YES Problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = x0 + 6, [head](x0) = 4x0 + 2, [s](x0) = 2x0, [odds] = 0, [pairs] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [incr](x0) = 2x0, [0] = 0, [active](x0) = x0, [nats] = 0 orientation: active(nats()) = 0 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 0 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 0 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = 2X + 2XS >= 2X + 2XS = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 4X + 4XS + 2 >= X = mark(X) active(tail(cons(X,XS))) = X + XS + 6 >= XS = mark(XS) mark(nats()) = 0 >= 0 = active(nats()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = 2X >= 2X = active(incr(mark(X))) mark(pairs()) = 0 >= 0 = active(pairs()) mark(odds()) = 0 >= 0 = active(odds()) mark(s(X)) = 2X >= 2X = active(s(mark(X))) mark(head(X)) = 4X + 2 >= 4X + 2 = active(head(mark(X))) mark(tail(X)) = X + 6 >= X + 6 = active(tail(mark(X))) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) incr(mark(X)) = 2X >= 2X = incr(X) incr(active(X)) = 2X >= 2X = incr(X) s(mark(X)) = 2X >= 2X = s(X) s(active(X)) = 2X >= 2X = s(X) head(mark(X)) = 4X + 2 >= 4X + 2 = head(X) head(active(X)) = 4X + 2 >= 4X + 2 = head(X) tail(mark(X)) = X + 6 >= X + 6 = tail(X) tail(active(X)) = X + 6 >= X + 6 = tail(X) problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) DP Processor: DPs: active#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) active#(nats()) -> mark#(cons(0(),incr(nats()))) active#(pairs()) -> incr#(odds()) active#(pairs()) -> cons#(0(),incr(odds())) active#(pairs()) -> mark#(cons(0(),incr(odds()))) active#(odds()) -> incr#(pairs()) active#(odds()) -> mark#(incr(pairs())) active#(incr(cons(X,XS))) -> incr#(XS) active#(incr(cons(X,XS))) -> s#(X) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(nats()) -> active#(nats()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> incr#(mark(X)) mark#(incr(X)) -> active#(incr(mark(X))) mark#(pairs()) -> active#(pairs()) mark#(odds()) -> active#(odds()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> head#(mark(X)) mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> tail#(mark(X)) mark#(tail(X)) -> active#(tail(mark(X))) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) incr#(active(X)) -> incr#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) head#(mark(X)) -> head#(X) head#(active(X)) -> head#(X) tail#(mark(X)) -> tail#(X) tail#(active(X)) -> tail#(X) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) TDG Processor: DPs: active#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) active#(nats()) -> mark#(cons(0(),incr(nats()))) active#(pairs()) -> incr#(odds()) active#(pairs()) -> cons#(0(),incr(odds())) active#(pairs()) -> mark#(cons(0(),incr(odds()))) active#(odds()) -> incr#(pairs()) active#(odds()) -> mark#(incr(pairs())) active#(incr(cons(X,XS))) -> incr#(XS) active#(incr(cons(X,XS))) -> s#(X) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(nats()) -> active#(nats()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> incr#(mark(X)) mark#(incr(X)) -> active#(incr(mark(X))) mark#(pairs()) -> active#(pairs()) mark#(odds()) -> active#(odds()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> head#(mark(X)) mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> tail#(mark(X)) mark#(tail(X)) -> active#(tail(mark(X))) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) incr#(active(X)) -> incr#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) head#(mark(X)) -> head#(X) head#(active(X)) -> head#(X) tail#(mark(X)) -> tail#(X) tail#(active(X)) -> tail#(X) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) graph: tail#(mark(X)) -> tail#(X) -> tail#(active(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) tail#(active(X)) -> tail#(X) -> tail#(active(X)) -> tail#(X) tail#(active(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) head#(mark(X)) -> head#(X) -> head#(active(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(mark(X)) -> head#(X) head#(active(X)) -> head#(X) -> head#(active(X)) -> head#(X) head#(active(X)) -> head#(X) -> head#(mark(X)) -> head#(X) s#(mark(X)) -> s#(X) -> s#(active(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) -> s#(active(X)) -> s#(X) s#(active(X)) -> s#(X) -> s#(mark(X)) -> s#(X) mark#(tail(X)) -> tail#(mark(X)) -> tail#(active(X)) -> tail#(X) mark#(tail(X)) -> tail#(mark(X)) -> tail#(mark(X)) -> tail#(X) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> tail#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> head#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(tail(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> incr#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(incr(cons(X,XS))) -> s#(X) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(odds()) -> mark#(incr(pairs())) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(odds()) -> incr#(pairs()) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(pairs()) -> incr#(odds()) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(nats()) -> incr#(nats()) mark#(head(X)) -> head#(mark(X)) -> head#(active(X)) -> head#(X) mark#(head(X)) -> head#(mark(X)) -> head#(mark(X)) -> head#(X) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> tail#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> head#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(head(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> incr#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(head(X)) -> active#(head(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(head(X)) -> active#(head(mark(X))) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(head(X)) -> active#(head(mark(X))) -> active#(incr(cons(X,XS))) -> s#(X) mark#(head(X)) -> active#(head(mark(X))) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(head(X)) -> active#(head(mark(X))) -> active#(odds()) -> mark#(incr(pairs())) mark#(head(X)) -> active#(head(mark(X))) -> active#(odds()) -> incr#(pairs()) mark#(head(X)) -> active#(head(mark(X))) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(head(X)) -> active#(head(mark(X))) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(head(X)) -> active#(head(mark(X))) -> active#(pairs()) -> incr#(odds()) mark#(head(X)) -> active#(head(mark(X))) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(head(X)) -> active#(head(mark(X))) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(head(X)) -> active#(head(mark(X))) -> active#(nats()) -> incr#(nats()) mark#(s(X)) -> s#(mark(X)) -> s#(active(X)) -> s#(X) mark#(s(X)) -> s#(mark(X)) -> s#(mark(X)) -> s#(X) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> tail#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> head#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(s(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> incr#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(s(X)) -> active#(s(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> active#(s(mark(X))) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(s(X)) -> active#(s(mark(X))) -> active#(incr(cons(X,XS))) -> s#(X) mark#(s(X)) -> active#(s(mark(X))) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(s(X)) -> active#(s(mark(X))) -> active#(odds()) -> mark#(incr(pairs())) mark#(s(X)) -> active#(s(mark(X))) -> active#(odds()) -> incr#(pairs()) mark#(s(X)) -> active#(s(mark(X))) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(s(X)) -> active#(s(mark(X))) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(s(X)) -> active#(s(mark(X))) -> active#(pairs()) -> incr#(odds()) mark#(s(X)) -> active#(s(mark(X))) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(s(X)) -> active#(s(mark(X))) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(s(X)) -> active#(s(mark(X))) -> active#(nats()) -> incr#(nats()) mark#(odds()) -> active#(odds()) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(odds()) -> active#(odds()) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(odds()) -> active#(odds()) -> active#(incr(cons(X,XS))) -> s#(X) mark#(odds()) -> active#(odds()) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(odds()) -> active#(odds()) -> active#(odds()) -> mark#(incr(pairs())) mark#(odds()) -> active#(odds()) -> active#(odds()) -> incr#(pairs()) mark#(odds()) -> active#(odds()) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(odds()) -> active#(odds()) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(odds()) -> active#(odds()) -> active#(pairs()) -> incr#(odds()) mark#(odds()) -> active#(odds()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(odds()) -> active#(odds()) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(odds()) -> active#(odds()) -> active#(nats()) -> incr#(nats()) mark#(pairs()) -> active#(pairs()) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(pairs()) -> active#(pairs()) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(pairs()) -> active#(pairs()) -> active#(incr(cons(X,XS))) -> s#(X) mark#(pairs()) -> active#(pairs()) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(pairs()) -> active#(pairs()) -> active#(odds()) -> mark#(incr(pairs())) mark#(pairs()) -> active#(pairs()) -> active#(odds()) -> incr#(pairs()) mark#(pairs()) -> active#(pairs()) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(pairs()) -> active#(pairs()) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(pairs()) -> active#(pairs()) -> active#(pairs()) -> incr#(odds()) mark#(pairs()) -> active#(pairs()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(pairs()) -> active#(pairs()) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(pairs()) -> active#(pairs()) -> active#(nats()) -> incr#(nats()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> active#(head(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> head#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> s#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(odds()) -> active#(odds()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pairs()) -> active#(pairs()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> incr#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats()) -> active#(nats()) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) -> cons#(active(X1),X2) -> cons#(X1,X2) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) -> cons#(X1,mark(X2)) -> cons#(X1,X2) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(incr(cons(X,XS))) -> s#(X) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(odds()) -> mark#(incr(pairs())) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(odds()) -> incr#(pairs()) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(pairs()) -> incr#(odds()) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(nats()) -> incr#(nats()) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> tail#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> head#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(incr(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> incr#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(incr(X)) -> incr#(mark(X)) -> incr#(active(X)) -> incr#(X) mark#(incr(X)) -> incr#(mark(X)) -> incr#(mark(X)) -> incr#(X) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> s#(X) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(odds()) -> incr#(pairs()) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(pairs()) -> incr#(odds()) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(nats()) -> incr#(nats()) mark#(0()) -> active#(0()) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(0()) -> active#(0()) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(0()) -> active#(0()) -> active#(incr(cons(X,XS))) -> s#(X) mark#(0()) -> active#(0()) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(0()) -> active#(0()) -> active#(odds()) -> mark#(incr(pairs())) mark#(0()) -> active#(0()) -> active#(odds()) -> incr#(pairs()) mark#(0()) -> active#(0()) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(0()) -> active#(0()) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(0()) -> active#(0()) -> active#(pairs()) -> incr#(odds()) mark#(0()) -> active#(0()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(0()) -> active#(0()) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(0()) -> active#(0()) -> active#(nats()) -> incr#(nats()) mark#(nats()) -> active#(nats()) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(nats()) -> active#(nats()) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(nats()) -> active#(nats()) -> active#(incr(cons(X,XS))) -> s#(X) mark#(nats()) -> active#(nats()) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(nats()) -> active#(nats()) -> active#(odds()) -> mark#(incr(pairs())) mark#(nats()) -> active#(nats()) -> active#(odds()) -> incr#(pairs()) mark#(nats()) -> active#(nats()) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(nats()) -> active#(nats()) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(nats()) -> active#(nats()) -> active#(pairs()) -> incr#(odds()) mark#(nats()) -> active#(nats()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(nats()) -> active#(nats()) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(nats()) -> active#(nats()) -> active#(nats()) -> incr#(nats()) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(active(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) -> cons#(active(X1),X2) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) -> cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) -> cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) -> cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) -> cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) -> cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) -> incr#(active(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) incr#(active(X)) -> incr#(X) -> incr#(active(X)) -> incr#(X) incr#(active(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(odds()) -> mark#(incr(pairs())) -> mark#(tail(X)) -> tail#(mark(X)) active#(odds()) -> mark#(incr(pairs())) -> mark#(tail(X)) -> mark#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(head(X)) -> active#(head(mark(X))) active#(odds()) -> mark#(incr(pairs())) -> mark#(head(X)) -> head#(mark(X)) active#(odds()) -> mark#(incr(pairs())) -> mark#(head(X)) -> mark#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(s(X)) -> active#(s(mark(X))) active#(odds()) -> mark#(incr(pairs())) -> mark#(s(X)) -> s#(mark(X)) active#(odds()) -> mark#(incr(pairs())) -> mark#(s(X)) -> mark#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) -> mark#(pairs()) -> active#(pairs()) active#(odds()) -> mark#(incr(pairs())) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(odds()) -> mark#(incr(pairs())) -> mark#(incr(X)) -> incr#(mark(X)) active#(odds()) -> mark#(incr(pairs())) -> mark#(incr(X)) -> mark#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(0()) -> active#(0()) active#(odds()) -> mark#(incr(pairs())) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(odds()) -> mark#(incr(pairs())) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) active#(odds()) -> mark#(incr(pairs())) -> mark#(cons(X1,X2)) -> mark#(X1) active#(odds()) -> mark#(incr(pairs())) -> mark#(nats()) -> active#(nats()) active#(odds()) -> incr#(pairs()) -> incr#(active(X)) -> incr#(X) active#(odds()) -> incr#(pairs()) -> incr#(mark(X)) -> incr#(X) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(tail(X)) -> tail#(mark(X)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(tail(X)) -> mark#(X) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(head(X)) -> active#(head(mark(X))) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(head(X)) -> head#(mark(X)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(head(X)) -> mark#(X) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(s(X)) -> active#(s(mark(X))) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(s(X)) -> s#(mark(X)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(s(X)) -> mark#(X) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(odds()) -> active#(odds()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(incr(X)) -> incr#(mark(X)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(incr(X)) -> mark#(X) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(0()) -> active#(0()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(nats()) -> active#(nats()) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(X1,active(X2)) -> cons#(X1,X2) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(active(X1),X2) -> cons#(X1,X2) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(X1,mark(X2)) -> cons#(X1,X2) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(pairs()) -> incr#(odds()) -> incr#(active(X)) -> incr#(X) active#(pairs()) -> incr#(odds()) -> incr#(mark(X)) -> incr#(X) active#(incr(cons(X,XS))) -> s#(X) -> s#(active(X)) -> s#(X) active#(incr(cons(X,XS))) -> s#(X) -> s#(mark(X)) -> s#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(tail(X)) -> tail#(mark(X)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(tail(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(head(X)) -> active#(head(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(head(X)) -> head#(mark(X)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(head(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(s(X)) -> active#(s(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(s(X)) -> s#(mark(X)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(s(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(odds()) -> active#(odds()) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(pairs()) -> active#(pairs()) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(incr(X)) -> incr#(mark(X)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(incr(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(0()) -> active#(0()) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(nats()) -> active#(nats()) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(X1,active(X2)) -> cons#(X1,X2) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(active(X1),X2) -> cons#(X1,X2) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(X1,mark(X2)) -> cons#(X1,X2) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(incr(cons(X,XS))) -> incr#(XS) -> incr#(active(X)) -> incr#(X) active#(incr(cons(X,XS))) -> incr#(XS) -> incr#(mark(X)) -> incr#(X) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(tail(X)) -> tail#(mark(X)) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(tail(X)) -> mark#(X) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(head(X)) -> active#(head(mark(X))) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(head(X)) -> head#(mark(X)) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(head(X)) -> mark#(X) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(s(X)) -> active#(s(mark(X))) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(s(X)) -> s#(mark(X)) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(s(X)) -> mark#(X) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(odds()) -> active#(odds()) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(pairs()) -> active#(pairs()) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(incr(X)) -> incr#(mark(X)) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(incr(X)) -> mark#(X) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(0()) -> active#(0()) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(nats()) -> active#(nats()) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(X1,active(X2)) -> cons#(X1,X2) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(active(X1),X2) -> cons#(X1,X2) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(X1,mark(X2)) -> cons#(X1,X2) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(nats()) -> incr#(nats()) -> incr#(active(X)) -> incr#(X) active#(nats()) -> incr#(nats()) -> incr#(mark(X)) -> incr#(X) SCC Processor: #sccs: 6 #rules: 30 #arcs: 345/1849 DPs: mark#(tail(X)) -> mark#(X) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(0()) -> active#(0()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(pairs()) -> active#(pairs()) mark#(odds()) -> active#(odds()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> active#(tail(mark(X))) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) EDG Processor: DPs: mark#(tail(X)) -> mark#(X) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(0()) -> active#(0()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(pairs()) -> active#(pairs()) mark#(odds()) -> active#(odds()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> active#(tail(mark(X))) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) graph: mark#(tail(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(tail(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(tail(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(head(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(head(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(head(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(head(X)) -> active#(head(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(s(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(s(X)) -> active#(s(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(odds()) -> active#(odds()) -> active#(odds()) -> mark#(incr(pairs())) mark#(pairs()) -> active#(pairs()) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats()) -> active#(nats()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pairs()) -> active#(pairs()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(odds()) -> active#(odds()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> active#(head(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(incr(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(incr(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(incr(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(nats()) -> active#(nats()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) active#(odds()) -> mark#(incr(pairs())) -> mark#(incr(X)) -> mark#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(incr(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(s(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(s(X)) -> active#(s(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(head(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(head(X)) -> active#(head(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(tail(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) SCC Processor: #sccs: 1 #rules: 17 #arcs: 94/324 DPs: mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Bounds Processor: bound: 0 enrichment: match-dp automaton: final states: {50} transitions: 00() -> 53* mark{#,0}(54) -> 50* mark{#,0}(53) -> 50* active{#,0}(81) -> 50* nats0() -> 51* active0(53) -> 80* mark0(53) -> 80* cons0(53,52) -> 81,54 cons0(80,52) -> 81* incr0(51) -> 52* problem: DPs: mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(nats()) -> active#(nats()) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) SCC Processor: #sccs: 1 #rules: 15 #arcs: 89/256 DPs: mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Bounds Processor: bound: 0 enrichment: match-dp automaton: final states: {37} transitions: 00() -> 40* mark{#,0}(40) -> 37* mark{#,0}(41) -> 37* active{#,0}(67) -> 37* active0(40) -> 66* mark0(40) -> 66* cons0(40,39) -> 67,41 cons0(66,39) -> 67* incr0(38) -> 39* odds0() -> 38* problem: DPs: mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) SCC Processor: #sccs: 1 #rules: 13 #arcs: 81/196 DPs: mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Semantic Labeling Processor: dimension: 1 usable rules: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) interpretation: [tail](x0) = x0 + 1, [head](x0) = 0, [s](x0) = 0, [odds] = 0, [pairs] = 0, [mark](x0) = x0, [cons](x0, x1) = 0, [incr](x0) = 0, [0] = 0, [active](x0) = 1, [nats] = 0 labeled: odds pairs nats 0 usable (for model): odds pairs nats 0 argument filtering: pi(nats) = [] pi(active) = 0 pi(0) = [] pi(incr) = [0] pi(cons) = [0] pi(mark) = 0 pi(pairs) = [] pi(odds) = [] pi(s) = 0 pi(head) = 0 pi(tail) = 0 pi(active#) = 0 pi(mark#) = 0 precedence: odds > pairs > nats > 0 > incr > mark# ~ active# ~ tail ~ head ~ s ~ mark ~ cons ~ active problem: DPs: mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) mark#(incr(X)) -> active#(incr(mark(X))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Restore Modifier: DPs: mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) mark#(incr(X)) -> active#(incr(mark(X))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) SCC Processor: #sccs: 1 #rules: 3 #arcs: 73/81 DPs: mark#(tail(X)) -> mark#(X) mark#(head(X)) -> mark#(X) mark#(s(X)) -> mark#(X) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Bounds Processor: bound: 0 enrichment: match-dp automaton: final states: {5} transitions: f1110() -> 6* mark{#,0}(6) -> 5* problem: DPs: mark#(tail(X)) -> mark#(X) mark#(head(X)) -> mark#(X) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Bounds Processor: bound: 0 enrichment: match-dp automaton: final states: {3} transitions: mark{#,0}(4) -> 3* f1130() -> 4* problem: DPs: mark#(tail(X)) -> mark#(X) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Bounds Processor: bound: 0 enrichment: match-dp automaton: final states: {1} transitions: mark{#,0}(2) -> 1* f1150() -> 2* problem: DPs: TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed DPs: incr#(mark(X)) -> incr#(X) incr#(active(X)) -> incr#(X) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(incr#) = 0 problem: DPs: TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed DPs: s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed DPs: head#(mark(X)) -> head#(X) head#(active(X)) -> head#(X) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(head#) = 0 problem: DPs: TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed DPs: tail#(mark(X)) -> tail#(X) tail#(active(X)) -> tail#(X) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(tail#) = 0 problem: DPs: TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed