MAYBE Problem: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(U12(tt(),M,N)) -> plus#(N,M) active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) s#(mark(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) s#(ok(X)) -> s#(X) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(U12(tt(),M,N)) -> plus#(N,M) active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) s#(mark(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) s#(ok(X)) -> s#(X) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) top#(ok(X)) -> active#(X) -> active#(U12(X1,X2,X3)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) top#(ok(X)) -> active#(X) -> active#(U11(X1,X2,X3)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) top#(ok(X)) -> active#(X) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) top#(ok(X)) -> active#(X) -> active#(U12(tt(),M,N)) -> plus#(N,M) top#(ok(X)) -> active#(X) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X3) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(s(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) -> plus#(ok(X1),ok(X2)) -> plus#(X1,X2) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) -> plus#(X1,mark(X2)) -> plus#(X1,X2) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) -> plus#(mark(X1),X2) -> plus#(X1,X2) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(s(X)) -> s#(proper(X)) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(s(X)) -> proper#(X) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) -> U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) -> U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) -> U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) -> U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) -> U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) -> U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) -> U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) -> U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) -> plus#(ok(X1),ok(X2)) -> plus#(X1,X2) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) -> plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) -> plus#(mark(X1),X2) -> plus#(X1,X2) plus#(mark(X1),X2) -> plus#(X1,X2) -> plus#(ok(X1),ok(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#(X1,mark(X2)) -> plus#(X1,X2) -> plus#(ok(X1),ok(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) U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) -> U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) -> U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) -> U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) -> U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X) active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(s(X)) -> active#(X) -> active#(U12(X1,X2,X3)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(s(X)) -> active#(X) -> active#(U11(X1,X2,X3)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(s(X)) -> active#(X) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(s(X)) -> active#(X) -> active#(U12(tt(),M,N)) -> plus#(N,M) active#(s(X)) -> active#(X) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(plus(X1,X2)) -> plus#(active(X1),X2) -> plus#(ok(X1),ok(X2)) -> plus#(X1,X2) active#(plus(X1,X2)) -> plus#(active(X1),X2) -> plus#(X1,mark(X2)) -> plus#(X1,X2) active#(plus(X1,X2)) -> plus#(active(X1),X2) -> plus#(mark(X1),X2) -> plus#(X1,X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) -> plus#(ok(X1),ok(X2)) -> plus#(X1,X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) -> plus#(X1,mark(X2)) -> plus#(X1,X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) -> plus#(mark(X1),X2) -> plus#(X1,X2) active#(plus(X1,X2)) -> active#(X2) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(plus(X1,X2)) -> active#(X2) -> active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> active#(X2) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X2) -> active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> active#(X2) -> active#(s(X)) -> s#(active(X)) active#(plus(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X) active#(plus(X1,X2)) -> active#(X2) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(plus(X1,X2)) -> active#(X2) -> active#(U12(X1,X2,X3)) -> active#(X1) active#(plus(X1,X2)) -> active#(X2) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(plus(X1,X2)) -> active#(X2) -> active#(U11(X1,X2,X3)) -> active#(X1) active#(plus(X1,X2)) -> active#(X2) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(plus(X1,X2)) -> active#(X2) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(plus(X1,X2)) -> active#(X2) -> active#(U12(tt(),M,N)) -> plus#(N,M) active#(plus(X1,X2)) -> active#(X2) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(plus(X1,X2)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(plus(X1,X2)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(plus(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(plus(X1,X2)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(plus(X1,X2)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> active#(X1) active#(plus(X1,X2)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(plus(X1,X2)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> active#(X1) active#(plus(X1,X2)) -> active#(X1) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(plus(X1,X2)) -> active#(X1) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(plus(X1,X2)) -> active#(X1) -> active#(U12(tt(),M,N)) -> plus#(N,M) active#(plus(X1,X2)) -> active#(X1) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(plus(N,s(M))) -> U11#(tt(),M,N) -> U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) active#(plus(N,s(M))) -> U11#(tt(),M,N) -> U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) active#(U12(tt(),M,N)) -> s#(plus(N,M)) -> s#(ok(X)) -> s#(X) active#(U12(tt(),M,N)) -> s#(plus(N,M)) -> s#(mark(X)) -> s#(X) active#(U12(tt(),M,N)) -> plus#(N,M) -> plus#(ok(X1),ok(X2)) -> plus#(X1,X2) active#(U12(tt(),M,N)) -> plus#(N,M) -> plus#(X1,mark(X2)) -> plus#(X1,X2) active#(U12(tt(),M,N)) -> plus#(N,M) -> plus#(mark(X1),X2) -> plus#(X1,X2) active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) -> U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) -> U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X2) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X1) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U12(tt(),M,N)) -> plus#(N,M) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(U11(tt(),M,N)) -> U12#(tt(),M,N) -> U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) active#(U11(tt(),M,N)) -> U12#(tt(),M,N) -> U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) -> U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) -> U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X2) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X1) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U12(tt(),M,N)) -> plus#(N,M) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) EDG Processor: DPs: active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(U12(tt(),M,N)) -> plus#(N,M) active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) s#(mark(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) s#(ok(X)) -> s#(X) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> active#(X) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) top#(ok(X)) -> active#(X) -> active#(U12(tt(),M,N)) -> plus#(N,M) top#(ok(X)) -> active#(X) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) top#(ok(X)) -> active#(X) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) top#(ok(X)) -> active#(X) -> active#(U11(X1,X2,X3)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) top#(ok(X)) -> active#(X) -> active#(U12(X1,X2,X3)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X3) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X3) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X2) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X1) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) -> plus#(mark(X1),X2) -> plus#(X1,X2) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) -> plus#(X1,mark(X2)) -> plus#(X1,X2) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) -> plus#(ok(X1),ok(X2)) -> plus#(X1,X2) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(s(X)) -> proper#(X) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(s(X)) -> s#(proper(X)) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) -> U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) -> U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X3) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X2) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3)) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X1) -> proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) -> U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) -> U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) -> U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) -> U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) -> U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) -> U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) -> plus#(mark(X1),X2) -> plus#(X1,X2) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) -> plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) -> plus#(ok(X1),ok(X2)) -> plus#(X1,X2) plus#(mark(X1),X2) -> plus#(X1,X2) -> plus#(mark(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#(ok(X1),ok(X2)) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) -> plus#(mark(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#(ok(X1),ok(X2)) -> plus#(X1,X2) U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) -> U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) -> U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) -> U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) -> U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X) active#(s(X)) -> active#(X) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(s(X)) -> active#(X) -> active#(U12(tt(),M,N)) -> plus#(N,M) active#(s(X)) -> active#(X) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(s(X)) -> active#(X) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(s(X)) -> active#(X) -> active#(U11(X1,X2,X3)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(s(X)) -> active#(X) -> active#(U12(X1,X2,X3)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(plus(X1,X2)) -> plus#(active(X1),X2) -> plus#(mark(X1),X2) -> plus#(X1,X2) active#(plus(X1,X2)) -> plus#(active(X1),X2) -> plus#(ok(X1),ok(X2)) -> plus#(X1,X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) -> plus#(X1,mark(X2)) -> plus#(X1,X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) -> plus#(ok(X1),ok(X2)) -> plus#(X1,X2) active#(plus(X1,X2)) -> active#(X2) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(plus(X1,X2)) -> active#(X2) -> active#(U12(tt(),M,N)) -> plus#(N,M) active#(plus(X1,X2)) -> active#(X2) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(plus(X1,X2)) -> active#(X2) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(plus(X1,X2)) -> active#(X2) -> active#(U11(X1,X2,X3)) -> active#(X1) active#(plus(X1,X2)) -> active#(X2) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(plus(X1,X2)) -> active#(X2) -> active#(U12(X1,X2,X3)) -> active#(X1) active#(plus(X1,X2)) -> active#(X2) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(plus(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X) active#(plus(X1,X2)) -> active#(X2) -> active#(s(X)) -> s#(active(X)) active#(plus(X1,X2)) -> active#(X2) -> active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> active#(X2) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X2) -> active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> active#(X2) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(plus(X1,X2)) -> active#(X1) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(plus(X1,X2)) -> active#(X1) -> active#(U12(tt(),M,N)) -> plus#(N,M) active#(plus(X1,X2)) -> active#(X1) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(plus(X1,X2)) -> active#(X1) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(plus(X1,X2)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> active#(X1) active#(plus(X1,X2)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(plus(X1,X2)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> active#(X1) active#(plus(X1,X2)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(plus(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(plus(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(plus(X1,X2)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(U12(tt(),M,N)) -> s#(plus(N,M)) -> s#(mark(X)) -> s#(X) active#(U12(tt(),M,N)) -> s#(plus(N,M)) -> s#(ok(X)) -> s#(X) active#(U12(tt(),M,N)) -> plus#(N,M) -> plus#(mark(X1),X2) -> plus#(X1,X2) active#(U12(tt(),M,N)) -> plus#(N,M) -> plus#(X1,mark(X2)) -> plus#(X1,X2) active#(U12(tt(),M,N)) -> plus#(N,M) -> plus#(ok(X1),ok(X2)) -> plus#(X1,X2) active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) -> U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) -> U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U12(tt(),M,N)) -> plus#(N,M) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X1) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X2) active#(U12(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) -> U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) -> U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U11(tt(),M,N)) -> U12#(tt(),M,N) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U12(tt(),M,N)) -> plus#(N,M) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U12(tt(),M,N)) -> s#(plus(N,M)) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(plus(N,s(M))) -> U11#(tt(),M,N) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X1) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> active#(X2) active#(U11(X1,X2,X3)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(X1,active(X2)) SCC Processor: #sccs: 7 #rules: 25 #arcs: 267/1600 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) proper#(U12(X1,X2,X3)) -> proper#(X1) proper#(U12(X1,X2,X3)) -> proper#(X2) proper#(U12(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X3) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [0] = 1, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [mark](x0) = x0, [U12](x0, x1, x2) = x0 + x1 + x2 + 1, [active](x0) = x0, [U11](x0, x1, x2) = x0 + x1 + x2 + 1, [tt] = 0 orientation: proper#(plus(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(plus(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(s(X)) = X + 1 >= X + 1 = proper#(X) proper#(U12(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(U12(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(U12(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) proper#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) active(U11(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(plus(N,0())) = N + 2 >= N = mark(N) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U12(X1,X2,X3)) s(mark(X)) = X >= X = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X >= X = s(proper(X)) proper(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(proper(X1),proper(X2)) proper(0()) = 1 >= 0 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U12(X1,X2,X3)) s(ok(X)) = 0 >= 0 = ok(s(X)) plus(ok(X1),ok(X2)) = 1 >= 0 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(s(X)) -> proper#(X) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [0] = 0, [s](x0) = x0 + 1, [plus](x0, x1) = 0, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: proper#(s(X)) = X + 2 >= X + 1 = proper#(X) active(U11(tt(),M,N)) = 0 >= 0 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(plus(N,0())) = 0 >= 0 = mark(N) active(plus(N,s(M))) = 0 >= 0 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = 0 >= 0 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = 0 >= 0 = U12(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = 0 >= 0 = plus(active(X1),X2) active(plus(X1,X2)) = 0 >= 0 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = 0 >= 0 = mark(U12(X1,X2,X3)) s(mark(X)) = 1 >= 0 = mark(s(X)) plus(mark(X1),X2) = 0 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2,X3)) = 0 >= 0 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(U12(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> active#(X1) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [mark](x0) = 1, [U12](x0, x1, x2) = x0 + 1, [active](x0) = x0, [U11](x0, x1, x2) = x0 + 1, [tt] = 0 orientation: active#(plus(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(plus(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(s(X)) = X + 1 >= X + 1 = active#(X) active#(U12(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U11(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1) active(U11(tt(),M,N)) = 1 >= 1 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = 1 >= 1 = mark(s(plus(N,M))) active(plus(N,0())) = N + 1 >= 1 = mark(N) active(plus(N,s(M))) = M + N + 1 >= 1 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = 2 >= 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = 2 >= 1 = mark(U12(X1,X2,X3)) s(mark(X)) = 1 >= 1 = mark(s(X)) plus(mark(X1),X2) = X2 + 2 >= 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 2 >= 1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X >= X = s(proper(X)) proper(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U12(X1,X2,X3)) s(ok(X)) = X >= X = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: active#(s(X)) -> active#(X) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [0] = 0, [s](x0) = x0 + 1, [plus](x0, x1) = 0, [mark](x0) = 0, [U12](x0, x1, x2) = 0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: active#(s(X)) = X + 2 >= X + 1 = active#(X) active(U11(tt(),M,N)) = 0 >= 0 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(plus(N,0())) = 0 >= 0 = mark(N) active(plus(N,s(M))) = 0 >= 0 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = 0 >= 0 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = 0 >= 0 = U12(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = 0 >= 0 = plus(active(X1),X2) active(plus(X1,X2)) = 0 >= 0 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = 0 >= 0 = mark(U12(X1,X2,X3)) s(mark(X)) = 1 >= 0 = mark(s(X)) plus(mark(X1),X2) = 0 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2,X3)) = 0 >= 0 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1, x2) = x2 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0, [mark](x0) = x0, [U12](x0, x1, x2) = x2, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: U11#(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 1 = U11#(X1,X2,X3) U11#(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11#(X1,X2,X3) active(U11(tt(),M,N)) = N >= N = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = N >= N = mark(s(plus(N,M))) active(plus(N,0())) = N >= N = mark(N) active(plus(N,s(M))) = N >= N = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X3 >= X3 = U12(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X1 >= X1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 >= X1 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X3 >= X3 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = X3 >= X3 = mark(U12(X1,X2,X3)) s(mark(X)) = X >= X = mark(s(X)) plus(mark(X1),X2) = X1 >= X1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 >= X1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2,X3)) = 