MAYBE Problem: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(f(X)) -> g#(X) active#(f(X)) -> f#(g(X)) active#(f(X)) -> cons#(X,f(g(X))) active#(g(0())) -> s#(0()) active#(g(s(X))) -> g#(X) active#(g(s(X))) -> s#(g(X)) active#(g(s(X))) -> s#(s(g(X))) active#(sel(s(X),cons(Y,Z))) -> sel#(X,Z) active#(f(X)) -> active#(X) active#(f(X)) -> f#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(g(X)) -> active#(X) active#(g(X)) -> g#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) f#(mark(X)) -> f#(X) cons#(mark(X1),X2) -> cons#(X1,X2) g#(mark(X)) -> g#(X) s#(mark(X)) -> s#(X) sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(g(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) f#(ok(X)) -> f#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) g#(ok(X)) -> g#(X) s#(ok(X)) -> s#(X) sel#(ok(X1),ok(X2)) -> sel#(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(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) EDG Processor: DPs: active#(f(X)) -> g#(X) active#(f(X)) -> f#(g(X)) active#(f(X)) -> cons#(X,f(g(X))) active#(g(0())) -> s#(0()) active#(g(s(X))) -> g#(X) active#(g(s(X))) -> s#(g(X)) active#(g(s(X))) -> s#(s(g(X))) active#(sel(s(X),cons(Y,Z))) -> sel#(X,Z) active#(f(X)) -> active#(X) active#(f(X)) -> f#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(g(X)) -> active#(X) active#(g(X)) -> g#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) f#(mark(X)) -> f#(X) cons#(mark(X1),X2) -> cons#(X1,X2) g#(mark(X)) -> g#(X) s#(mark(X)) -> s#(X) sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(g(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) f#(ok(X)) -> f#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) g#(ok(X)) -> g#(X) s#(ok(X)) -> s#(X) sel#(ok(X1),ok(X2)) -> sel#(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(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(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#(f(X)) -> g#(X) top#(ok(X)) -> active#(X) -> active#(f(X)) -> f#(g(X)) top#(ok(X)) -> active#(X) -> active#(f(X)) -> cons#(X,f(g(X))) top#(ok(X)) -> active#(X) -> active#(g(0())) -> s#(0()) top#(ok(X)) -> active#(X) -> active#(g(s(X))) -> g#(X) top#(ok(X)) -> active#(X) -> active#(g(s(X))) -> s#(g(X)) top#(ok(X)) -> active#(X) -> active#(g(s(X))) -> s#(s(g(X))) top#(ok(X)) -> active#(X) -> active#(sel(s(X),cons(Y,Z))) -> sel#(X,Z) top#(ok(X)) -> active#(X) -> active#(f(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(g(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(g(X)) -> g#(active(X)) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(f(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(f(X)) -> f#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(g(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(g(X)) -> g#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(f(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(f(X)) -> f#(proper(X)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(g(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(g(X)) -> g#(proper(X)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) -> sel#(mark(X1),X2) -> sel#(X1,X2) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) -> sel#(X1,mark(X2)) -> sel#(X1,X2) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) proper#(s(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(f(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(f(X)) -> f#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(g(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(g(X)) -> g#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(f(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(f(X)) -> f#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(g(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(g(X)) -> g#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(mark(X1),X2) -> cons#(X1,X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(g(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(g(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(g(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(g(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(g(X)) -> g#(proper(X)) -> g#(mark(X)) -> g#(X) proper#(g(X)) -> g#(proper(X)) -> g#(ok(X)) -> g#(X) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(f(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(f(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(f(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(f(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(f(X)) -> f#(proper(X)) -> f#(mark(X)) -> f#(X) proper#(f(X)) -> f#(proper(X)) -> f#(ok(X)) -> f#(X) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) -> sel#(mark(X1),X2) -> sel#(X1,X2) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) -> sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) sel#(mark(X1),X2) -> sel#(X1,X2) -> sel#(mark(X1),X2) -> sel#(X1,X2) sel#(mark(X1),X2) -> sel#(X1,X2) -> sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(mark(X1),X2) -> sel#(X1,X2) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) -> sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) -> sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(mark(X1),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#(mark(X1),X2) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) f#(ok(X)) -> f#(X) -> f#(mark(X)) -> f#(X) f#(ok(X)) -> f#(X) -> f#(ok(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(mark(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(ok(X)) -> f#(X) g#(ok(X)) -> g#(X) -> g#(mark(X)) -> g#(X) g#(ok(X)) -> g#(X) -> g#(ok(X)) -> g#(X) g#(mark(X)) -> g#(X) -> g#(mark(X)) -> g#(X) g#(mark(X)) -> g#(X) -> g#(ok(X)) -> g#(X) active#(sel(s(X),cons(Y,Z))) -> sel#(X,Z) -> sel#(X1,mark(X2)) -> sel#(X1,X2) active#(sel(X1,X2)) -> sel#(active(X1),X2) -> sel#(mark(X1),X2) -> sel#(X1,X2) active#(sel(X1,X2)) -> sel#(active(X1),X2) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) -> sel#(X1,mark(X2)) -> sel#(X1,X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) active#(sel(X1,X2)) -> active#(X2) -> active#(f(X)) -> g#(X) active#(sel(X1,X2)) -> active#(X2) -> active#(f(X)) -> f#(g(X)) active#(sel(X1,X2)) -> active#(X2) -> active#(f(X)) -> cons#(X,f(g(X))) active#(sel(X1,X2)) -> active#(X2) -> active#(g(0())) -> s#(0()) active#(sel(X1,X2)) -> active#(X2) -> active#(g(s(X))) -> g#(X) active#(sel(X1,X2)) -> active#(X2) -> active#(g(s(X))) -> s#(g(X)) active#(sel(X1,X2)) -> active#(X2) -> active#(g(s(X))) -> s#(s(g(X))) active#(sel(X1,X2)) -> active#(X2) -> active#(sel(s(X),cons(Y,Z))) -> sel#(X,Z) active#(sel(X1,X2)) -> active#(X2) -> active#(f(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X2) -> active#(f(X)) -> f#(active(X)) active#(sel(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) -> active#(g(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X2) -> active#(g(X)) -> g#(active(X)) active#(sel(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X2) -> active#(s(X)) -> s#(active(X)) active#(sel(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(sel(X1,X2)) -> active#(X1) -> active#(f(X)) -> g#(X) active#(sel(X1,X2)) -> active#(X1) -> active#(f(X)) -> f#(g(X)) active#(sel(X1,X2)) -> active#(X1) -> active#(f(X)) -> cons#(X,f(g(X))) active#(sel(X1,X2)) -> active#(X1) -> active#(g(0())) -> s#(0()) active#(sel(X1,X2)) -> active#(X1) -> active#(g(s(X))) -> g#(X) active#(sel(X1,X2)) -> active#(X1) -> active#(g(s(X))) -> s#(g(X)) active#(sel(X1,X2)) -> active#(X1) -> active#(g(s(X))) -> s#(s(g(X))) active#(sel(X1,X2)) -> active#(X1) -> active#(sel(s(X),cons(Y,Z))) -> sel#(X,Z) active#(sel(X1,X2)) -> active#(X1) -> active#(f(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X1) -> active#(f(X)) -> f#(active(X)) active#(sel(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(sel(X1,X2)) -> active#(X1) -> active#(g(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X1) -> active#(g(X)) -> g#(active(X)) active#(sel(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(sel(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(sel(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X) active#(s(X)) -> active#(X) -> active#(f(X)) -> g#(X) active#(s(X)) -> active#(X) -> active#(f(X)) -> f#(g(X)) active#(s(X)) -> active#(X) -> active#(f(X)) -> cons#(X,f(g(X))) active#(s(X)) -> active#(X) -> active#(g(0())) -> s#(0()) active#(s(X)) -> active#(X) -> active#(g(s(X))) -> g#(X) active#(s(X)) -> active#(X) -> active#(g(s(X))) -> s#(g(X)) active#(s(X)) -> active#(X) -> active#(g(s(X))) -> s#(s(g(X))) active#(s(X)) -> active#(X) -> active#(sel(s(X),cons(Y,Z))) -> sel#(X,Z) active#(s(X)) -> active#(X) -> active#(f(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(g(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(g(X)) -> g#(active(X)) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(cons(X1,X2)) -> active#(X1) -> active#(f(X)) -> g#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(f(X)) -> f#(g(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(f(X)) -> cons#(X,f(g(X))) active#(cons(X1,X2)) -> active#(X1) -> active#(g(0())) -> s#(0()) active#(cons(X1,X2)) -> active#(X1) -> active#(g(s(X))) -> g#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(g(s(X))) -> s#(g(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(g(s(X))) -> s#(s(g(X))) active#(cons(X1,X2)) -> active#(X1) -> active#(sel(s(X),cons(Y,Z))) -> sel#(X,Z) active#(cons(X1,X2)) -> active#(X1) -> active#(f(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(f(X)) -> f#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(g(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(g(X)) -> g#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(g(X)) -> g#(active(X)) -> g#(mark(X)) -> g#(X) active#(g(X)) -> g#(active(X)) -> g#(ok(X)) -> g#(X) active#(g(X)) -> active#(X) -> active#(f(X)) -> g#(X) active#(g(X)) -> active#(X) -> active#(f(X)) -> f#(g(X)) active#(g(X)) -> active#(X) -> active#(f(X)) -> cons#(X,f(g(X))) active#(g(X)) -> active#(X) -> active#(g(0())) -> s#(0()) active#(g(X)) -> active#(X) -> active#(g(s(X))) -> g#(X) active#(g(X)) -> active#(X) -> active#(g(s(X))) -> s#(g(X)) active#(g(X)) -> active#(X) -> active#(g(s(X))) -> s#(s(g(X))) active#(g(X)) -> active#(X) -> active#(sel(s(X),cons(Y,Z))) -> sel#(X,Z) active#(g(X)) -> active#(X) -> active#(f(X)) -> active#(X) active#(g(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) active#(g(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(g(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(g(X)) -> active#(X) -> active#(g(X)) -> active#(X) active#(g(X)) -> active#(X) -> active#(g(X)) -> g#(active(X)) active#(g(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(g(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(g(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X1) active#(g(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(g(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X2) active#(g(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(f(X)) -> f#(active(X)) -> f#(mark(X)) -> f#(X) active#(f(X)) -> f#(active(X)) -> f#(ok(X)) -> f#(X) active#(f(X)) -> active#(X) -> active#(f(X)) -> g#(X) active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(g(X)) active#(f(X)) -> active#(X) -> active#(f(X)) -> cons#(X,f(g(X))) active#(f(X)) -> active#(X) -> active#(g(0())) -> s#(0()) active#(f(X)) -> active#(X) -> active#(g(s(X))) -> g#(X) active#(f(X)) -> active#(X) -> active#(g(s(X))) -> s#(g(X)) active#(f(X)) -> active#(X) -> active#(g(s(X))) -> s#(s(g(X))) active#(f(X)) -> active#(X) -> active#(sel(s(X),cons(Y,Z))) -> sel#(X,Z) active#(f(X)) -> active#(X) -> active#(f(X)) -> active#(X) active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) active#(f(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(f(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(f(X)) -> active#(X) -> active#(g(X)) -> active#(X) active#(f(X)) -> active#(X) -> active#(g(X)) -> g#(active(X)) active#(f(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(f(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(f(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X1) active#(f(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(f(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X2) active#(f(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) SCC Processor: #sccs: 8 #rules: 26 #arcs: 293/2209 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) proper#(g(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) proper#(f(X)) -> proper#(X) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(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) = 0, [proper](x0) = x0, [sel](x0, x1) = x0 + x1 + 1, [s](x0) = x0, [0] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + x1, [g](x0) = x0 + 1, [active](x0) = x0, [f](x0) = x0 orientation: proper#(sel(X1,X2)) = X1 + X2 + 1 >= X1 = proper#(X1) proper#(sel(X1,X2)) = X1 + X2 + 1 >= X2 = proper#(X2) proper#(s(X)) = X >= X = proper#(X) proper#(g(X)) = X + 1 >= X = proper#(X) proper#(cons(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(f(X)) = X >= X = proper#(X) active(f(X)) = X >= 0 = mark(cons(X,f(g(X)))) active(g(0())) = 1 >= 0 = mark(s(0())) active(g(s(X))) = X + 1 >= 0 = mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) = X + Y + 1 >= 0 = mark(X) active(sel(s(X),cons(Y,Z))) = X + Y + Z + 1 >= 0 = mark(sel(X,Z)) active(f(X)) = X >= X = f(active(X)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(g(X)) = X + 1 >= X + 1 = g(active(X)) active(s(X)) = X >= X = s(active(X)) active(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(X1,active(X2)) f(mark(X)) = 0 >= 0 = mark(f(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) g(mark(X)) = 1 >= 0 = mark(g(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) sel(mark(X1),X2) = X2 + 1 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 + 1 >= 0 = mark(sel(X1,X2)) proper(f(X)) = X >= X = f(proper(X)) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(g(X)) = X + 1 >= X + 1 = g(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X >= X = s(proper(X)) proper(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(proper(X1),proper(X2)) f(ok(X)) = 0 >= 0 = ok(f(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) g(ok(X)) = 1 >= 0 = ok(g(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) sel(ok(X1),ok(X2)) = 1 >= 0 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) proper#(f(X)) -> proper#(X) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(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, [sel](x0, x1) = x1, [s](x0) = x0, [0] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [g](x0) = 0, [active](x0) = x0, [f](x0) = x0 + 1 orientation: proper#(s(X)) = X >= X = proper#(X) proper#(cons(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(f(X)) = X + 1 >= X = proper#(X) active(f(X)) = X + 1 >= X + 1 = mark(cons(X,f(g(X)))) active(g(0())) = 0 >= 0 = mark(s(0())) active(g(s(X))) = 0 >= 0 = mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) = X + Y >= X = mark(X) active(sel(s(X),cons(Y,Z))) = Y + Z >= Z = mark(sel(X,Z)) active(f(X)) = X + 1 >= X + 1 = f(active(X)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(g(X)) = 0 >= 0 = g(active(X)) active(s(X)) = X >= X = s(active(X)) active(sel(X1,X2)) = X2 >= X2 = sel(active(X1),X2) active(sel(X1,X2)) = X2 >= X2 = sel(X1,active(X2)) f(mark(X)) = X + 1 >= X + 1 = mark(f(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) g(mark(X)) = 0 >= 0 = mark(g(X)) s(mark(X)) = X >= X = mark(s(X)) sel(mark(X1),X2) = X2 >= X2 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X2 >= X2 = mark(sel(X1,X2)) proper(f(X)) = X + 1 >= X + 1 = f(proper(X)) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(g(X)) = 0 >= 0 = g(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X >= X = s(proper(X)) proper(sel(X1,X2)) = X2 >= X2 = sel(proper(X1),proper(X2)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) g(ok(X)) = 0 >= 0 = ok(g(X)) s(ok(X)) = X >= X = ok(s(X)) sel(ok(X1),ok(X2)) = X2 >= X2 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(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, [sel](x0, x1) = 0, [s](x0) = x0 + 1, [0] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + x1 + 1, [g](x0) = 0, [active](x0) = x0, [f](x0) = 0 orientation: proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(f(X)) = 0 >= 0 = mark(cons(X,f(g(X)))) active(g(0())) = 0 >= 0 = mark(s(0())) active(g(s(X))) = 0 >= 0 = mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) = 0 >= 0 = mark(X) active(sel(s(X),cons(Y,Z))) = 0 >= 0 = mark(sel(X,Z)) active(f(X)) = 0 >= 0 = f(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(g(X)) = 0 >= 0 = g(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(sel(X1,X2)) = 0 >= 0 = sel(active(X1),X2) active(sel(X1,X2)) = 0 >= 0 = sel(X1,active(X2)) f(mark(X)) = 0 >= 0 = mark(f(X)) cons(mark(X1),X2) = X2 + 1 >= 0 = mark(cons(X1,X2)) g(mark(X)) = 0 >= 0 = mark(g(X)) s(mark(X)) = 1 >= 0 = mark(s(X)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 0 >= 0 = mark(sel(X1,X2)) proper(f(X)) = 0 >= 0 = f(proper(X)) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(g(X)) = 0 >= 0 = g(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(sel(X1,X2)) = 0 >= 0 = sel(proper(X1),proper(X2)) f(ok(X)) = 0 >= 0 = ok(f(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) g(ok(X)) = 0 >= 0 = ok(g(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) sel(ok(X1),ok(X2)) = 0 >= 0 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(g(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(f(X)) -> active#(X) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(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) = 1, [ok](x0) = x0, [proper](x0) = x0, [sel](x0, x1) = x0 + x1 + 1, [s](x0) = x0 + 1, [0] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [g](x0) = x0, [active](x0) = x0, [f](x0) = x0 + 1 orientation: active#(sel(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(sel(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(g(X)) = X + 1 >= X + 1 = active#(X) active#(cons(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active#(f(X)) = X + 2 >= X + 1 = active#(X) active(f(X)) = X + 1 >= 0 = mark(cons(X,f(g(X)))) active(g(0())) = 0 >= 0 = mark(s(0())) active(g(s(X))) = X + 1 >= 0 = mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) = X + 1 >= 0 = mark(X) active(sel(s(X),cons(Y,Z))) = X + Y + 2 >= 0 = mark(sel(X,Z)) active(f(X)) = X + 1 >= X + 1 = f(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(g(X)) = X >= X = g(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(X1,active(X2)) f(mark(X)) = 1 >= 0 = mark(f(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) g(mark(X)) = 0 >= 0 = mark(g(X)) s(mark(X)) = 1 >= 0 = mark(s(X)) sel(mark(X1),X2) = X2 + 1 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 + 1 >= 0 = mark(sel(X1,X2)) proper(f(X)) = X + 1 >= X + 1 = f(proper(X)) proper(cons(X1,X2)) = X1 >= X1 = cons(proper(X1),proper(X2)) proper(g(X)) = X >= X = g(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(proper(X1),proper(X2)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) cons(ok(X1),ok(X2)) = X1 >= X1 = ok(cons(X1,X2)) g(ok(X)) = X >= X = ok(g(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) sel(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(sel(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: active#(g(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(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, [sel](x0, x1) = 0, [s](x0) = x0, [0] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [g](x0) = x0 + 1, [active](x0) = x0, [f](x0) = 0 orientation: active#(g(X)) = X + 2 >= X + 1 = active#(X) active#(cons(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active(f(X)) = 0 >= 0 = mark(cons(X,f(g(X)))) active(g(0())) = 2 >= 0 = mark(s(0())) active(g(s(X))) = X + 1 >= 0 = mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) = 0 >= 0 = mark(X) active(sel(s(X),cons(Y,Z))) = 0 >= 0 = mark(sel(X,Z)) active(f(X)) = 0 >= 0 = f(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(g(X)) = X + 1 >= X + 1 = g(active(X)) active(s(X)) = X >= X = s(active(X)) active(sel(X1,X2)) = 0 >= 0 = sel(active(X1),X2) active(sel(X1,X2)) = 0 >= 0 = sel(X1,active(X2)) f(mark(X)) = 0 >= 0 = mark(f(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) g(mark(X)) = 1 >= 0 = mark(g(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 0 >= 0 = mark(sel(X1,X2)) proper(f(X)) = 0 >= 0 = f(proper(X)) proper(cons(X1,X2)) = X1 >= X1 = cons(proper(X1),proper(X2)) proper(g(X)) = X + 1 >= X + 1 = g(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X >= X = s(proper(X)) proper(sel(X1,X2)) = 0 >= 0 = sel(proper(X1),proper(X2)) f(ok(X)) = 0 >= 0 = ok(f(X)) cons(ok(X1),ok(X2)) = X1 >= X1 = ok(cons(X1,X2)) g(ok(X)) = X + 1 >= X + 1 = ok(g(X)) s(ok(X)) = X >= X = ok(s(X)) sel(ok(X1),ok(X2)) = 0 >= 0 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: active#(cons(X1,X2)) -> active#(X1) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(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) = 1, [proper](x0) = x0, [sel](x0, x1) = 1, [s](x0) = 1, [0] = 1, [mark](x0) = 1, [cons](x0, x1) = x0 + 1, [g](x0) = 1, [active](x0) = x0, [f](x0) = 1 orientation: active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(f(X)) = 1 >= 1 = mark(cons(X,f(g(X)))) active(g(0())) = 1 >= 1 = mark(s(0())) active(g(s(X))) = 1 >= 1 = mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) = 1 >= 1 = mark(X) active(sel(s(X),cons(Y,Z))) = 1 >= 1 = mark(sel(X,Z)) active(f(X)) = 1 >= 1 = f(active(X)) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) active(g(X)) = 1 >= 1 = g(active(X)) active(s(X)) = 1 >= 1 = s(active(X)) active(sel(X1,X2)) = 1 >= 1 = sel(active(X1),X2) active(sel(X1,X2)) = 1 >= 1 = sel(X1,active(X2)) f(mark(X)) = 1 >= 1 = mark(f(X)) cons(mark(X1),X2) = 2 >= 1 = mark(cons(X1,X2)) g(mark(X)) = 1 >= 1 = mark(g(X)) s(mark(X)) = 1 >= 1 = mark(s(X)) sel(mark(X1),X2) = 1 >= 1 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 1 >= 1 = mark(sel(X1,X2)) proper(f(X)) = 1 >= 1 = f(proper(X)) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(g(X)) = 1 >= 1 = g(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(sel(X1,X2)) = 1 >= 1 = sel(proper(X1),proper(X2)) f(ok(X)) = 1 >= 1 = ok(f(X)) cons(ok(X1),ok(X2)) = 2 >= 1 = ok(cons(X1,X2)) g(ok(X)) = 1 >= 1 = ok(g(X)) s(ok(X)) = 1 >= 1 = ok(s(X)) sel(ok(X1),ok(X2)) = 1 >= 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: f#(ok(X)) -> f#(X) f#(mark(X)) -> f#(X) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [sel](x0, x1) = x1, [s](x0) = x0, [0] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [g](x0) = x0, [active](x0) = x0, [f](x0) = x0 orientation: cons#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = cons#(X1,X2) cons#(mark(X1),X2) = X2 + 1 >= X2 + 1 = cons#(X1,X2) active(f(X)) = X >= 0 = mark(cons(X,f(g(X)))) active(g(0())) = 0 >= 0 = mark(s(0())) active(g(s(X))) = X >= 0 = mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) = Y >= 0 = mark(X) active(sel(s(X),cons(Y,Z))) = Z >= 0 = mark(sel(X,Z)) active(f(X)) = X >= X = f(active(X)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(g(X)) = X >= X = g(active(X)) active(s(X)) = X >= X = s(active(X)) active(sel(X1,X2)) = X2 >= X2 = sel(active(X1),X2) active(sel(X1,X2)) = X2 >= X2 = sel(X1,active(X2)) f(mark(X)) = 0 >= 0 = mark(f(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) g(mark(X)) = 0 >= 0 = mark(g(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) sel(mark(X1),X2) = X2 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 0 >= 0 = mark(sel(X1,X2)) proper(f(X)) = 1 >= 1 = f(proper(X)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(g(X)) = 1 >= 1 = g(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(sel(X1,X2)) = 1 >= 1 = sel(proper(X1),proper(X2)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) g(ok(X)) = X + 1 >= X + 1 = ok(g(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) sel(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: g#(ok(X)) -> g#(X) g#(mark(X)) -> g#(X) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: sel#(ok(X1),ok(X2)) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(mark(X1),X2) -> sel#(X1,X2) TRS: active(f(X)) -> mark(cons(X,f(g(X)))) active(g(0())) -> mark(s(0())) active(g(s(X))) -> mark(s(s(g(X)))) active(sel(0(),cons(X,Y))) -> mark(X) active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(g(X)) -> g(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) g(mark(X)) -> mark(g(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(f(X)) -> f(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(g(X)) -> g(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) g(ok(X)) -> ok(g(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open