YES Problem: active(f(0())) -> mark(cons(0(),f(s(0())))) active(f(s(0()))) -> mark(f(p(s(0())))) active(p(s(0()))) -> mark(0()) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(p(X)) -> p(active(X)) f(mark(X)) -> mark(f(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) p(mark(X)) -> mark(p(X)) proper(f(X)) -> f(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(p(X)) -> p(proper(X)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) p(ok(X)) -> ok(p(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [top](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [ok](x0) = x0 , [proper](x0) = x0 , [1 0 0] [p](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 1] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 1] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 1] [1], [active](x0) = x0 , [1 0 0] [0] [f](x0) = [0 0 0]x0 + [0] [1 1 1] [1], [0] [0] = [0] [0] orientation: [0] [0] active(f(0())) = [0] >= [0] = mark(cons(0(),f(s(0())))) [1] [1] [0] [0] active(f(s(0()))) = [0] >= [0] = mark(f(p(s(0())))) [3] [3] [0] [0] active(p(s(0()))) = [0] >= [0] = mark(0()) [1] [1] [1 0 0] [0] [1 0 0] [0] active(f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(active(X)) [1 1 1] [1] [1 1 1] [1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 1] [0] [1 0 1] [0] active(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(active(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1 0 0] active(p(X)) = [0 0 0]X >= [0 0 0]X = p(active(X)) [0 0 1] [0 0 1] [1 1 0] [0] [1 0 0] [0] f(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(f(X)) [1 1 1] [2] [1 1 1] [2] [1 1 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 1 1] [1] [1 0 1] [1] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(s(X)) [0 0 1] [2] [0 0 1] [2] [1 1 0] [0] [1 0 0] [0] p(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(p(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [0] [1 0 0] [0] proper(f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(proper(X)) [1 1 1] [1] [1 1 1] [1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 1] [0] [1 0 1] [0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1 0 0] proper(p(X)) = [0 0 0]X >= [0 0 0]X = p(proper(X)) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 0] [0] f(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(f(X)) [1 1 1] [1] [1 1 1] [1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 1] [0] [1 0 1] [0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(s(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1 0 0] p(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(p(X)) [0 0 1] [0 0 1] [1 1 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X)) [0 0 0] [1] [0 0 0] [1] problem: active(f(0())) -> mark(cons(0(),f(s(0())))) active(f(s(0()))) -> mark(f(p(s(0())))) active(p(s(0()))) -> mark(0()) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(p(X)) -> p(active(X)) f(mark(X)) -> mark(f(X)) s(mark(X)) -> mark(s(X)) p(mark(X)) -> mark(p(X)) proper(f(X)) -> f(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(p(X)) -> p(proper(X)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) p(ok(X)) -> ok(p(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [0] [top](x0) = [1 1 1]x0 + [0] [0 0 0] [1], [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 0 0] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [p](x0) = x0 , [1 1 1] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [cons](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [s](x0) = x0 , [1 0 1] [0] [active](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 1] [0] [f](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [0] [0] = [0] [0] orientation: [1] [0] active(f(0())) = [0] >= [0] = mark(cons(0(),f(s(0())))) [1] [1] [1] [1] active(f(s(0()))) = [0] >= [0] = mark(f(p(s(0())))) [1] [1] [0] [0] active(p(s(0()))) = [0] >= [0] = mark(0()) [1] [1] [1 0 1] [1] [1 0 1] [1] active(f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] [0] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [0] [1 0 1] [0] active(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 1] [0] active(p(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = p(active(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1] [1 1 1] [1] f(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(f(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] p(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(p(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 1] [0] proper(f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(proper(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] proper(0()) = [1] >= [1] = ok(0()) [0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 0 1] [0] [0 0 1] [0] [1 0 0] [0] [1 0 0] [0] proper(p(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = p(proper(X)) [0 0 1] [0] [0 0 1] [0] [1 0 1] [0] [1 0 1] [0] f(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(f(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(cons(X1,X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(s(X)) [0 0 1] [0] [0 0 1] [0] [1 0 0] [0] [1 0 0] [0] p(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(p(X)) [0 0 1] [0] [0 0 1] [0] [1 1 1] [1] [1 0 1] [1] top(mark(X)) = [1 1 1]X + [1] >= [1 0 1]X + [1] = top(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [1] top(ok(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = top(active(X)) [0 0 0] [1] [0 0 0] [1] problem: active(f(s(0()))) -> mark(f(p(s(0())))) active(p(s(0()))) -> mark(0()) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(p(X)) -> p(active(X)) f(mark(X)) -> mark(f(X)) s(mark(X)) -> mark(s(X)) p(mark(X)) -> mark(p(X)) proper(f(X)) -> f(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(p(X)) -> p(proper(X)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) p(ok(X)) -> ok(p(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [top](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [ok](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [proper](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [p](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [mark](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [1 0 0] [1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [1], [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [active](x0) = [0 0 1]x0 [0 0 1] , [1 0 1] [f](x0) = [1 0 0]x0 [1 0 0] , [0] [0] = [0] [0] orientation: [1] [0] active(f(s(0()))) = [0] >= [0] = mark(f(p(s(0())))) [0] [0] [0] [0] active(p(s(0()))) = [0] >= [0] = mark(0()) [0] [0] [1 0 1] [1 0 1] active(f(X)) = [1 0 0]X >= [1 0 0]X = f(active(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] active(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] active(p(X)) = [0 0 0]X >= [0 0 0]X = p(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] f(mark(X)) = [1 0 0]X >= [1 0 0]X = mark(f(X)) [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(s(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] p(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(p(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] proper(f(X)) = [1 0 0]X >= [1 0 0]X = f(proper(X)) [1 0 0] [1 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] proper(p(X)) = [0 0 0]X >= [0 0 0]X = p(proper(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] f(ok(X)) = [1 0 0]X >= [1 0 0]X = ok(f(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(s(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] p(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(p(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [1 0 0] [1 0 0] [1 0 1] [1 0 1] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [1 0 0] [1 0 0] problem: active(p(s(0()))) -> mark(0()) active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(p(X)) -> p(active(X)) f(mark(X)) -> mark(f(X)) s(mark(X)) -> mark(s(X)) p(mark(X)) -> mark(p(X)) proper(f(X)) -> f(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(p(X)) -> p(proper(X)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) p(ok(X)) -> ok(p(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [ok](x0) = [0 1 0]x0 [0 1 0] , [1 0 1] [proper](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [p](x0) = [0 0 0]x0 [0 1 0] , [1 0 0] [mark](x0) = [0 1 0]x0 [0 1 1] , [1 0 0] [1 0 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 1] [1] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [active](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1] [f](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [0] = [0] [0] orientation: [1] [0] active(p(s(0()))) = [0] >= [0] = mark(0()) [0] [0] [1 0 1] [1] [1 0 0] [1] active(f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(active(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1] [1 0 0] [1] active(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(active(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 0] active(p(X)) = [0 0 0]X >= [0 0 0]X = p(active(X)) [0 0 0] [0 0 0] [1 1 1] [1] [1 0 