1 >= 1 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(plus(X1,X2)) = 1 >= 1 = plus(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1, x2) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [0] = 1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [U12](x0, x1, x2) = x0 + x1 + x2, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 1 orientation: U11#(mark(X1),X2,X3) = X1 + 2 >= X1 + 1 = U11#(X1,X2,X3) active(U11(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(s(plus(N,M))) active(plus(N,0())) = N + 2 >= N + 1 = mark(N) active(plus(N,s(M))) = M + N + 2 >= M + N + 2 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(active(X1),X2,X3) active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U12(X1,X2,X3)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U12#](x0, x1, x2) = x2 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0, [mark](x0) = x0, [U12](x0, x1, x2) = x2, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: U12#(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 1 = U12#(X1,X2,X3) U12#(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U12#(X1,X2,X3) active(U11(tt(),M,N)) = N >= N = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = N >= N = mark(s(plus(N,M))) active(plus(N,0())) = N >= N = mark(N) active(plus(N,s(M))) = N >= N = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X3 >= X3 = U12(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X1 >= X1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 >= X1 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X3 >= X3 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = X3 >= X3 = mark(U12(X1,X2,X3)) s(mark(X)) = X >= X = mark(s(X)) plus(mark(X1),X2) = X1 >= X1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 >= X1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2,X3)) = 1 >= 1 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(plus(X1,X2)) = 1 >= 1 = plus(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U12#](x0, x1, x2) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [0] = 1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [U12](x0, x1, x2) = x0 + x1 + x2, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 1 orientation: U12#(mark(X1),X2,X3) = X1 + 2 >= X1 + 1 = U12#(X1,X2,X3) active(U11(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(s(plus(N,M))) active(plus(N,0())) = N + 2 >= N + 1 = mark(N) active(plus(N,s(M))) = M + N + 2 >= M + N + 2 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(active(X1),X2,X3) active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U12(X1,X2,X3)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0, [mark](x0) = x0, [U12](x0, x1, x2) = x2, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: s#(ok(X)) = X + 2 >= X + 1 = s#(X) s#(mark(X)) = X + 1 >= X + 1 = s#(X) active(U11(tt(),M,N)) = N + 1 >= N = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = N + 1 >= N = mark(s(plus(N,M))) active(plus(N,0())) = N + 1 >= N = mark(N) active(plus(N,s(M))) = N + 1 >= N = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X3 + 1 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X3 + 1 >= X3 = U12(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + 1 >= X1 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 1 >= X1 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X3 >= X3 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = X3 >= X3 = mark(U12(X1,X2,X3)) s(mark(X)) = X >= X = mark(s(X)) plus(mark(X1),X2) = X1 >= X1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 >= X1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2,X3)) = 1 >= 1 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(plus(X1,X2)) = 1 >= 1 = plus(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: s#(mark(X)) -> s#(X) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [U12](x0, x1, x2) = x0 + x1 + x2, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 0 orientation: s#(mark(X)) = X + 2 >= X + 1 = s#(X) active(U11(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(plus(N,0())) = N + 1 >= N + 1 = mark(N) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U12(X1,X2,X3)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2,X3)) = 0 >= 0 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U12(X1,X2,X3)) s(ok(X)) = 0 >= 0 = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: plus#(ok(X1),ok(X2)) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(mark(X1),X2) -> plus#(X1,X2) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [plus#](x0, x1) = x0 + x1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + 1, [mark](x0) = x0, [U12](x0, x1, x2) = x2 + 1, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: plus#(ok(X1),ok(X2)) = X1 + X2 + 2 >= X1 + X2 = plus#(X1,X2) plus#(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus#(X1,X2) plus#(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus#(X1,X2) active(U11(tt(),M,N)) = N + 1 >= N + 1 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = N + 2 >= N + 1 = mark(s(plus(N,M))) active(plus(N,0())) = N + 2 >= N = mark(N) active(plus(N,s(M))) = N + 2 >= N = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X3 + 1 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X3 + 2 >= X3 + 1 = U12(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 2 >= X1 + 1 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X3 >= X3 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = mark(U12(X1,X2,X3)) s(mark(X)) = X >= X = mark(s(X)) plus(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 1 >= X1 + 1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2,X3)) = X3 + 2 >= X3 + 2 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(mark(X1),X2) -> plus#(X1,X2) TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [plus#](x0, x1) = x0 + x1 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [U12](x0, x1, x2) = x0 + x1 + x2, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 0 orientation: plus#(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = plus#(X1,X2) plus#(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = plus#(X1,X2) active(U11(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(plus(N,0())) = N + 1 >= N + 1 = mark(N) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U12(X1,X2,X3)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2,X3)) = 0 >= 0 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U12(X1,X2,X3)) s(ok(X)) = 0 >= 0 = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed