MAYBE Problem: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(h(X)) -> g#(X,X) active#(g(a(),X)) -> f#(b(),X) active#(f(X,X)) -> h#(a()) active#(h(X)) -> active#(X) active#(h(X)) -> h#(active(X)) active#(g(X1,X2)) -> active#(X1) active#(g(X1,X2)) -> g#(active(X1),X2) active#(f(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> f#(active(X1),X2) h#(mark(X)) -> h#(X) g#(mark(X1),X2) -> g#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) proper#(h(X)) -> proper#(X) proper#(h(X)) -> h#(proper(X)) proper#(g(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) h#(ok(X)) -> h#(X) g#(ok(X1),ok(X2)) -> g#(X1,X2) f#(ok(X1),ok(X2)) -> f#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(h(X)) -> g#(X,X) active#(g(a(),X)) -> f#(b(),X) active#(f(X,X)) -> h#(a()) active#(h(X)) -> active#(X) active#(h(X)) -> h#(active(X)) active#(g(X1,X2)) -> active#(X1) active#(g(X1,X2)) -> g#(active(X1),X2) active#(f(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> f#(active(X1),X2) h#(mark(X)) -> h#(X) g#(mark(X1),X2) -> g#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) proper#(h(X)) -> proper#(X) proper#(h(X)) -> h#(proper(X)) proper#(g(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) h#(ok(X)) -> h#(X) g#(ok(X1),ok(X2)) -> g#(X1,X2) f#(ok(X1),ok(X2)) -> f#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> f#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(g(X1,X2)) -> g#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(g(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(h(X)) -> h#(active(X)) top#(ok(X)) -> active#(X) -> active#(h(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(f(X,X)) -> h#(a()) top#(ok(X)) -> active#(X) -> active#(g(a(),X)) -> f#(b(),X) top#(ok(X)) -> active#(X) -> active#(h(X)) -> g#(X,X) 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#(f(X1,X2)) -> f#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(g(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(g(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X2) -> proper#(h(X)) -> h#(proper(X)) proper#(f(X1,X2)) -> proper#(X2) -> proper#(h(X)) -> proper#(X) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) -> proper#(h(X)) -> h#(proper(X)) proper#(f(X1,X2)) -> proper#(X1) -> proper#(h(X)) -> proper#(X) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) -> f#(mark(X1),X2) -> f#(X1,X2) proper#(g(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(g(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(g(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X2) -> proper#(h(X)) -> h#(proper(X)) proper#(g(X1,X2)) -> proper#(X2) -> proper#(h(X)) -> proper#(X) proper#(g(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(g(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(g(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X1) -> proper#(h(X)) -> h#(proper(X)) proper#(g(X1,X2)) -> proper#(X1) -> proper#(h(X)) -> proper#(X) proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) -> g#(ok(X1),ok(X2)) -> g#(X1,X2) proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) -> g#(mark(X1),X2) -> g#(X1,X2) proper#(h(X)) -> proper#(X) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(h(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1) proper#(h(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2) proper#(h(X)) -> proper#(X) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(h(X)) -> proper#(X) -> proper#(g(X1,X2)) -> proper#(X1) proper#(h(X)) -> proper#(X) -> proper#(g(X1,X2)) -> proper#(X2) proper#(h(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(h(X)) -> h#(proper(X)) -> h#(ok(X)) -> h#(X) proper#(h(X)) -> h#(proper(X)) -> h#(mark(X)) -> h#(X) h#(ok(X)) -> h#(X) -> h#(ok(X)) -> h#(X) h#(ok(X)) -> h#(X) -> h#(mark(X)) -> h#(X) h#(mark(X)) -> h#(X) -> h#(ok(X)) -> h#(X) h#(mark(X)) -> h#(X) -> h#(mark(X)) -> h#(X) f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2) g#(ok(X1),ok(X2)) -> g#(X1,X2) -> g#(ok(X1),ok(X2)) -> g#(X1,X2) g#(ok(X1),ok(X2)) -> g#(X1,X2) -> g#(mark(X1),X2) -> g#(X1,X2) g#(mark(X1),X2) -> g#(X1,X2) -> g#(ok(X1),ok(X2)) -> g#(X1,X2) g#(mark(X1),X2) -> g#(X1,X2) -> g#(mark(X1),X2) -> g#(X1,X2) active#(f(X1,X2)) -> f#(active(X1),X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) active#(f(X1,X2)) -> f#(active(X1),X2) -> f#(mark(X1),X2) -> f#(X1,X2) active#(f(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> f#(active(X1),X2) active#(f(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> active#(X1) -> active#(g(X1,X2)) -> g#(active(X1),X2) active#(f(X1,X2)) -> active#(X1) -> active#(g(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> active#(X1) -> active#(h(X)) -> h#(active(X)) active#(f(X1,X2)) -> active#(X1) -> active#(h(X)) -> active#(X) active#(f(X1,X2)) -> active#(X1) -> active#(f(X,X)) -> h#(a()) active#(f(X1,X2)) -> active#(X1) -> active#(g(a(),X)) -> f#(b(),X) active#(f(X1,X2)) -> active#(X1) -> active#(h(X)) -> g#(X,X) active#(f(X,X)) -> h#(a()) -> h#(ok(X)) -> h#(X) active#(f(X,X)) -> h#(a()) -> h#(mark(X)) -> h#(X) active#(g(a(),X)) -> f#(b(),X) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) active#(g(a(),X)) -> f#(b(),X) -> f#(mark(X1),X2) -> f#(X1,X2) active#(g(X1,X2)) -> g#(active(X1),X2) -> g#(ok(X1),ok(X2)) -> g#(X1,X2) active#(g(X1,X2)) -> g#(active(X1),X2) -> g#(mark(X1),X2) -> g#(X1,X2) active#(g(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> f#(active(X1),X2) active#(g(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> active#(X1) active#(g(X1,X2)) -> active#(X1) -> active#(g(X1,X2)) -> g#(active(X1),X2) active#(g(X1,X2)) -> active#(X1) -> active#(g(X1,X2)) -> active#(X1) active#(g(X1,X2)) -> active#(X1) -> active#(h(X)) -> h#(active(X)) active#(g(X1,X2)) -> active#(X1) -> active#(h(X)) -> active#(X) active#(g(X1,X2)) -> active#(X1) -> active#(f(X,X)) -> h#(a()) active#(g(X1,X2)) -> active#(X1) -> active#(g(a(),X)) -> f#(b(),X) active#(g(X1,X2)) -> active#(X1) -> active#(h(X)) -> g#(X,X) active#(h(X)) -> h#(active(X)) -> h#(ok(X)) -> h#(X) active#(h(X)) -> h#(active(X)) -> h#(mark(X)) -> h#(X) active#(h(X)) -> g#(X,X) -> g#(ok(X1),ok(X2)) -> g#(X1,X2) active#(h(X)) -> g#(X,X) -> g#(mark(X1),X2) -> g#(X1,X2) active#(h(X)) -> active#(X) -> active#(f(X1,X2)) -> f#(active(X1),X2) active#(h(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1) active#(h(X)) -> active#(X) -> active#(g(X1,X2)) -> g#(active(X1),X2) active#(h(X)) -> active#(X) -> active#(g(X1,X2)) -> active#(X1) active#(h(X)) -> active#(X) -> active#(h(X)) -> h#(active(X)) active#(h(X)) -> active#(X) -> active#(h(X)) -> active#(X) active#(h(X)) -> active#(X) -> active#(f(X,X)) -> h#(a()) active#(h(X)) -> active#(X) -> active#(g(a(),X)) -> f#(b(),X) active#(h(X)) -> active#(X) -> active#(h(X)) -> g#(X,X) EDG Processor: DPs: active#(h(X)) -> g#(X,X) active#(g(a(),X)) -> f#(b(),X) active#(f(X,X)) -> h#(a()) active#(h(X)) -> active#(X) active#(h(X)) -> h#(active(X)) active#(g(X1,X2)) -> active#(X1) active#(g(X1,X2)) -> g#(active(X1),X2) active#(f(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> f#(active(X1),X2) h#(mark(X)) -> h#(X) g#(mark(X1),X2) -> g#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) proper#(h(X)) -> proper#(X) proper#(h(X)) -> h#(proper(X)) proper#(g(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) h#(ok(X)) -> h#(X) g#(ok(X1),ok(X2)) -> g#(X1,X2) f#(ok(X1),ok(X2)) -> f#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> active#(X) -> active#(h(X)) -> g#(X,X) top#(ok(X)) -> active#(X) -> active#(g(a(),X)) -> f#(b(),X) top#(ok(X)) -> active#(X) -> active#(f(X,X)) -> h#(a()) top#(ok(X)) -> active#(X) -> active#(h(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(h(X)) -> h#(active(X)) top#(ok(X)) -> active#(X) -> active#(g(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(g(X1,X2)) -> g#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> f#(active(X1),X2) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(g(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(g(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X2) -> proper#(h(X)) -> proper#(X) proper#(f(X1,X2)) -> proper#(X2) -> proper#(h(X)) -> h#(proper(X)) proper#(f(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X1) -> proper#(h(X)) -> proper#(X) proper#(f(X1,X2)) -> proper#(X1) -> proper#(h(X)) -> h#(proper(X)) proper#(f(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X2) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) -> f#(mark(X1),X2) -> f#(X1,X2) proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) proper#(g(X1,X2)) -> proper#(X2) -> proper#(h(X)) -> proper#(X) proper#(g(X1,X2)) -> proper#(X2) -> proper#(h(X)) -> h#(proper(X)) proper#(g(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> proper#(X2) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(g(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> proper#(X2) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(g(X1,X2)) -> proper#(X1) -> proper#(h(X)) -> proper#(X) proper#(g(X1,X2)) -> proper#(X1) -> proper#(h(X)) -> h#(proper(X)) proper#(g(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> proper#(X1) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(g(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> proper#(X1) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) -> g#(mark(X1),X2) -> g#(X1,X2) proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) -> g#(ok(X1),ok(X2)) -> g#(X1,X2) proper#(h(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(g(X1,X2)) -> proper#(X2) proper#(h(X)) -> proper#(X) -> proper#(g(X1,X2)) -> proper#(X1) proper#(h(X)) -> proper#(X) -> proper#(g(X1,X2)) -> g#(proper(X1),proper(X2)) proper#(h(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2) proper#(h(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1) proper#(h(X)) -> proper#(X) -> proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) proper#(h(X)) -> h#(proper(X)) -> h#(mark(X)) -> h#(X) proper#(h(X)) -> h#(proper(X)) -> h#(ok(X)) -> h#(X) h#(ok(X)) -> h#(X) -> h#(mark(X)) -> h#(X) h#(ok(X)) -> h#(X) -> h#(ok(X)) -> h#(X) h#(mark(X)) -> h#(X) -> h#(mark(X)) -> h#(X) h#(mark(X)) -> h#(X) -> h#(ok(X)) -> h#(X) f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2) f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) g#(ok(X1),ok(X2)) -> g#(X1,X2) -> g#(mark(X1),X2) -> g#(X1,X2) g#(ok(X1),ok(X2)) -> g#(X1,X2) -> g#(ok(X1),ok(X2)) -> g#(X1,X2) g#(mark(X1),X2) -> g#(X1,X2) -> g#(mark(X1),X2) -> g#(X1,X2) g#(mark(X1),X2) -> g#(X1,X2) -> g#(ok(X1),ok(X2)) -> g#(X1,X2) active#(f(X1,X2)) -> f#(active(X1),X2) -> f#(mark(X1),X2) -> f#(X1,X2) active#(f(X1,X2)) -> f#(active(X1),X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2) active#(f(X1,X2)) -> active#(X1) -> active#(h(X)) -> g#(X,X) active#(f(X1,X2)) -> active#(X1) -> active#(g(a(),X)) -> f#(b(),X) active#(f(X1,X2)) -> active#(X1) -> active#(f(X,X)) -> h#(a()) active#(f(X1,X2)) -> active#(X1) -> active#(h(X)) -> active#(X) active#(f(X1,X2)) -> active#(X1) -> active#(h(X)) -> h#(active(X)) active#(f(X1,X2)) -> active#(X1) -> active#(g(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> active#(X1) -> active#(g(X1,X2)) -> g#(active(X1),X2) active#(f(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> active#(X1) active#(f(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> f#(active(X1),X2) active#(g(X1,X2)) -> g#(active(X1),X2) -> g#(mark(X1),X2) -> g#(X1,X2) active#(g(X1,X2)) -> g#(active(X1),X2) -> g#(ok(X1),ok(X2)) -> g#(X1,X2) active#(g(X1,X2)) -> active#(X1) -> active#(h(X)) -> g#(X,X) active#(g(X1,X2)) -> active#(X1) -> active#(g(a(),X)) -> f#(b(),X) active#(g(X1,X2)) -> active#(X1) -> active#(f(X,X)) -> h#(a()) active#(g(X1,X2)) -> active#(X1) -> active#(h(X)) -> active#(X) active#(g(X1,X2)) -> active#(X1) -> active#(h(X)) -> h#(active(X)) active#(g(X1,X2)) -> active#(X1) -> active#(g(X1,X2)) -> active#(X1) active#(g(X1,X2)) -> active#(X1) -> active#(g(X1,X2)) -> g#(active(X1),X2) active#(g(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> active#(X1) active#(g(X1,X2)) -> active#(X1) -> active#(f(X1,X2)) -> f#(active(X1),X2) active#(h(X)) -> h#(active(X)) -> h#(mark(X)) -> h#(X) active#(h(X)) -> h#(active(X)) -> h#(ok(X)) -> h#(X) active#(h(X)) -> active#(X) -> active#(h(X)) -> g#(X,X) active#(h(X)) -> active#(X) -> active#(g(a(),X)) -> f#(b(),X) active#(h(X)) -> active#(X) -> active#(f(X,X)) -> h#(a()) active#(h(X)) -> active#(X) -> active#(h(X)) -> active#(X) active#(h(X)) -> active#(X) -> active#(h(X)) -> h#(active(X)) active#(h(X)) -> active#(X) -> active#(g(X1,X2)) -> active#(X1) active#(h(X)) -> active#(X) -> active#(g(X1,X2)) -> g#(active(X1),X2) active#(h(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1) active#(h(X)) -> active#(X) -> active#(f(X1,X2)) -> f#(active(X1),X2) SCC Processor: #sccs: 6 #rules: 16 #arcs: 116/729 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> proper#(X2) proper#(h(X)) -> proper#(X) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [f](x0, x1) = x0 + x1, [b] = 0, [a] = 0, [mark](x0) = 1, [g](x0, x1) = x0 + x1, [active](x0) = x0 + 1, [h](x0) = x0 + 1 orientation: proper#(f(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(f(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(g(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(g(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(h(X)) = X + 1 >= X = proper#(X) active(h(X)) = X + 2 >= 1 = mark(g(X,X)) active(g(a(),X)) = X + 1 >= 1 = mark(f(b(),X)) active(f(X,X)) = 2X + 1 >= 1 = mark(h(a())) active(a()) = 1 >= 1 = mark(b()) active(h(X)) = X + 2 >= X + 2 = h(active(X)) active(g(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = g(active(X1),X2) active(f(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = f(active(X1),X2) h(mark(X)) = 2 >= 1 = mark(h(X)) g(mark(X1),X2) = X2 + 1 >= 1 = mark(g(X1,X2)) f(mark(X1),X2) = X2 + 1 >= 1 = mark(f(X1,X2)) proper(h(X)) = X + 1 >= X + 1 = h(proper(X)) proper(g(X1,X2)) = X1 + X2 >= X1 + X2 = g(proper(X1),proper(X2)) proper(a()) = 0 >= 0 = ok(a()) proper(f(X1,X2)) = X1 + X2 >= X1 + X2 = f(proper(X1),proper(X2)) proper(b()) = 0 >= 0 = ok(b()) h(ok(X)) = X + 1 >= X + 1 = ok(h(X)) g(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(g(X1,X2)) f(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(f(X1,X2)) -> proper#(X1) proper#(f(X1,X2)) -> proper#(X2) proper#(g(X1,X2)) -> proper#(X1) proper#(g(X1,X2)) -> proper#(X2) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [f](x0, x1) = x0 + x1 + 1, [b] = 1, [a] = 0, [mark](x0) = 0, [g](x0, x1) = x0 + x1 + 1, [active](x0) = x0, [h](x0) = 0 orientation: proper#(f(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(f(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(g(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(g(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(h(X)) = 0 >= 0 = mark(g(X,X)) active(g(a(),X)) = X + 1 >= 0 = mark(f(b(),X)) active(f(X,X)) = 2X + 1 >= 0 = mark(h(a())) active(a()) = 0 >= 0 = mark(b()) active(h(X)) = 0 >= 0 = h(active(X)) active(g(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = g(active(X1),X2) active(f(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = f(active(X1),X2) h(mark(X)) = 0 >= 0 = mark(h(X)) g(mark(X1),X2) = X2 + 1 >= 0 = mark(g(X1,X2)) f(mark(X1),X2) = X2 + 1 >= 0 = mark(f(X1,X2)) proper(h(X)) = 0 >= 0 = h(proper(X)) proper(g(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = g(proper(X1),proper(X2)) proper(a()) = 0 >= 0 = ok(a()) proper(f(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = f(proper(X1),proper(X2)) proper(b()) = 1 >= 1 = ok(b()) h(ok(X)) = 0 >= 0 = ok(h(X)) g(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(g(X1,X2)) f(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(f(X1,X2)) -> active#(X1) active#(g(X1,X2)) -> active#(X1) active#(h(X)) -> active#(X) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = x0, [f](x0, x1) = x0 + 1, [b] = 1, [a] = 1, [mark](x0) = 1, [g](x0, x1) = x0 + x1, [active](x0) = x0 + 1, [h](x0) = x0 orientation: active#(f(X1,X2)) = X1 + 1 >= X1 = active#(X1) active#(g(X1,X2)) = X1 + X2 >= X1 = active#(X1) active#(h(X)) = X >= X = active#(X) active(h(X)) = X + 1 >= 1 = mark(g(X,X)) active(g(a(),X)) = X + 2 >= 1 = mark(f(b(),X)) active(f(X,X)) = X + 2 >= 1 = mark(h(a())) active(a()) = 2 >= 1 = mark(b()) active(h(X)) = X + 1 >= X + 1 = h(active(X)) active(g(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = g(active(X1),X2) active(f(X1,X2)) = X1 + 2 >= X1 + 2 = f(active(X1),X2) h(mark(X)) = 1 >= 1 = mark(h(X)) g(mark(X1),X2) = X2 + 1 >= 1 = mark(g(X1,X2)) f(mark(X1),X2) = 2 >= 1 = mark(f(X1,X2)) proper(h(X)) = X >= X = h(proper(X)) proper(g(X1,X2)) = X1 + X2 >= X1 + X2 = g(proper(X1),proper(X2)) proper(a()) = 1 >= 0 = ok(a()) proper(f(X1,X2)) = X1 + 1 >= X1 + 1 = f(proper(X1),proper(X2)) proper(b()) = 1 >= 0 = ok(b()) h(ok(X)) = 0 >= 0 = ok(h(X)) g(ok(X1),ok(X2)) = 0 >= 0 = ok(g(X1,X2)) f(ok(X1),ok(X2)) = 1 >= 0 = ok(f(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: active#(g(X1,X2)) -> active#(X1) active#(h(X)) -> active#(X) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [f](x0, x1) = x0 + 1, [b] = 0, [a] = 0, [mark](x0) = x0 + 1, [g](x0, x1) = x0 + 1, [active](x0) = x0 + 1, [h](x0) = x0 + 1 orientation: active#(g(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(h(X)) = X + 2 >= X + 1 = active#(X) active(h(X)) = X + 2 >= X + 2 = mark(g(X,X)) active(g(a(),X)) = 2 >= 2 = mark(f(b(),X)) active(f(X,X)) = X + 2 >= 2 = mark(h(a())) active(a()) = 1 >= 1 = mark(b()) active(h(X)) = X + 2 >= X + 2 = h(active(X)) active(g(X1,X2)) = X1 + 2 >= X1 + 2 = g(active(X1),X2) active(f(X1,X2)) = X1 + 2 >= X1 + 2 = f(active(X1),X2) h(mark(X)) = X + 2 >= X + 2 = mark(h(X)) g(mark(X1),X2) = X1 + 2 >= X1 + 2 = mark(g(X1,X2)) f(mark(X1),X2) = X1 + 2 >= X1 + 2 = mark(f(X1,X2)) proper(h(X)) = X + 1 >= X + 1 = h(proper(X)) proper(g(X1,X2)) = X1 + 1 >= X1 + 1 = g(proper(X1),proper(X2)) proper(a()) = 0 >= 0 = ok(a()) proper(f(X1,X2)) = X1 + 1 >= X1 + 1 = f(proper(X1),proper(X2)) proper(b()) = 0 >= 0 = ok(b()) h(ok(X)) = X + 1 >= X + 1 = ok(h(X)) g(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(g(X1,X2)) f(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: h#(ok(X)) -> h#(X) h#(mark(X)) -> h#(X) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [h#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [f](x0, x1) = x0, [b] = 0, [a] = 0, [mark](x0) = x0 + 1, [g](x0, x1) = x0, [active](x0) = x0 + 1, [h](x0) = x0 orientation: h#(ok(X)) = X + 2 >= X + 1 = h#(X) h#(mark(X)) = X + 2 >= X + 1 = h#(X) active(h(X)) = X + 1 >= X + 1 = mark(g(X,X)) active(g(a(),X)) = 1 >= 1 = mark(f(b(),X)) active(f(X,X)) = X + 1 >= 1 = mark(h(a())) active(a()) = 1 >= 1 = mark(b()) active(h(X)) = X + 1 >= X + 1 = h(active(X)) active(g(X1,X2)) = X1 + 1 >= X1 + 1 = g(active(X1),X2) active(f(X1,X2)) = X1 + 1 >= X1 + 1 = f(active(X1),X2) h(mark(X)) = X + 1 >= X + 1 = mark(h(X)) g(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(g(X1,X2)) f(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(f(X1,X2)) proper(h(X)) = 1 >= 1 = h(proper(X)) proper(g(X1,X2)) = 1 >= 1 = g(proper(X1),proper(X2)) proper(a()) = 1 >= 1 = ok(a()) proper(f(X1,X2)) = 1 >= 1 = f(proper(X1),proper(X2)) proper(b()) = 1 >= 1 = ok(b()) h(ok(X)) = X + 1 >= X + 1 = ok(h(X)) g(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(g(X1,X2)) f(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: g#(ok(X1),ok(X2)) -> g#(X1,X2) g#(mark(X1),X2) -> g#(X1,X2) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [g#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [f](x0, x1) = x1, [b] = 1, [a] = 1, [mark](x0) = 0, [g](x0, x1) = x0, [active](x0) = x0 + 1, [h](x0) = x0 orientation: g#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = g#(X1,X2) g#(mark(X1),X2) = X2 + 1 >= X2 + 1 = g#(X1,X2) active(h(X)) = X + 1 >= 0 = mark(g(X,X)) active(g(a(),X)) = 2 >= 0 = mark(f(b(),X)) active(f(X,X)) = X + 1 >= 0 = mark(h(a())) active(a()) = 2 >= 0 = mark(b()) active(h(X)) = X + 1 >= X + 1 = h(active(X)) active(g(X1,X2)) = X1 + 1 >= X1 + 1 = g(active(X1),X2) active(f(X1,X2)) = X2 + 1 >= X2 = f(active(X1),X2) h(mark(X)) = 0 >= 0 = mark(h(X)) g(mark(X1),X2) = 0 >= 0 = mark(g(X1,X2)) f(mark(X1),X2) = X2 >= 0 = mark(f(X1,X2)) proper(h(X)) = X + 1 >= X + 1 = h(proper(X)) proper(g(X1,X2)) = X1 + 1 >= X1 + 1 = g(proper(X1),proper(X2)) proper(a()) = 2 >= 2 = ok(a()) proper(f(X1,X2)) = X2 + 1 >= X2 + 1 = f(proper(X1),proper(X2)) proper(b()) = 2 >= 2 = ok(b()) h(ok(X)) = X + 1 >= X + 1 = ok(h(X)) g(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(g(X1,X2)) f(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: g#(mark(X1),X2) -> g#(X1,X2) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [g#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [f](x0, x1) = x0, [b] = 0, [a] = 0, [mark](x0) = x0 + 1, [g](x0, x1) = x0, [active](x0) = x0 + 1, [h](x0) = x0 orientation: g#(mark(X1),X2) = X1 + 2 >= X1 + 1 = g#(X1,X2) active(h(X)) = X + 1 >= X + 1 = mark(g(X,X)) active(g(a(),X)) = 1 >= 1 = mark(f(b(),X)) active(f(X,X)) = X + 1 >= 1 = mark(h(a())) active(a()) = 1 >= 1 = mark(b()) active(h(X)) = X + 1 >= X + 1 = h(active(X)) active(g(X1,X2)) = X1 + 1 >= X1 + 1 = g(active(X1),X2) active(f(X1,X2)) = X1 + 1 >= X1 + 1 = f(active(X1),X2) h(mark(X)) = X + 1 >= X + 1 = mark(h(X)) g(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(g(X1,X2)) f(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(f(X1,X2)) proper(h(X)) = 0 >= 0 = h(proper(X)) proper(g(X1,X2)) = 0 >= 0 = g(proper(X1),proper(X2)) proper(a()) = 0 >= 0 = ok(a()) proper(f(X1,X2)) = 0 >= 0 = f(proper(X1),proper(X2)) proper(b()) = 0 >= 0 = ok(b()) h(ok(X)) = 0 >= 0 = ok(h(X)) g(ok(X1),ok(X2)) = 0 >= 0 = ok(g(X1,X2)) f(ok(X1),ok(X2)) = 0 >= 0 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: f#(ok(X1),ok(X2)) -> f#(X1,X2) f#(mark(X1),X2) -> f#(X1,X2) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [f#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [f](x0, x1) = x1, [b] = 1, [a] = 1, [mark](x0) = 0, [g](x0, x1) = x0, [active](x0) = x0 + 1, [h](x0) = x0 orientation: f#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = f#(X1,X2) f#(mark(X1),X2) = X2 + 1 >= X2 + 1 = f#(X1,X2) active(h(X)) = X + 1 >= 0 = mark(g(X,X)) active(g(a(),X)) = 2 >= 0 = mark(f(b(),X)) active(f(X,X)) = X + 1 >= 0 = mark(h(a())) active(a()) = 2 >= 0 = mark(b()) active(h(X)) = X + 1 >= X + 1 = h(active(X)) active(g(X1,X2)) = X1 + 1 >= X1 + 1 = g(active(X1),X2) active(f(X1,X2)) = X2 + 1 >= X2 = f(active(X1),X2) h(mark(X)) = 0 >= 0 = mark(h(X)) g(mark(X1),X2) = 0 >= 0 = mark(g(X1,X2)) f(mark(X1),X2) = X2 >= 0 = mark(f(X1,X2)) proper(h(X)) = X + 1 >= X + 1 = h(proper(X)) proper(g(X1,X2)) = X1 + 1 >= X1 + 1 = g(proper(X1),proper(X2)) proper(a()) = 2 >= 2 = ok(a()) proper(f(X1,X2)) = X2 + 1 >= X2 + 1 = f(proper(X1),proper(X2)) proper(b()) = 2 >= 2 = ok(b()) h(ok(X)) = X + 1 >= X + 1 = ok(h(X)) g(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(g(X1,X2)) f(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: f#(mark(X1),X2) -> f#(X1,X2) TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [f#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [f](x0, x1) = x0, [b] = 0, [a] = 0, [mark](x0) = x0 + 1, [g](x0, x1) = x0, [active](x0) = x0 + 1, [h](x0) = x0 orientation: f#(mark(X1),X2) = X1 + 2 >= X1 + 1 = f#(X1,X2) active(h(X)) = X + 1 >= X + 1 = mark(g(X,X)) active(g(a(),X)) = 1 >= 1 = mark(f(b(),X)) active(f(X,X)) = X + 1 >= 1 = mark(h(a())) active(a()) = 1 >= 1 = mark(b()) active(h(X)) = X + 1 >= X + 1 = h(active(X)) active(g(X1,X2)) = X1 + 1 >= X1 + 1 = g(active(X1),X2) active(f(X1,X2)) = X1 + 1 >= X1 + 1 = f(active(X1),X2) h(mark(X)) = X + 1 >= X + 1 = mark(h(X)) g(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(g(X1,X2)) f(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(f(X1,X2)) proper(h(X)) = 0 >= 0 = h(proper(X)) proper(g(X1,X2)) = 0 >= 0 = g(proper(X1),proper(X2)) proper(a()) = 0 >= 0 = ok(a()) proper(f(X1,X2)) = 0 >= 0 = f(proper(X1),proper(X2)) proper(b()) = 0 >= 0 = ok(b()) h(ok(X)) = 0 >= 0 = ok(h(X)) g(ok(X1),ok(X2)) = 0 >= 0 = ok(g(X1,X2)) f(ok(X1),ok(X2)) = 0 >= 0 = ok(f(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(h(X)) -> mark(g(X,X)) active(g(a(),X)) -> mark(f(b(),X)) active(f(X,X)) -> mark(h(a())) active(a()) -> mark(b()) active(h(X)) -> h(active(X)) active(g(X1,X2)) -> g(active(X1),X2) active(f(X1,X2)) -> f(active(X1),X2) h(mark(X)) -> mark(h(X)) g(mark(X1),X2) -> mark(g(X1,X2)) f(mark(X1),X2) -> mark(f(X1,X2)) proper(h(X)) -> h(proper(X)) proper(g(X1,X2)) -> g(proper(X1),proper(X2)) proper(a()) -> ok(a()) proper(f(X1,X2)) -> f(proper(X1),proper(X2)) proper(b()) -> ok(b()) h(ok(X)) -> ok(h(X)) g(ok(X1),ok(X2)) -> ok(g(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed