YES Problem: active(f(b(),X,c())) -> mark(f(X,c(),X)) active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=2 interpretation: [2 0] [2] [top](x0) = [1 1]x0 + [0], [ok](x0) = x0, [proper](x0) = x0, [1 1] [mark](x0) = [0 0]x0, [1 0] [active](x0) = [0 0]x0, [1 2] [3 2] [2 0] [f](x0, x1, x2) = [0 0]x0 + [0 1]x1 + [0 0]x2, [1] [c] = [0], [0] [b] = [1] orientation: [3 2] [4] [3 2] [3] active(f(b(),X,c())) = [0 0]X + [0] >= [0 0]X + [0] = mark(f(X,c(),X)) [1] [1] active(c()) = [0] >= [0] = mark(b()) [1 2] [3 2] [2 0] [1 2] [3 0] [2 0] active(f(X1,X2,X3)) = [0 0]X1 + [0 0]X2 + [0 0]X3 >= [0 0]X1 + [0 0]X2 + [0 0]X3 = f(X1,active(X2),X3) [1 2] [3 3] [2 0] [1 2] [3 3] [2 0] f(X1,mark(X2),X3) = [0 0]X1 + [0 0]X2 + [0 0]X3 >= [0 0]X1 + [0 0]X2 + [0 0]X3 = mark(f(X1,X2,X3)) [1 2] [3 2] [2 0] [1 2] [3 2] [2 0] proper(f(X1,X2,X3)) = [0 0]X1 + [0 1]X2 + [0 0]X3 >= [0 0]X1 + [0 1]X2 + [0 0]X3 = f(proper(X1),proper(X2),proper(X3)) [0] [0] proper(b()) = [1] >= [1] = ok(b()) [1] [1] proper(c()) = [0] >= [0] = ok(c()) [1 2] [3 2] [2 0] [1 2] [3 2] [2 0] f(ok(X1),ok(X2),ok(X3)) = [0 0]X1 + [0 1]X2 + [0 0]X3 >= [0 0]X1 + [0 1]X2 + [0 0]X3 = ok(f(X1,X2,X3)) [2 2] [2] [2 0] [2] top(mark(X)) = [1 1]X + [0] >= [1 1]X + [0] = top(proper(X)) [2 0] [2] [2 0] [2] top(ok(X)) = [1 1]X + [0] >= [1 0]X + [0] = top(active(X)) problem: active(c()) -> mark(b()) active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [ok](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [0] [proper](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [active](x0) = x0 , [1 1 0] [1 0 0] [1 0 0] [f](x0, x1, x2) = [0 0 0]x0 + [1 0 0]x1 + [0 0 0]x2 [0 0 1] [0 0 0] [0 0 0] , [1] [c] = [0] [0], [0] [b] = [0] [0] orientation: [1] [0] active(c()) = [0] >= [0] = mark(b()) [0] [0] [1 1 0] [1 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] active(f(X1,X2,X3)) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 = f(X1,active(X2),X3) [0 0 1] [0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] f(X1,mark(X2),X3) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 = mark(f(X1,X2,X3)) [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 0 0] [0] [1 1 0] [1 0 0] [1 0 0] [0] proper(f(X1,X2,X3)) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = f(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [0] [0] proper(b()) = [0] >= [0] = ok(b()) [1] [0] [1] [1] proper(c()) = [0] >= [0] = ok(c()) [1] [0] [1 1 0] [1 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] f(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 = ok(f(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [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] [1 0 0] [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(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [top](x0) = [1 0 1]x0 [1 0 0] , [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [mark](x0) = [0 1 1]x0 + [1] [0 0 1] [0], [1 0 0] [active](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 1] [1 0 1] [1] [f](x0, x1, x2) = [0 0 0]x0 + [0 1 0]x1 + [0 0 0]x2 + [0] [0 0 0] [0 0 0] [0 0 0] [0], [0] [c] = [0] [0], [1] [b] = [0] [0] orientation: [1 0 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 0] [1 0 1] [1] active(f(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = f(X1,active(X2),X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] f(X1,mark(X2),X3) = [0 0 0]X1 + [0 1 1]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] = mark(f(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 0] [1 0 0] [1] proper(f(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = f(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1] [1] proper(b()) = [0] >= [0] = ok(b()) [0] [0] [0] [0] proper(c()) = [0] >= [0] = ok(c()) [0] [0] [1 0 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 1] [1 0 1] [1] f(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = ok(f(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 1] [1] [1 0 0] top(mark(X)) = [1 0 1]X + [0] >= [1 0 0]X = top(proper(X)) [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] top(ok(X)) = [1 0 1]X >= [1 0 0]X = top(active(X)) [1 0 0] [1 0 0] problem: active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0, [ok](x0) = 7x0 + 1, [proper](x0) = 7x0 + 1, [mark](x0) = x0, [active](x0) = x0, [f](x0, x1, x2) = 2x0 + 4x1 + x2 + 1, [c] = 0, [b] = 2 orientation: active(f(X1,X2,X3)) = 2X1 + 4X2 + X3 + 1 >= 2X1 + 4X2 + X3 + 1 = f(X1,active(X2),X3) f(X1,mark(X2),X3) = 2X1 + 4X2 + X3 + 1 >= 2X1 + 4X2 + X3 + 1 = mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) = 14X1 + 28X2 + 7X3 + 8 >= 14X1 + 28X2 + 7X3 + 8 = f(proper(X1),proper(X2),proper(X3)) proper(b()) = 15 >= 15 = ok(b()) proper(c()) = 1 >= 1 = ok(c()) f(ok(X1),ok(X2),ok(X3)) = 14X1 + 28X2 + 7X3 + 8 >= 14X1 + 28X2 + 7X3 + 8 = ok(f(X1,X2,X3)) top(ok(X)) = 7X + 1 >= X = top(active(X)) problem: active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) Matrix Interpretation Processor: dim=3 interpretation: [ok](x0) = x0 , [1 0 1] [proper](x0) = [0 1 0]x0 [0 1 0] , [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [active](x0) = [0 1 0]x0 [0 0 1] , [1 0 1] [1 1 0] [0] [f](x0, x1, x2) = [0 1 0]x0 + x1 + [0 1 0]x2 + [1] [0 1 0] [0 0 1] [1], [0] [c] = [0] [0], [0] [b] = [0] [0] orientation: [1 1 1] [1 0 1] [1 1 1] [1] [1 0 1] [1 0 1] [1 1 0] [0] active(f(X1,X2,X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = f(X1,active(X2),X3) [0 1 0] [0 0 1] [0 0 1] [1] [0 1 0] [0 0 1] [0 0 1] [1] [1 0 1] [1 0 0] [1 1 0] [0] [1 0 1] [1 0 0] [1 1 0] [0] f(X1,mark(X2),X3) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = mark(f(X1,X2,X3)) [0 1 0] [0 0 0] [0 0 1] [1] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 1] [1 0 1] [1 1 1] [1] [1 1 1] [1 0 1] [1 1 1] [0] proper(f(X1,X2,X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = f(proper(X1),proper(X2),proper(X3)) [0 1 0] [0 1 0] [0 1 0] [1] [0 1 0] [0 1 0] [0 1 0] [1] [0] [0] proper(b()) = [0] >= [0] = ok(b()) [0] [0] [0] [0] proper(c()) = [0] >= [0] = ok(c()) [0] [0] [1 0 1] [1 1 0] [0] [1 0 1] [1 1 0] [0] f(ok(X1),ok(X2),ok(X3)) = [0 1 0]X1 + X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + X2 + [0 1 0]X3 + [1] = ok(f(X1,X2,X3)) [0 1 0] [0 0 1] [1] [0 1 0] [0 0 1] [1] problem: f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(b()) -> ok(b()) proper(c()) -> ok(c()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 1 1]x0 [0 0 1] , [1 0 0] [1] [proper](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 1 0] [1 0 1] [1 0 0] [f](x0, x1, x2) = [0 0 0]x0 + [0 1 0]x1 + [0 0 0]x2 [0 0 0] [0 0 1] [0 0 0] , [0] [c] = [0] [0], [0] [b] = [0] [0] orientation: [1 1 0] [1 0 1] [1 0 0] [1] [1 1 0] [1 0 1] [1 0 0] [0] f(X1,mark(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = mark(f(X1,X2,X3)) [0 0 0] [0 0 1] [0 0 0] [1] [0 0 0] [0 0 1] [0 0 0] [1] [1] [0] proper(b()) = [0] >= [0] = ok(b()) [0] [0] [1] [0] proper(c()) = [0] >= [0] = ok(c()) [0] [0] [1 1 1] [1 0 1] [1 0 0] [1 1 0] [1 0 1] [1 0 0] f(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 1 1]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 1 1]X2 + [0 0 0]X3 = ok(f(X1,X2,X3)) [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 1] [0 0 0] problem: f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [ok](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 1 0] [1 0 0] [f](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 1 0]x2 [1 0 0] [0 0 0] [0 0 0] orientation: [1 1 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [0] f(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = ok(f(X1,X2,X3)) [1 1 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] problem: Qed