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) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [top](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [proper](x0) = [0 0 1]x0 [1 0 0] , [1 0 0] [1] [tail](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [head](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [0] [odds] = [0] [1], [0] [pairs] = [0] [1], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [incr](x0) = [0 0 1]x0 [1 0 0] , [0] [0] = [0] [0], [1 0 0] [active](x0) = [0 0 1]x0 [1 0 0] , [0] [nats] = [0] [0] orientation: [0] [0] active(nats()) = [0] >= [0] = mark(cons(0(),incr(nats()))) [0] [0] [0] [0] active(pairs()) = [1] >= [0] = mark(cons(0(),incr(odds()))) [0] [0] [0] [0] active(odds()) = [1] >= [0] = mark(incr(pairs())) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(incr(cons(X,XS))) = [1 0 0]X + [1 0 0]XS >= [0 0 0]X + [0 0 0]XS = mark(cons(s(X),incr(XS))) [1 0 0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] active(head(cons(X,XS))) = [0 0 0]X + [0 0 0]XS >= [0 0 0]X = mark(X) [1 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] active(tail(cons(X,XS))) = [0 0 0]X + [0 0 0]XS + [0] >= [0 0 0]XS = mark(XS) [1 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [1 0 0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(incr(X)) = [1 0 0]X >= [1 0 0]X = incr(active(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] active(head(X)) = [0 0 0]X >= [0 0 0]X = head(active(X)) [1 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] active(tail(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(active(X)) [1 0 0] [1] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(incr(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(s(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] head(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(head(X)) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] tail(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(tail(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] proper(nats()) = [0] >= [0] = ok(nats()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [1 0 0] [1 0 0] [0 0 0] [0 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] proper(incr(X)) = [1 0 0]X >= [1 0 0]X = incr(proper(X)) [1 0 0] [1 0 0] [0] [0] proper(pairs()) = [1] >= [0] = ok(pairs()) [0] [0] [0] [0] proper(odds()) = [1] >= [0] = ok(odds()) [0] [0] [1 0 0] [1 0 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] proper(head(X)) = [0 0 0]X >= [0 0 0]X = head(proper(X)) [1 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] proper(tail(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(proper(X)) [1 0 0] [1] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] incr(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(incr(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] head(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(head(X)) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] tail(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(tail(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X)) [0 0 0] [1] [0 0 0] [1] 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(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) DP Processor: DPs: active#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) active#(pairs()) -> incr#(odds()) active#(pairs()) -> cons#(0(),incr(odds())) active#(odds()) -> 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#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(incr(X)) -> active#(X) active#(incr(X)) -> incr#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) s#(mark(X)) -> s#(X) head#(mark(X)) -> head#(X) tail#(mark(X)) -> tail#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(incr(X)) -> proper#(X) proper#(incr(X)) -> incr#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) incr#(ok(X)) -> incr#(X) s#(ok(X)) -> s#(X) head#(ok(X)) -> head#(X) tail#(ok(X)) -> tail#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) active#(pairs()) -> incr#(odds()) active#(pairs()) -> cons#(0(),incr(odds())) active#(odds()) -> 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#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(incr(X)) -> active#(X) active#(incr(X)) -> incr#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) s#(mark(X)) -> s#(X) head#(mark(X)) -> head#(X) tail#(mark(X)) -> tail#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(incr(X)) -> proper#(X) proper#(incr(X)) -> incr#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) incr#(ok(X)) -> incr#(X) s#(ok(X)) -> s#(X) head#(ok(X)) -> head#(X) tail#(ok(X)) -> tail#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) 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#(tail(X)) -> tail#(active(X)) top#(ok(X)) -> active#(X) -> active#(tail(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) top#(ok(X)) -> active#(X) -> active#(head(X)) -> active#(X) 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#(incr(X)) -> incr#(active(X)) top#(ok(X)) -> active#(X) -> active#(incr(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) top#(ok(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) top#(ok(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) top#(ok(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) top#(ok(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) top#(ok(X)) -> active#(X) -> active#(nats()) -> incr#(nats()) 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#(tail(X)) -> tail#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) 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#(incr(X)) -> incr#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(tail(X)) -> tail#(proper(X)) -> tail#(ok(X)) -> tail#(X) proper#(tail(X)) -> tail#(proper(X)) -> tail#(mark(X)) -> tail#(X) proper#(head(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(head(X)) -> head#(proper(X)) -> head#(ok(X)) -> head#(X) proper#(head(X)) -> head#(proper(X)) -> head#(mark(X)) -> head#(X) proper#(s(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) 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#(incr(X)) -> incr#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tail(X)) -> tail#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(head(X)) -> head#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(head(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(incr(X)) -> incr#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tail(X)) -> tail#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(head(X)) -> head#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(head(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(incr(X)) -> incr#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(mark(X1),X2) -> cons#(X1,X2) proper#(incr(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(incr(X)) -> incr#(proper(X)) -> incr#(ok(X)) -> incr#(X) proper#(incr(X)) -> incr#(proper(X)) -> incr#(mark(X)) -> incr#(X) tail#(ok(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X) tail#(ok(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) head#(ok(X)) -> head#(X) -> head#(ok(X)) -> head#(X) head#(ok(X)) -> head#(X) -> head#(mark(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(ok(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(mark(X)) -> head#(X) 