#(plus(N,0())) -> mark#(N) -> mark#(tt()) -> active#(tt()) active#(plus(N,0())) -> mark#(N) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(plus(N,0())) -> mark#(N) -> mark#(and(X1,X2)) -> and#(mark(X1),X2) active#(plus(N,0())) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) 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#(0()) -> active#(0()) 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#(tt()) -> active#(tt()) 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) SCC Processor: #sccs: 4 #rules: 22 #arcs: 161/729 DPs: mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(tt()) -> active#(tt()) active#(plus(N,0())) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(0()) -> active#(0()) mark#(s(X)) -> active#(s(mark(X))) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [active#](x0) = x0 + 1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1 + 1, [0] = 0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 1 orientation: mark#(s(X)) = X + 2 >= X + 1 = mark#(X) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(and(mark(X1),X2)) active#(and(tt(),X)) = X + 2 >= X + 1 = mark#(X) mark#(tt()) = 2 >= 2 = active#(tt()) active#(plus(N,0())) = N + 2 >= N + 1 = mark#(N) mark#(plus(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = M + N + 3 >= M + N + 3 = mark#(s(plus(N,M))) mark#(0()) = 1 >= 1 = active#(0()) mark#(s(X)) = X + 2 >= X + 2 = active#(s(mark(X))) active(and(tt(),X)) = X + 1 >= X = mark(X) active(plus(N,0())) = N + 1 >= N = mark(N) active(plus(N,s(M))) = M + N + 2 >= M + N + 2 = mark(s(plus(N,M))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) 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) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) s(mark(X)) = X + 1 >= X + 1 = s(X) s(active(X)) = X + 1 >= X + 1 = s(X) problem: DPs: mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(0()) -> active#(0()) mark#(s(X)) -> active#(s(mark(X))) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [s](x0) = 0, [plus](x0, x1) = x0, [0] = 1, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= 0 = active#(and(mark(X1),X2)) mark#(tt()) = 0 >= 0 = active#(tt()) mark#(plus(X1,X2)) = X1 >= 0 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = 0 >= 0 = mark#(s(plus(N,M))) mark#(0()) = 1 >= 0 = active#(0()) mark#(s(X)) = 0 >= 0 = active#(s(mark(X))) active(and(tt(),X)) = X >= X = mark(X) active(plus(N,0())) = N >= N = mark(N) active(plus(N,s(M))) = N >= 0 = mark(s(plus(N,M))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(plus(X1,X2)) = X1 >= X1 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) 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) plus(mark(X1),X2) = X1 >= X1 = plus(X1,X2) plus(X1,mark(X2)) = X1 >= X1 = plus(X1,X2) plus(active(X1),X2) = X1 >= X1 = plus(X1,X2) plus(X1,active(X2)) = X1 >= X1 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(s(X)) -> active#(s(mark(X))) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [active#](x0) = x0 + 1, [s](x0) = 1, [plus](x0, x1) = x0 + 1, [0] = 1, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1 + 1, [tt] = 1 orientation: mark#(and(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = active#(and(mark(X1),X2)) mark#(tt()) = 2 >= 2 = active#(tt()) mark#(plus(X1,X2)) = X1 + 2 >= X1 + 2 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = N + 2 >= 2 = mark#(s(plus(N,M))) mark#(s(X)) = 2 >= 2 = active#(s(mark(X))) active(and(tt(),X)) = X + 2 >= X = mark(X) active(plus(N,0())) = N + 1 >= N = mark(N) active(plus(N,s(M))) = N + 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(plus(X1,X2)) = X1 + 1 >= X1 + 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1 >= 1 = active(s(mark(X))) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) and(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) and(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) plus(mark(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) problem: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) mark#(s(X)) -> active#(s(mark(X))) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [s](x0) = 0, [plus](x0, x1) = 1, [0] = 0, [mark](x0) = 0, [active](x0) = 0, [and](x0, x1) = 1, [tt] = 1 orientation: mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) mark#(tt()) = 1 >= 1 = active#(tt()) mark#(plus(X1,X2)) = 1 >= 1 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = 1 >= 1 = mark#(s(plus(N,M))) mark#(s(X)) = 1 >= 0 = active#(s(mark(X))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(plus(N,0())) = 0 >= 0 = mark(N) active(plus(N,s(M))) = 0 >= 0 = mark(s(plus(N,M))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) -> mark#(s(plus(N,M))) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 1, [s](x0) = 0, [plus](x0, x1) = 1, [0] = 0, [mark](x0) = 0, [active](x0) = 0, [and](x0, x1) = 1, [tt] = 1 orientation: mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) mark#(tt()) = 1 >= 1 = active#(tt()) mark#(plus(X1,X2)) = 1 >= 1 = active#(plus(mark(X1),mark(X2))) active#(plus(N,s(M))) = 1 >= 0 = mark#(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(plus(N,0())) = 0 >= 0 = mark(N) active(plus(N,s(M))) = 0 >= 0 = mark(s(plus(N,M))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) plus(mark(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,mark(X2)) = 1 >= 1 = plus(X1,X2) plus(active(X1),X2) = 1 >= 1 = plus(X1,X2) plus(X1,active(X2)) = 1 >= 1 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(plus(X1,X2)) -> active#(plus(mark(X1),mark(X2))) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [s](x0) = 0, [plus](x0, x1) = 0, [0] = 0, [mark](x0) = 0, [active](x0) = 0, [and](x0, x1) = 1, [tt] = 1 orientation: mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) mark#(tt()) = 1 >= 1 = active#(tt()) mark#(plus(X1,X2)) = 1 >= 0 = active#(plus(mark(X1),mark(X2))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(plus(N,0())) = 0 >= 0 = mark(N) active(plus(N,s(M))) = 0 >= 0 = mark(s(plus(N,M))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(tt()) -> active#(tt()) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [s](x0) = 0, [plus](x0, x1) = 0, [0] = 0, [mark](x0) = 0, [active](x0) = 0, [and](x0, x1) = 0, [tt] = 1 orientation: mark#(and(X1,X2)) = 0 >= 0 = active#(and(mark(X1),X2)) mark#(tt()) = 1 >= 0 = active#(tt()) active(and(tt(),X)) = 0 >= 0 = mark(X) active(plus(N,0())) = 0 >= 0 = mark(N) active(plus(N,s(M))) = 0 >= 0 = mark(s(plus(N,M))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) and(mark(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,mark(X2)) = 0 >= 0 = and(X1,X2) and(active(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,active(X2)) = 0 >= 0 = and(X1,X2) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [s](x0) = 0, [plus](x0, x1) = 0, [0] = 0, [mark](x0) = 0, [active](x0) = 0, [and](x0, x1) = 1, [tt] = 0 orientation: mark#(and(X1,X2)) = 1 >= 0 = active#(and(mark(X1),X2)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(plus(N,0())) = 0 >= 0 = mark(N) active(plus(N,s(M))) = 0 >= 0 = mark(s(plus(N,M))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(plus(X1,X2)) = 0 >= 0 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Qed 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(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [plus#](x0, x1) = x1, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [0] = 0, [mark](x0) = x0 + 1, [active](x0) = x0, [and](x0, x1) = x1 + 1, [tt] = 0 orientation: plus#(mark(X1),X2) = X2 >= X2 = plus#(X1,X2) plus#(X1,mark(X2)) = X2 + 1 >= X2 = plus#(X1,X2) plus#(active(X1),X2) = X2 >= X2 = plus#(X1,X2) plus#(X1,active(X2)) = X2 >= X2 = plus#(X1,X2) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(plus(N,0())) = N + 1 >= N + 1 = mark(N) active(plus(N,s(M))) = N + 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = X2 + 2 >= X2 + 1 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 0 = active(tt()) mark(plus(X1,X2)) = X1 + 2 >= X1 + 2 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 0 = active(0()) mark(s(X)) = 1 >= 0 = active(s(mark(X))) and(mark(X1),X2) = X2 + 1 >= X2 + 1 = and(X1,X2) and(X1,mark(X2)) = X2 + 2 >= X2 + 1 = and(X1,X2) and(active(X1),X2) = X2 + 1 >= X2 + 1 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 + 1 = and(X1,X2) plus(mark(X1),X2) = X1 + 2 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: plus#(mark(X1),X2) -> plus#(X1,X2) plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [plus#](x0, x1) = x0, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [0] = 1, [mark](x0) = x0 + 1, [active](x0) = x0, [and](x0, x1) = x0 + x1 + 1, [tt] = 0 orientation: plus#(mark(X1),X2) = X1 + 1 >= X1 = plus#(X1,X2) plus#(active(X1),X2) = X1 >= X1 = plus#(X1,X2) plus#(X1,active(X2)) = X1 >= X1 = plus#(X1,X2) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(plus(N,0())) = N + 1 >= N + 1 = mark(N) active(plus(N,s(M))) = N + 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 0 = active(tt()) mark(plus(X1,X2)) = X1 + 2 >= X1 + 2 = active(plus(mark(X1),mark(X2))) mark(0()) = 2 >= 1 = active(0()) mark(s(X)) = 1 >= 0 = active(s(mark(X))) and(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = and(X1,X2) and(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) and(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) plus(mark(X1),X2) = X1 + 2 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: plus#(active(X1),X2) -> plus#(X1,X2) plus#(X1,active(X2)) -> plus#(X1,X2) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [plus#](x0, x1) = x0, [s](x0) = 0, [plus](x0, x1) = 0, [0] = 0, [mark](x0) = 1, [active](x0) = x0 + 1, [and](x0, x1) = 0, [tt] = 0 orientation: plus#(active(X1),X2) = X1 + 1 >= X1 = plus#(X1,X2) plus#(X1,active(X2)) = X1 >= X1 = plus#(X1,X2) active(and(tt(),X)) = 1 >= 1 = mark(X) active(plus(N,0())) = 1 >= 1 = mark(N) active(plus(N,s(M))) = 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = 1 >= 1 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(plus(X1,X2)) = 1 >= 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1 >= 1 = active(s(mark(X))) and(mark(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,mark(X2)) = 0 >= 0 = and(X1,X2) and(active(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,active(X2)) = 0 >= 0 = and(X1,X2) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: plus#(X1,active(X2)) -> plus#(X1,X2) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [plus#](x0, x1) = x1, [s](x0) = 0, [plus](x0, x1) = 0, [0] = 0, [mark](x0) = 1, [active](x0) = x0 + 1, [and](x0, x1) = 0, [tt] = 0 orientation: plus#(X1,active(X2)) = X2 + 1 >= X2 = plus#(X1,X2) active(and(tt(),X)) = 1 >= 1 = mark(X) active(plus(N,0())) = 1 >= 1 = mark(N) active(plus(N,s(M))) = 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = 1 >= 1 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(plus(X1,X2)) = 1 >= 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1 >= 1 = active(s(mark(X))) and(mark(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,mark(X2)) = 0 >= 0 = and(X1,X2) and(active(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,active(X2)) = 0 >= 0 = and(X1,X2) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Qed DPs: s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [0] = 1, [mark](x0) = x0 + 1, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 1 orientation: s#(mark(X)) = X + 2 >= X + 1 = s#(X) s#(active(X)) = X + 1 >= X + 1 = s#(X) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(plus(N,0())) = N + 1 >= N + 1 = mark(N) active(plus(N,s(M))) = N + 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(and(mark(X1),X2)) mark(tt()) = 2 >= 1 = active(tt()) mark(plus(X1,X2)) = X1 + 2 >= X1 + 2 = active(plus(mark(X1),mark(X2))) mark(0()) = 2 >= 1 = active(0()) mark(s(X)) = 1 >= 0 = active(s(mark(X))) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 + 1 >= 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) plus(mark(X1),X2) = X1 + 2 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: s#(active(X)) -> s#(X) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [s](x0) = 0, [plus](x0, x1) = 0, [0] = 0, [mark](x0) = 1, [active](x0) = x0 + 1, [and](x0, x1) = 0, [tt] = 0 orientation: s#(active(X)) = X + 2 >= X + 1 = s#(X) active(and(tt(),X)) = 1 >= 1 = mark(X) active(plus(N,0())) = 1 >= 1 = mark(N) active(plus(N,s(M))) = 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = 1 >= 1 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(plus(X1,X2)) = 1 >= 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1 >= 1 = active(s(mark(X))) and(mark(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,mark(X2)) = 0 >= 0 = and(X1,X2) and(active(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,active(X2)) = 0 >= 0 = and(X1,X2) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(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(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x1, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [0] = 0, [mark](x0) = x0 + 1, [active](x0) = x0, [and](x0, x1) = x1 + 1, [tt] = 0 orientation: and#(mark(X1),X2) = X2 >= X2 = and#(X1,X2) and#(X1,mark(X2)) = X2 + 1 >= X2 = and#(X1,X2) and#(active(X1),X2) = X2 >= X2 = and#(X1,X2) and#(X1,active(X2)) = X2 >= X2 = and#(X1,X2) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(plus(N,0())) = N + 1 >= N + 1 = mark(N) active(plus(N,s(M))) = N + 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = X2 + 2 >= X2 + 1 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 0 = active(tt()) mark(plus(X1,X2)) = X1 + 2 >= X1 + 2 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 0 = active(0()) mark(s(X)) = 1 >= 0 = active(s(mark(X))) and(mark(X1),X2) = X2 + 1 >= X2 + 1 = and(X1,X2) and(X1,mark(X2)) = X2 + 2 >= X2 + 1 = and(X1,X2) and(active(X1),X2) = X2 + 1 >= X2 + 1 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 + 1 = and(X1,X2) plus(mark(X1),X2) = X1 + 2 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x0, [s](x0) = 0, [plus](x0, x1) = x0 + 1, [0] = 1, [mark](x0) = x0 + 1, [active](x0) = x0, [and](x0, x1) = x0 + x1 + 1, [tt] = 0 orientation: and#(mark(X1),X2) = X1 + 1 >= X1 = and#(X1,X2) and#(active(X1),X2) = X1 >= X1 = and#(X1,X2) and#(X1,active(X2)) = X1 >= X1 = and#(X1,X2) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(plus(N,0())) = N + 1 >= N + 1 = mark(N) active(plus(N,s(M))) = N + 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 0 = active(tt()) mark(plus(X1,X2)) = X1 + 2 >= X1 + 2 = active(plus(mark(X1),mark(X2))) mark(0()) = 2 >= 1 = active(0()) mark(s(X)) = 1 >= 0 = active(s(mark(X))) and(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = and(X1,X2) and(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) and(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) plus(mark(X1),X2) = X1 + 2 >= X1 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + 1 >= X1 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + 1 >= X1 + 1 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x0, [s](x0) = 0, [plus](x0, x1) = 0, [0] = 0, [mark](x0) = 1, [active](x0) = x0 + 1, [and](x0, x1) = 0, [tt] = 0 orientation: and#(active(X1),X2) = X1 + 1 >= X1 = and#(X1,X2) and#(X1,active(X2)) = X1 >= X1 = and#(X1,X2) active(and(tt(),X)) = 1 >= 1 = mark(X) active(plus(N,0())) = 1 >= 1 = mark(N) active(plus(N,s(M))) = 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = 1 >= 1 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(plus(X1,X2)) = 1 >= 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1 >= 1 = active(s(mark(X))) and(mark(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,mark(X2)) = 0 >= 0 = and(X1,X2) and(active(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,active(X2)) = 0 >= 0 = and(X1,X2) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: and#(X1,active(X2)) -> and#(X1,X2) TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x1, [s](x0) = 0, [plus](x0, x1) = 0, [0] = 0, [mark](x0) = 1, [active](x0) = x0 + 1, [and](x0, x1) = 0, [tt] = 0 orientation: and#(X1,active(X2)) = X2 + 1 >= X2 = and#(X1,X2) active(and(tt(),X)) = 1 >= 1 = mark(X) active(plus(N,0())) = 1 >= 1 = mark(N) active(plus(N,s(M))) = 1 >= 1 = mark(s(plus(N,M))) mark(and(X1,X2)) = 1 >= 1 = active(and(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(plus(X1,X2)) = 1 >= 1 = active(plus(mark(X1),mark(X2))) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1 >= 1 = active(s(mark(X))) and(mark(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,mark(X2)) = 0 >= 0 = and(X1,X2) and(active(X1),X2) = 0 >= 0 = and(X1,X2) and(X1,active(X2)) = 0 >= 0 = and(X1,X2) plus(mark(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,mark(X2)) = 0 >= 0 = plus(X1,X2) plus(active(X1),X2) = 0 >= 0 = plus(X1,X2) plus(X1,active(X2)) = 0 >= 0 = plus(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) problem: DPs: TRS: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) 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) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Qed