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()) mark(f(X)) -> active(f(mark(X))) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(p(X)) -> active(p(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) p(mark(X)) -> p(X) p(active(X)) -> p(X) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 1 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [f](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [0] [mark](x0) = x0 + [1] [0], [0] [0] = [0] [0], [1 0 0] [p](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [active](x0) = [0 0 0]x0 + [1] [0 1 1] [0] orientation: [0] [0] active(f(0())) = [1] >= [1] = mark(cons(0(),f(s(0())))) [0] [0] [0] [0] active(f(s(0()))) = [1] >= [1] = mark(f(p(s(0())))) [0] [0] [0] [0] active(p(s(0()))) = [1] >= [1] = mark(0()) [0] [0] [1 0 1] [0] [1 0 1] [0] mark(f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(f(mark(X))) [0 0 0] [0] [0 0 0] [0] [0] [0] mark(0()) = [1] >= [1] = active(0()) [0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(s(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] mark(p(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(p(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 1] f(mark(X)) = [0 0 0]X >= [0 0 0]X = f(X) [0 0 0] [0 0 0] [1 1 1] [1 0 1] f(active(X)) = [0 0 0]X >= [0 0 0]X = f(X) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1] [1 0 0] [1 1 1] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1] [1 0 0] [1 1 1] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] p(mark(X)) = [0 0 0]X >= [0 0 0]X = p(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] p(active(X)) = [0 0 0]X >= [0 0 0]X = p(X) [0 0 0] [0 0 0] 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()) mark(f(X)) -> active(f(mark(X))) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(p(X)) -> active(p(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) p(mark(X)) -> p(X) p(active(X)) -> p(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 1 1] [0] [f](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 1 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [mark](x0) = [0 0 1]x0 [0 0 0] , [0] [0] = [0] [0], [1 1 1] [p](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [active](x0) = [0 1 0]x0 [0 0 0] orientation: [1] [0] active(f(0())) = [1] >= [0] = mark(cons(0(),f(s(0())))) [0] [0] [1] [1] active(f(s(0()))) = [1] >= [1] = mark(f(p(s(0())))) [0] [0] [0] [0] active(p(s(0()))) = [0] >= [0] = mark(0()) [0] [0] [1 1 1] [1] [1 1 1] [1] mark(f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(f(mark(X))) [0 0 0] [0] [0 0 0] [0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 1 1] [1 0 0] [1 1 1] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(mark(X))) [0 0 0] [0 0 0] [1 1 1] [1 1 1] mark(p(X)) = [0 0 0]X >= [0 0 0]X = active(p(mark(X))) [0 0 0] [0 0 0] [1 1 1] [0] [1 1 1] [0] f(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(X) [0 0 0] [1] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] f(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(X) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1 0 0] [1 1 1] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 1 1] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 1] [1 1 1] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 1] [1 1 1] p(mark(X)) = [0 0 0]X >= [0 0 0]X = p(X) [0 0 0] [0 0 0] [1 1 1] [1 1 1] p(active(X)) = [0 0 0]X >= [0 0 0]X = p(X) [0 0 0] [0 0 0] problem: active(f(s(0()))) -> mark(f(p(s(0())))) active(p(s(0()))) -> mark(0()) mark(f(X)) -> active(f(mark(X))) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(p(X)) -> active(p(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) p(mark(X)) -> p(X) p(active(X)) -> p(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 1 0] [f](x0) = [0 0 0]x0 [0 1 0] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [mark](x0) = x0 + [0] [1], [0] [0] = [0] [0], [1 0 0] [p](x0) = [0 0 0]x0 [0 1 1] , [active](x0) = x0 orientation: [1] [0] 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 1 0] [0] [1 1 0] mark(f(X)) = [0 0 0]X + [0] >= [0 0 0]X = active(f(mark(X))) [0 1 0] [1] [0 1 0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(s(mark(X))) [0 0 0] [1] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] mark(p(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(p(mark(X))) [0 1 1] [1] [0 1 1] [1] [1 1 0] [1 1 0] f(mark(X)) = [0 0 0]X >= [0 0 0]X = f(X) [0 1 0] [0 1 0] [1 1 0] [1 1 0] f(active(X)) = [0 0 0]X >= [0 0 0]X = f(X) [0 1 0] [0 1 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] s(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] p(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = p(X) [0 1 1] [1] [0 1 1] [1 0 0] [1 0 0] p(active(X)) = [0 0 0]X >= [0 0 0]X = p(X) [0 1 1] [0 1 1] problem: active(p(s(0()))) -> mark(0()) mark(f(X)) -> active(f(mark(X))) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(p(X)) -> active(p(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) p(mark(X)) -> p(X) p(active(X)) -> p(X) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = 4x0 + 4x1, [f](x0) = 2x0, [s](x0) = 2x0 + 1, [mark](x0) = x0, [0] = 0, [p](x0) = 2x0, [active](x0) = x0 orientation: active(p(s(0()))) = 2 >= 0 = mark(0()) mark(f(X)) = 2X >= 2X = active(f(mark(X))) mark(0()) = 0 >= 0 = active(0()) mark(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = active(cons(mark(X1),X2)) mark(s(X)) = 2X + 1 >= 2X + 1 = active(s(mark(X))) mark(p(X)) = 2X >= 2X = active(p(mark(X))) f(mark(X)) = 2X >= 2X = f(X) f(active(X)) = 2X >= 2X = f(X) cons(mark(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(X1,X2) cons(active(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(X1,X2) s(mark(X)) = 2X + 1 >= 2X + 1 = s(X) s(active(X)) = 2X + 1 >= 2X + 1 = s(X) p(mark(X)) = 2X >= 2X = p(X) p(active(X)) = 2X >= 2X = p(X) problem: mark(f(X)) -> active(f(mark(X))) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(p(X)) -> active(p(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) p(mark(X)) -> p(X) p(active(X)) -> p(X) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = x0 + 2x1 + 4, [f](x0) = 4x0, [s](x0) = x0 + 4, [mark](x0) = 2x0, [0] = 1, [p](x0) = x0 + 2, [active](x0) = x0 orientation: mark(f(X)) = 8X >= 8X = active(f(mark(X))) mark(0()) = 2 >= 1 = active(0()) mark(cons(X1,X2)) = 2X1 + 4X2 + 8 >= 2X1 + 2X2 + 4 = active(cons(mark(X1),X2)) mark(s(X)) = 2X + 8 >= 2X + 4 = active(s(mark(X))) mark(p(X)) = 2X + 4 >= 2X + 2 = active(p(mark(X))) f(mark(X)) = 8X >= 4X = f(X) f(active(X)) = 4X >= 4X = f(X) cons(mark(X1),X2) = 2X1 + 2X2 + 4 >= X1 + 2X2 + 4 = cons(X1,X2) cons(active(X1),X2) = X1 + 2X2 + 4 >= X1 + 2X2 + 4 = cons(X1,X2) s(mark(X)) = 2X + 4 >= X + 4 = s(X) s(active(X)) = X + 4 >= X + 4 = s(X) p(mark(X)) = 2X + 2 >= X + 2 = p(X) p(active(X)) = X + 2 >= X + 2 = p(X) problem: mark(f(X)) -> active(f(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) p(mark(X)) -> p(X) p(active(X)) -> p(X) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = 4x0 + 2x1 + 4, [f](x0) = x0 + 6, [s](x0) = 4x0 + 1, [mark](x0) = 4x0, [p](x0) = x0 + 6, [active](x0) = x0 + 2 orientation: mark(f(X)) = 4X + 24 >= 4X + 8 = active(f(mark(X))) f(mark(X)) = 4X + 6 >= X + 6 = f(X) f(active(X)) = X + 8 >= X + 6 = f(X) cons(mark(X1),X2) = 16X1 + 2X2 + 4 >= 4X1 + 2X2 + 4 = cons(X1,X2) cons(active(X1),X2) = 4X1 + 2X2 + 12 >= 4X1 + 2X2 + 4 = cons(X1,X2) s(mark(X)) = 16X + 1 >= 4X + 1 = s(X) s(active(X)) = 4X + 9 >= 4X + 1 = s(X) p(mark(X)) = 4X + 6 >= X + 6 = p(X) p(active(X)) = X + 8 >= X + 6 = p(X) problem: f(mark(X)) -> f(X) cons(mark(X1),X2) -> cons(X1,X2) s(mark(X)) -> s(X) p(mark(X)) -> p(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [f](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [p](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1] [1 0 0] f(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = f(X) [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(X) [0 0 0] [0] [0 0 0] [1 0 0] [1] [1 0 0] p(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = p(X) [0 0 0] [0] [0 0 0] problem: Qed