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) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) incr#(ok(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X) incr#(ok(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) active#(tail(X)) -> tail#(active(X)) -> tail#(ok(X)) -> tail#(X) active#(tail(X)) -> tail#(active(X)) -> tail#(mark(X)) -> tail#(X) active#(tail(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(tail(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(tail(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(tail(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(tail(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(tail(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(tail(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(tail(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(tail(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(tail(X)) -> active#(X) -> active#(nats()) -> incr#(nats()) active#(head(X)) -> head#(active(X)) -> head#(ok(X)) -> head#(X) active#(head(X)) -> head#(active(X)) -> head#(mark(X)) -> head#(X) active#(head(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(head(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(head(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(head(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(head(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(head(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(head(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(head(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(head(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(head(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(head(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(head(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(head(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(head(X)) -> active#(X) -> active#(nats()) -> incr#(nats()) 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#(tail(X)) -> tail#(active(X)) active#(s(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(s(X)) -> active#(X) -> active#(head(X)) -> active#(X) 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#(incr(X)) -> incr#(active(X)) active#(s(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(s(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(s(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(s(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(s(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(s(X)) -> active#(X) -> active#(nats()) -> incr#(nats()) active#(odds()) -> incr#(pairs()) -> incr#(ok(X)) -> incr#(X) active#(odds()) -> incr#(pairs()) -> incr#(mark(X)) -> incr#(X) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(pairs()) -> incr#(odds()) -> incr#(ok(X)) -> incr#(X) active#(pairs()) -> incr#(odds()) -> incr#(mark(X)) -> incr#(X) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(cons(X1,X2)) -> active#(X1) -> active#(tail(X)) -> tail#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(tail(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(head(X)) -> head#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(head(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(X)) -> incr#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,XS))) -> s#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(cons(X1,X2)) -> active#(X1) -> active#(odds()) -> incr#(pairs()) active#(cons(X1,X2)) -> active#(X1) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(cons(X1,X2)) -> active#(X1) -> active#(pairs()) -> incr#(odds()) active#(cons(X1,X2)) -> active#(X1) -> active#(nats()) -> cons#(0(),incr(nats())) active#(cons(X1,X2)) -> active#(X1) -> active#(nats()) -> incr#(nats()) active#(incr(cons(X,XS))) -> s#(X) -> s#(ok(X)) -> s#(X) active#(incr(cons(X,XS))) -> s#(X) -> s#(mark(X)) -> s#(X) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(ok(X1),ok(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#(ok(X)) -> incr#(X) active#(incr(cons(X,XS))) -> incr#(XS) -> incr#(mark(X)) -> incr#(X) active#(incr(X)) -> incr#(active(X)) -> incr#(ok(X)) -> incr#(X) active#(incr(X)) -> incr#(active(X)) -> incr#(mark(X)) -> incr#(X) active#(incr(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(incr(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(incr(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(incr(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(incr(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(incr(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(incr(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(incr(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(incr(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(incr(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(incr(X)) -> active#(X) -> active#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(nats()) -> incr#(nats()) -> incr#(ok(X)) -> incr#(X) active#(nats()) -> incr#(nats()) -> incr#(mark(X)) -> incr#(X) SCC Processor: #sccs: 8 #rules: 23 #arcs: 249/1849 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) LPO Processor: 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(proper) = 0 pi(ok) = 0 pi(top) = 0 pi(top#) = 0 precedence: odds > incr > head ~ pairs ~ nats > top# ~ top ~ ok ~ proper ~ tail ~ s ~ mark ~ cons ~ 0 ~ active problem: DPs: top#(ok(X)) -> top#(active(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) KBO Processor: argument filtering: pi(nats) = [] pi(active) = 0 pi(0) = [] pi(incr) = [0] pi(cons) = 0 pi(mark) = [] pi(pairs) = [] pi(odds) = [] pi(s) = [0] pi(head) = [0] pi(tail) = 0 pi(proper) = [0] pi(ok) = [0] pi(top) = [] pi(top#) = 0 weight function: w0 = 1 w(top#) = w(top) = w(ok) = w(proper) = w(tail) = w(head) = w( s) = w(odds) = w(pairs) = w(mark) = w(incr) = w(0) = w(nats) = 1 w(cons) = w(active) = 0 precedence: top# ~ top ~ proper ~ odds ~ pairs ~ 0 ~ nats > head ~ s ~ incr > ok ~ tail ~ mark ~ cons ~ active 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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(cons(X1,X2)) -> active#(X1) active#(incr(X)) -> active#(X) active#(s(X)) -> active#(X) active#(head(X)) -> active#(X) active#(tail(X)) -> active#(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(active#) = 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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(incr(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(head(X)) -> proper#(X) proper#(tail(X)) -> proper#(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: tail#(mark(X)) -> tail#(X) tail#(ok(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: head#(mark(X)) -> head#(X) head#(ok(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) 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(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(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: incr#(mark(X)) -> incr#(X) incr#(ok(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: cons#(mark(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(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))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed