YES Problem: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0, [ok](x0) = 4x0 + 1, [proper](x0) = x0, [s](x0) = x0, [from](x0) = 4x0 + 1, [mark](x0) = x0, [active](x0) = 4x0, [2nd](x0) = x0, [cons](x0, x1) = 6x0 + x1 + 2 orientation: active(2nd(cons(X,cons(Y,Z)))) = 24X + 24Y + 4Z + 16 >= Y = mark(Y) active(from(X)) = 16X + 4 >= 10X + 3 = mark(cons(X,from(s(X)))) active(2nd(X)) = 4X >= 4X = 2nd(active(X)) active(cons(X1,X2)) = 24X1 + 4X2 + 8 >= 24X1 + X2 + 2 = cons(active(X1),X2) active(from(X)) = 16X + 4 >= 16X + 1 = from(active(X)) active(s(X)) = 4X >= 4X = s(active(X)) 2nd(mark(X)) = X >= X = mark(2nd(X)) cons(mark(X1),X2) = 6X1 + X2 + 2 >= 6X1 + X2 + 2 = mark(cons(X1,X2)) from(mark(X)) = 4X + 1 >= 4X + 1 = mark(from(X)) s(mark(X)) = X >= X = mark(s(X)) proper(2nd(X)) = X >= X = 2nd(proper(X)) proper(cons(X1,X2)) = 6X1 + X2 + 2 >= 6X1 + X2 + 2 = cons(proper(X1),proper(X2)) proper(from(X)) = 4X + 1 >= 4X + 1 = from(proper(X)) proper(s(X)) = X >= X = s(proper(X)) 2nd(ok(X)) = 4X + 1 >= 4X + 1 = ok(2nd(X)) cons(ok(X1),ok(X2)) = 24X1 + 4X2 + 9 >= 24X1 + 4X2 + 9 = ok(cons(X1,X2)) from(ok(X)) = 16X + 5 >= 16X + 5 = ok(from(X)) s(ok(X)) = 4X + 1 >= 4X + 1 = ok(s(X)) top(mark(X)) = 2X >= 2X = top(proper(X)) top(ok(X)) = 8X + 2 >= 8X = top(active(X)) problem: active(2nd(X)) -> 2nd(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [0] [ok](x0) = x0 + [1] [1], [proper](x0) = x0 , [1 0 0] [1] [s](x0) = [1 0 1]x0 + [0] [0 1 0] [0], [1 0 1] [from](x0) = [0 0 1]x0 [0 0 1] , [mark](x0) = x0 , [1 0 0] [0] [active](x0) = [1 0 1]x0 + [0] [0 1 0] [1], [1 0 0] [1] [2nd](x0) = [1 0 1]x0 + [0] [0 1 0] [0], [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1 [0 0 1] [0 0 0] orientation: [1 0 0] [1] [1 0 0] [1] active(2nd(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = 2nd(active(X)) [1 0 1] [1] [1 0 1] [0] [1 0 0] [1] [1 0 0] [1] active(s(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = s(active(X)) [1 0 1] [1] [1 0 1] [0] [1 0 0] [1] [1 0 0] [1] 2nd(mark(X)) = [1 0 1]X + [0] >= [1 0 1]X + [0] = mark(2nd(X)) [0 1 0] [0] [0 1 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 1]X2 >= [0 0 0]X1 + [0 0 1]X2 = mark(cons(X1,X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 1] [1 0 1] from(mark(X)) = [0 0 1]X >= [0 0 1]X = mark(from(X)) [0 0 1] [0 0 1] [1 0 0] [1] [1 0 0] [1] s(mark(X)) = [1 0 1]X + [0] >= [1 0 1]X + [0] = mark(s(X)) [0 1 0] [0] [0 1 0] [0] [1 0 0] [1] [1 0 0] [1] proper(2nd(X)) = [1 0 1]X + [0] >= [1 0 1]X + [0] = 2nd(proper(X)) [0 1 0] [0] [0 1 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 1]X2 >= [0 0 0]X1 + [0 0 1]X2 = cons(proper(X1),proper(X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 1] [1 0 1] proper(from(X)) = [0 0 1]X >= [0 0 1]X = from(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1] [1 0 0] [1] proper(s(X)) = [1 0 1]X + [0] >= [1 0 1]X + [0] = s(proper(X)) [0 1 0] [0] [0 1 0] [0] [1 0 0] [1] [1 0 0] [1] 2nd(ok(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = ok(2nd(X)) [0 1 0] [1] [0 1 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 1]X2 + [1] >= [0 0 0]X1 + [0 0 1]X2 + [1] = ok(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [0] from(ok(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = ok(from(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1] [1 0 0] [1] s(ok(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = ok(s(X)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [1 0 0] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] problem: active(2nd(X)) -> 2nd(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0 + 2, [ok](x0) = x0, [proper](x0) = x0, [s](x0) = x0 + 5, [from](x0) = x0 + 4, [mark](x0) = x0 + 5, [active](x0) = x0 + 1, [2nd](x0) = x0 + 7, [cons](x0, x1) = x0 + x1 + 4 orientation: active(2nd(X)) = X + 8 >= X + 8 = 2nd(active(X)) active(s(X)) = X + 6 >= X + 6 = s(active(X)) 2nd(mark(X)) = X + 12 >= X + 12 = mark(2nd(X)) cons(mark(X1),X2) = X1 + X2 + 9 >= X1 + X2 + 9 = mark(cons(X1,X2)) from(mark(X)) = X + 9 >= X + 9 = mark(from(X)) s(mark(X)) = X + 10 >= X + 10 = mark(s(X)) proper(2nd(X)) = X + 7 >= X + 7 = 2nd(proper(X)) proper(cons(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = cons(proper(X1),proper(X2)) proper(from(X)) = X + 4 >= X + 4 = from(proper(X)) proper(s(X)) = X + 5 >= X + 5 = s(proper(X)) 2nd(ok(X)) = X + 7 >= X + 7 = ok(2nd(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = ok(cons(X1,X2)) s(ok(X)) = X + 5 >= X + 5 = ok(s(X)) top(mark(X)) = X + 7 >= X + 2 = top(proper(X)) problem: active(2nd(X)) -> 2nd(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 2x0, [proper](x0) = 4x0, [s](x0) = x0, [from](x0) = 2x0 + 2, [mark](x0) = x0, [active](x0) = x0, [2nd](x0) = 4x0, [cons](x0, x1) = x0 + 2x1 orientation: active(2nd(X)) = 4X >= 4X = 2nd(active(X)) active(s(X)) = X >= X = s(active(X)) 2nd(mark(X)) = 4X >= 4X = mark(2nd(X)) cons(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(cons(X1,X2)) from(mark(X)) = 2X + 2 >= 2X + 2 = mark(from(X)) s(mark(X)) = X >= X = mark(s(X)) proper(2nd(X)) = 16X >= 16X = 2nd(proper(X)) proper(cons(X1,X2)) = 4X1 + 8X2 >= 4X1 + 8X2 = cons(proper(X1),proper(X2)) proper(from(X)) = 8X + 8 >= 8X + 2 = from(proper(X)) proper(s(X)) = 4X >= 4X = s(proper(X)) 2nd(ok(X)) = 8X >= 8X = ok(2nd(X)) cons(ok(X1),ok(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = ok(cons(X1,X2)) s(ok(X)) = 2X >= 2X = ok(s(X)) problem: active(2nd(X)) -> 2nd(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [ok](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [proper](x0) = x0 , [s](x0) = x0 , [1 0 0] [from](x0) = [0 0 0]x0 [0 0 0] , [mark](x0) = x0 , [1 1 0] [active](x0) = [1 0 1]x0 [0 0 1] , [1 0 1] [0] [2nd](x0) = [0 1 0]x0 + [1] [0 0 0] [1], [1 0 1] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [1] orientation: [1 1 1] [1] [1 1 1] [0] active(2nd(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = 2nd(active(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1 1 0] active(s(X)) = [1 0 1]X >= [1 0 1]X = s(active(X)) [0 0 1] [0 0 1] [1 0 1] [0] [1 0 1] [0] 2nd(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(2nd(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] [0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = mark(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] from(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(from(X)) [0 0 0] [0 0 0] s(mark(X)) = X >= X = mark(s(X)) [1 0 1] [0] [1 0 1] [0] proper(2nd(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = 2nd(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] [0] [1 0 1] [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 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] proper(s(X)) = X >= X = s(proper(X)) [1 0 1] [1] [1 0 1] [1] 2nd(ok(X)) = [0 0 1]X + [1] >= [0 0 0]X + [1] = ok(2nd(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 1] [1] [1 0 1] [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 1] [0] [1 0 1] [0] s(ok(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = ok(s(X)) [0 0 0] [1] [0 0 0] [1] problem: active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 4x0 + 4, [proper](x0) = x0, [s](x0) = 4x0 + 4, [from](x0) = x0 + 2, [mark](x0) = x0, [active](x0) = x0, [2nd](x0) = 2x0, [cons](x0, x1) = 2x0 + 2x1 + 4 orientation: active(s(X)) = 4X + 4 >= 4X + 4 = s(active(X)) 2nd(mark(X)) = 2X >= 2X = mark(2nd(X)) cons(mark(X1),X2) = 2X1 + 2X2 + 4 >= 2X1 + 2X2 + 4 = mark(cons(X1,X2)) from(mark(X)) = X + 2 >= X + 2 = mark(from(X)) s(mark(X)) = 4X + 4 >= 4X + 4 = mark(s(X)) proper(2nd(X)) = 2X >= 2X = 2nd(proper(X)) proper(cons(X1,X2)) = 2X1 + 2X2 + 4 >= 2X1 + 2X2 + 4 = cons(proper(X1),proper(X2)) proper(s(X)) = 4X + 4 >= 4X + 4 = s(proper(X)) 2nd(ok(X)) = 8X + 8 >= 8X + 4 = ok(2nd(X)) cons(ok(X1),ok(X2)) = 8X1 + 8X2 + 20 >= 8X1 + 8X2 + 20 = ok(cons(X1,X2)) s(ok(X)) = 16X + 20 >= 16X + 20 = ok(s(X)) problem: active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [0] [proper](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 1] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 0] [1], [1 0 1] [0] [from](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [1 0 1] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 1 0] [active](x0) = [1 1 0]x0 [0 0 1] , [1 0 1] [0] [2nd](x0) = [0 1 0]x0 + [1] [0 0 0] [1], [1 0 1] [1 0 0] [0] [cons](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [1] orientation: [1 1 1] [1] [1 1 1] [0] active(s(X)) = [1 1 1]X + [1] >= [1 1 0]X + [1] = s(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [1] 2nd(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(2nd(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [1] cons(mark(X1),X2) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1] [1 0 1] [1] from(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(from(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [1] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(s(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1] [1 1 0] [1] proper(2nd(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = 2nd(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1 1 0] [1] [1 1 0] [1 1 0] [1] proper(cons(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 1] [1] [1 1 0] [1] proper(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] [0] [1 0 1] [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 + [0] = ok(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [0] [1 0 1] [0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = ok(s(X)) [0 0 0] [1] [0 0 0] [1] problem: 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [s](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [from](x0) = [0 0 0]x0 [0 0 1] , [0] [mark](x0) = x0 + [0] [1], [1 0 1] [2nd](x0) = [0 0 0]x0 [0 1 1] , [1 0 1] [1 1 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] orientation: [1 0 1] [1] [1 0 1] [0] 2nd(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(2nd(X)) [0 1 1] [1] [0 1 1] [1] [1 0 1] [1 1 0] [1] [1 0 1] [1 1 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 0 1] [1] [1 0 1] [0] from(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(from(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1] [1 0 1] [0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1 0 0] proper(2nd(X)) = [0 0 0]X >= [0 0 0]X = 2nd(proper(X)) [0 0 0] [0 0 0] [1 0 1] [1 1 0] [1 0 0] [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 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 0] [1] [1 0 1] [1 1 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 0 1] [0] s(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(s(X)) [0 0 1] [1] [0 0 1] [1] problem: proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 1] , [0] [2nd](x0) = x0 + [0] [1], [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 1] orientation: [1 0 1] [1] [1 0 1] [0] proper(2nd(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = 2nd(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 1] [0 0 1] [0] [0 0 1] [0 0 1] [1 0 1] [0] [1 0 1] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X = s(proper(X)) [0 0 1] [0] [0 0 1] problem: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [proper](x0) = [0 0 1]x0 [0 1 0] , [1 1 0] [s](x0) = [0 0 1]x0 [0 1 0] , [0] [cons](x0, x1) = x0 + x1 + [1] [1] orientation: [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [0] proper(cons(X1,X2)) = [0 0 1]X1 + [0 0 1]X2 + [1] >= [0 0 1]X1 + [0 0 1]X2 + [1] = cons(proper(X1),proper(X2)) [0 1 0] [0 1 0] [1] [0 1 0] [0 1 0] [1] [1 1 1] [1 1 1] proper(s(X)) = [0 1 0]X >= [0 1 0]X = s(proper(X)) [0 0 1] [0 0 1] problem: proper(s(X)) -> s(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [proper](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 1] [1] orientation: [1 0 1] [1] [1 0 1] [0] proper(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(proper(X)) [0 0 1] [1] [0 0 1] [1] problem: Qed