YES 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) SCC Processor: #sccs: 7 #rules: 25 #arcs: 273/1600 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)) Matrix Interpretation Processor: dim=1 interpretation: [top#](x0) = 2x0, [top](x0) = 7/2, [ok](x0) = x0, [proper](x0) = x0, [0] = 3, [s](x0) = x0 + 3, [plus](x0, x1) = 2x0 + 3x1, [mark](x0) = x0 + 1/2, [U12](x0, x1, x2) = 2x0 + 3x1 + 2x2 + 1/2, [active](x0) = x0, [U11](x0, x1, x2) = 2x0 + 3x1 + 2x2 + 1, [tt] = 5/2 orientation: top#(ok(X)) = 2X >= 2X = top#(active(X)) top#(mark(X)) = 2X + 1 >= 2X = top#(proper(X)) active(U11(tt(),M,N)) = 3M + 2N + 6 >= 3M + 2N + 6 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = 3M + 2N + 11/2 >= 3M + 2N + 7/2 = mark(s(plus(N,M))) active(plus(N,0())) = 2N + 9 >= N + 1/2 = mark(N) active(plus(N,s(M))) = 3M + 2N + 9 >= 3M + 2N + 13/2 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = 2X1 + 3X2 + 2X3 + 1 >= 2X1 + 3X2 + 2X3 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = 2X1 + 3X2 + 2X3 + 1/2 >= 2X1 + 3X2 + 2X3 + 1/2 = U12(active(X1),X2,X3) active(s(X)) = X + 3 >= X + 3 = s(active(X)) active(plus(X1,X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = plus(active(X1),X2) active(plus(X1,X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = 2X1 + 3X2 + 2X3 + 2 >= 2X1 + 3X2 + 2X3 + 3/2 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = 2X1 + 3X2 + 2X3 + 3/2 >= 2X1 + 3X2 + 2X3 + 1 = mark(U12(X1,X2,X3)) s(mark(X)) = X + 7/2 >= X + 7/2 = mark(s(X)) plus(mark(X1),X2) = 2X1 + 3X2 + 1 >= 2X1 + 3X2 + 1/2 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 2X1 + 3X2 + 3/2 >= 2X1 + 3X2 + 1/2 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 2X1 + 3X2 + 2X3 + 1 >= 2X1 + 3X2 + 2X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 5/2 >= 5/2 = ok(tt()) proper(U12(X1,X2,X3)) = 2X1 + 3X2 + 2X3 + 1/2 >= 2X1 + 3X2 + 2X3 + 1/2 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 3 >= X + 3 = s(proper(X)) proper(plus(X1,X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = plus(proper(X1),proper(X2)) proper(0()) = 3 >= 3 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = 2X1 + 3X2 + 2X3 + 1 >= 2X1 + 3X2 + 2X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 2X1 + 3X2 + 2X3 + 1/2 >= 2X1 + 3X2 + 2X3 + 1/2 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 3 >= X + 3 = ok(s(X)) plus(ok(X1),ok(X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = ok(plus(X1,X2)) top(mark(X)) = 7/2 >= 7/2 = top(proper(X)) top(ok(X)) = 7/2 >= 7/2 = top(active(X)) problem: DPs: 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)) KBO Processor: argument filtering: pi(tt) = [] pi(U11) = 2 pi(active) = 0 pi(U12) = 2 pi(mark) = 0 pi(plus) = 0 pi(s) = 0 pi(0) = [] pi(proper) = [0] pi(ok) = [0] pi(top) = [] pi(top#) = [0] weight function: w0 = 1 w(top#) = w(top) = w(proper) = w(0) = w(s) = w(U11) = w(tt) = 1 w(ok) = w(plus) = w(mark) = w(U12) = w(active) = 0 precedence: top# ~ top ~ ok ~ proper ~ 0 ~ s ~ plus ~ mark ~ U12 ~ active ~ U11 ~ tt 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#(U11(X1,X2,X3)) -> active#(X1) active#(U12(X1,X2,X3)) -> active#(X1) active#(s(X)) -> active#(X) active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> active#(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)) Subterm Criterion Processor: simple projection: pi(active#) = 0 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: proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(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#(X1) proper#(s(X)) -> proper#(X) proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(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)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 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#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(ok(X1),ok(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)) Subterm Criterion Processor: simple projection: pi(plus#) = 1 problem: DPs: 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)) Subterm Criterion Processor: simple projection: pi(plus#) = 0 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#(mark(X)) -> s#(X) s#(ok(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)) Subterm Criterion Processor: simple projection: pi(s#) = 0 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#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(ok(X1),ok(X2),ok(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)) Subterm Criterion Processor: simple projection: pi(U12#) = 2 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)) Subterm Criterion Processor: simple projection: pi(U12#) = 0 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#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(ok(X1),ok(X2),ok(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)) Subterm Criterion Processor: simple projection: pi(U11#) = 2 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)) Subterm Criterion Processor: simple projection: pi(U11#) = 0 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