YES Problem: d(0()) -> 0() d(s(x)) -> s(s(d(x))) e(0(),x) -> x e(s(x),y) -> e(x,d(y)) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [1] [e](x0, x1) = [0 0 0]x0 + [1 1 0]x1 + [0] [0 0 0] [0 0 1] [1], [1 1 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [d](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0] orientation: [0] [0] d(0()) = [0] >= [0] = 0() [0] [0] [1 1 1] [1 0 0] d(s(x)) = [0 0 0]x >= [0 0 0]x = s(s(d(x))) [0 0 0] [0 0 0] [1 0 0] [1] e(0(),x) = [1 1 0]x + [0] >= x = x [0 0 1] [1] [1 1 1] [1 0 0] [1] [1 0 0] [1 0 0] [1] e(s(x),y) = [0 0 0]x + [1 1 0]y + [0] >= [0 0 0]x + [1 0 0]y + [0] = e(x,d(y)) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 0] [1] problem: d(0()) -> 0() d(s(x)) -> s(s(d(x))) e(s(x),y) -> e(x,d(y)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [1 0 0] [e](x0, x1) = [1 1 1]x0 + [0 0 0]x1 [0 1 1] [1 0 0] , [1 0 0] [0] [s](x0) = [0 0 1]x0 + [0] [0 1 0] [1], [1 0 0] [d](x0) = [0 1 1]x0 [1 1 1] , [0] [0] = [0] [1] orientation: [0] [0] d(0()) = [1] >= [0] = 0() [1] [1] [1 0 0] [0] [1 0 0] [0] d(s(x)) = [0 1 1]x + [1] >= [0 1 1]x + [1] = s(s(d(x))) [1 1 1] [1] [1 1 1] [1] [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] e(s(x),y) = [1 1 1]x + [0 0 0]y + [1] >= [1 1 1]x + [0 0 0]y = e(x,d(y)) [0 1 1] [1 0 0] [1] [0 1 1] [1 0 0] problem: d(0()) -> 0() d(s(x)) -> s(s(d(x))) Matrix Interpretation Processor: dim=3 interpretation: [s](x0) = x0 , [1 0 0] [1] [d](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [0] [0] = [0] [0] orientation: [1] [0] d(0()) = [0] >= [0] = 0() [1] [0] [1 0 0] [1] [1 0 0] [1] d(s(x)) = [0 1 0]x + [0] >= [0 1 0]x + [0] = s(s(d(x))) [0 0 0] [1] [0 0 0] [1] problem: d(s(x)) -> s(s(d(x))) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [s](x0) = [0 0 1]x0 + [0] [0 1 0] [1], [1 1 1] [d](x0) = [0 1 1]x0 [0 1 1] orientation: [1 1 1] [1] [1 1 1] [0] d(s(x)) = [0 1 1]x + [1] >= [0 1 1]x + [1] = s(s(d(x))) [0 1 1] [1] [0 1 1] [1] problem: Qed