1] [1] f(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(f(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 0 1] [1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 0 1] p(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(p(X)) [0 1 0] [0 1 0] [1 0 1] [1] [1 0 1] [1] proper(f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(proper(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1] [1 0 1] [1] proper(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 0 1] proper(p(X)) = [0 0 0]X >= [0 0 0]X = p(proper(X)) [0 0 0] [0 0 0] [1 1 1] [1] [1 0 1] [1] f(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(f(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 1 1] [1 0 1] [1 0 1] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 1 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1] [1 0 1] [1] s(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 1 1] p(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(p(X)) [0 1 0] [0 0 0] [1 1 1] [1 0 1] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 1 1] [1 0 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(p(X)) -> p(active(X)) f(mark(X)) -> mark(f(X)) s(mark(X)) -> mark(s(X)) p(mark(X)) -> mark(p(X)) proper(f(X)) -> f(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(p(X)) -> p(proper(X)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) p(ok(X)) -> ok(p(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 4x0 + 1, [ok](x0) = x0, [proper](x0) = 2x0 + 2, [p](x0) = 2x0 + 2, [mark](x0) = 2x0 + 4, [cons](x0, x1) = x0 + x1 + 3, [s](x0) = x0, [active](x0) = x0, [f](x0) = 2x0 + 2, [0] = 1 orientation: active(f(X)) = 2X + 2 >= 2X + 2 = f(active(X)) active(cons(X1,X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = cons(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(p(X)) = 2X + 2 >= 2X + 2 = p(active(X)) f(mark(X)) = 4X + 10 >= 4X + 8 = mark(f(X)) s(mark(X)) = 2X + 4 >= 2X + 4 = mark(s(X)) p(mark(X)) = 4X + 10 >= 4X + 8 = mark(p(X)) proper(f(X)) = 4X + 6 >= 4X + 6 = f(proper(X)) proper(0()) = 4 >= 1 = ok(0()) proper(cons(X1,X2)) = 2X1 + 2X2 + 8 >= 2X1 + 2X2 + 7 = cons(proper(X1),proper(X2)) proper(s(X)) = 2X + 2 >= 2X + 2 = s(proper(X)) proper(p(X)) = 4X + 6 >= 4X + 6 = p(proper(X)) f(ok(X)) = 2X + 2 >= 2X + 2 = ok(f(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) p(ok(X)) = 2X + 2 >= 2X + 2 = ok(p(X)) top(mark(X)) = 8X + 17 >= 8X + 9 = top(proper(X)) top(ok(X)) = 4X + 1 >= 4X + 1 = top(active(X)) problem: active(f(X)) -> f(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(p(X)) -> p(active(X)) s(mark(X)) -> mark(s(X)) proper(f(X)) -> f(proper(X)) proper(s(X)) -> s(proper(X)) proper(p(X)) -> p(proper(X)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) p(ok(X)) -> ok(p(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [top](x0) = [0 0 1]x0 [0 0 0] , [1 1 0] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 1 0] [proper](x0) = [1 1 0]x0 [0 0 0] , [p](x0) = x0 , [1 1 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 1] [0] [cons](x0, x1) = x0 + [0 0 0]x1 + [1] [0 0 0] [0], [1 0 1] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 1 0] [active](x0) = [0 1 0]x0 [0 0 1] , [1 0 1] [0] [f](x0) = [0 1 0]x0 + [1] [0 0 1] [0] orientation: [1 1 1] [1] [1 1 1] [0] active(f(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(active(X)) [0 0 1] [0] [0 0 1] [0] [1 1 0] [1 0 1] [1] [1 1 0] [1 0 1] [0] active(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = cons(active(X1),X2) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 1 1] [1] [1 1 1] [0] active(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(active(X)) [0 0 1] [0] [0 0 1] [0] [1 1 0] [1 1 0] active(p(X)) = [0 1 0]X >= [0 1 0]X = p(active(X)) [0 0 1] [0 0 1] [1 1 1] [1] [1 1 1] [1] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(s(X)) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 1 0] [0] proper(f(X)) = [1 1 1]X + [1] >= [1 1 0]X + [1] = f(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 0] [0] proper(s(X)) = [1 1 1]X + [1] >= [1 1 0]X + [1] = s(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] proper(p(X)) = [1 1 0]X >= [1 1 0]X = p(proper(X)) [0 0 0] [0 0 0] [1 1 1] [1] [1 1 1] [1] f(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = ok(f(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1 1 1] [1] [1 1 0] [1 0 1] [1] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 1 1] [1] [1 1 1] [1] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = ok(s(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [0] [1 1 0] [0] p(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(p(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [0] [1 1 0] top(ok(X)) = [0 0 1]X + [1] >= [0 0 1]X = top(active(X)) [0 0 0] [0] [0 0 0] problem: active(p(X)) -> p(active(X)) s(mark(X)) -> mark(s(X)) proper(p(X)) -> p(proper(X)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) p(ok(X)) -> ok(p(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [top](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [1] [proper](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [1 0 1] [0] [p](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 1 0] [0] [mark](x0) = [0 0 1]x0 + [0] [0 0 1] [1], [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 1] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 1 0] [active](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [f](x0) = [0 0 0]x0 [0 0 1] orientation: [1 1 1] [1] [1 1 0] [0] active(p(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = p(active(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 1] [1] s(mark(X)) = [0 0 1]X + [1] >= [0 0 1]X + [0] = mark(s(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1] [1 0 0] [1] proper(p(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = p(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [0] [1 0 0] [0] f(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(f(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1 1 0] [0] [1 0 0] [1 0 0] [0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 1 1] [1] [1 1 1] [1] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = ok(s(X)) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 1 1] [1] p(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = ok(p(X)) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 1 0] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X = top(active(X)) [0 0 0] [0] [0 0 0] problem: s(mark(X)) -> mark(s(X)) proper(p(X)) -> p(proper(X)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) p(ok(X)) -> ok(p(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [1] [proper](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [p](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [0] [mark](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 0] [0] [cons](x0, x1) = [1 1 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 1 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [f](x0) = [0 0 0]x0 [0 1 0] orientation: [1 1 0] [1] [1 1 0] [0] s(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1] [1 1 0] [1] proper(p(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = p(proper(X)) [0 0 0] [1] [0 0 0] [0] [1 1 0] [1 1 0] f(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(f(X)) [0 1 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(ok(X1),ok(X2)) = [1 1 0]X1 + [0 0 0]X2 + [1] >= [1 1 0]X1 + [0 0 0]X2 + [1] = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] s(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] p(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(p(X)) [0 0 0] [0 0 0] problem: proper(p(X)) -> p(proper(X)) f(ok(X)) -> ok(f(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) p(ok(X)) -> ok(p(X)) Matrix Interpretation Processor: dim=3 interpretation: [0] [ok](x0) = x0 + [1] [0], [1 0 1] [proper](x0) = [0 0 0]x0 [0 0 1] , [0] [p](x0) = x0 + [0] [1], [1 0 0] [1 1 0] [cons](x0, x1) = [0 0 0]x0 + [0 1 1]x1 [0 0 0] [0 0 0] , [1 0 0] [1] [s](x0) = [1 1 1]x0 + [0] [0 0 0] [0], [1 1 0] [f](x0) = [1 1 0]x0 [0 0 0] orientation: [1 0 1] [1] [1 0 1] [0] proper(p(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = p(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1] [1 1 0] [0] f(ok(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = ok(f(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 1 1]X2 + [1] >= [0 0 0]X1 + [0 1 1]X2 + [1] = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] s(ok(X)) = [1 1 1]X + [1] >= [1 1 1]X + [1] = ok(s(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] p(ok(X)) = X + [1] >= X + [1] = ok(p(X)) [1] [1] problem: s(ok(X)) -> ok(s(X)) p(ok(X)) -> ok(p(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [ok](x0) = [0 1 0]x0 + [1] [0 1 1] [0], [1 1 0] [0] [p](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [1 1 0] [s](x0) = [0 1 0]x0 [0 0 1] orientation: [1 1 0] [1] [1 1 0] [0] s(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(s(X)) [0 1 1] [0] [0 1 1] [0] [1 1 0] [1] [1 1 0] [0] p(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(p(X)) [0 1 1] [1] [0 1 1] [1] problem: Qed