YES Problem: active(f(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) mark(a()) -> active(a()) mark(b()) -> active(b()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) Proof: Matrix Interpretation Processor: dim=2 interpretation: [1 0] [0] [mark](x0) = [2 2]x0 + [1], [0] [b] = [0], [1 0] [0] [active](x0) = [0 2]x0 + [1], [2 0] [2 0] [1 2] [f](x0, x1, x2) = [0 0]x0 + [0 0]x1 + [2 0]x2, [0] [a] = [0] orientation: [3 2] [0] [2 0] [0] active(f(a(),X,X)) = [4 0]X + [1] >= [4 0]X + [1] = mark(f(X,b(),b())) [0] [0] active(b()) = [1] >= [1] = mark(a()) [2 0] [2 0] [1 2] [0] [2 0] [2 0] [1 2] [0] mark(f(X1,X2,X3)) = [4 0]X1 + [4 0]X2 + [6 4]X3 + [1] >= [0 0]X1 + [0 0]X2 + [4 0]X3 + [1] = active(f(X1,mark(X2),X3)) [0] [0] mark(a()) = [1] >= [1] = active(a()) [0] [0] mark(b()) = [1] >= [1] = active(b()) [2 0] [2 0] [1 2] [2 0] [2 0] [1 2] f(mark(X1),X2,X3) = [0 0]X1 + [0 0]X2 + [2 0]X3 >= [0 0]X1 + [0 0]X2 + [2 0]X3 = f(X1,X2,X3) [2 0] [2 0] [1 2] [2 0] [2 0] [1 2] f(X1,mark(X2),X3) = [0 0]X1 + [0 0]X2 + [2 0]X3 >= [0 0]X1 + [0 0]X2 + [2 0]X3 = f(X1,X2,X3) [2 0] [2 0] [5 4] [2] [2 0] [2 0] [1 2] f(X1,X2,mark(X3)) = [0 0]X1 + [0 0]X2 + [2 0]X3 + [0] >= [0 0]X1 + [0 0]X2 + [2 0]X3 = f(X1,X2,X3) [2 0] [2 0] [1 2] [2 0] [2 0] [1 2] f(active(X1),X2,X3) = [0 0]X1 + [0 0]X2 + [2 0]X3 >= [0 0]X1 + [0 0]X2 + [2 0]X3 = f(X1,X2,X3) [2 0] [2 0] [1 2] [2 0] [2 0] [1 2] f(X1,active(X2),X3) = [0 0]X1 + [0 0]X2 + [2 0]X3 >= [0 0]X1 + [0 0]X2 + [2 0]X3 = f(X1,X2,X3) [2 0] [2 0] [1 4] [2] [2 0] [2 0] [1 2] f(X1,X2,active(X3)) = [0 0]X1 + [0 0]X2 + [2 0]X3 + [0] >= [0 0]X1 + [0 0]X2 + [2 0]X3 = f(X1,X2,X3) problem: active(f(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) mark(a()) -> active(a()) mark(b()) -> active(b()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) Matrix Interpretation Processor: dim=2 interpretation: [2 0] [0] [mark](x0) = [0 1]x0 + [1], [0] [b] = [0], [0] [active](x0) = x0 + [1], [1 1] [1 0] [1 2] [f](x0, x1, x2) = [0 0]x0 + [0 0]x1 + [1 0]x2, [0] [a] = [0] orientation: [2 2] [0] [2 2] [0] active(f(a(),X,X)) = [1 0]X + [1] >= [0 0]X + [1] = mark(f(X,b(),b())) [0] [0] active(b()) = [1] >= [1] = mark(a()) [2 2] [2 0] [2 4] [0] [1 1] [2 0] [1 2] [0] mark(f(X1,X2,X3)) = [0 0]X1 + [0 0]X2 + [1 0]X3 + [1] >= [0 0]X1 + [0 0]X2 + [1 0]X3 + [1] = active(f(X1,mark(X2),X3)) [0] [0] mark(a()) = [1] >= [1] = active(a()) [0] [0] mark(b()) = [1] >= [1] = active(b()) [2 1] [1 0] [1 2] [1] [1 1] [1 0] [1 2] f(mark(X1),X2,X3) = [0 0]X1 + [0 0]X2 + [1 0]X3 + [0] >= [0 0]X1 + [0 0]X2 + [1 0]X3 = f(X1,X2,X3) [1 1] [2 0] [1 2] [1 1] [1 0] [1 2] f(X1,mark(X2),X3) = [0 0]X1 + [0 0]X2 + [1 0]X3 >= [0 0]X1 + [0 0]X2 + [1 0]X3 = f(X1,X2,X3) [1 1] [1 0] [1 2] [1] [1 1] [1 0] [1 2] f(active(X1),X2,X3) = [0 0]X1 + [0 0]X2 + [1 0]X3 + [0] >= [0 0]X1 + [0 0]X2 + [1 0]X3 = f(X1,X2,X3) [1 1] [1 0] [1 2] [1 1] [1 0] [1 2] f(X1,active(X2),X3) = [0 0]X1 + [0 0]X2 + [1 0]X3 >= [0 0]X1 + [0 0]X2 + [1 0]X3 = f(X1,X2,X3) problem: active(f(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) mark(a()) -> active(a()) mark(b()) -> active(b()) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) Matrix Interpretation Processor: dim=2 interpretation: [1 0] [0] [mark](x0) = [0 0]x0 + [2], [0] [b] = [0], [1 0] [0] [active](x0) = [0 0]x0 + [2], [2 2] [2 0] [1 2] [f](x0, x1, x2) = [0 0]x0 + [0 0]x1 + [0 0]x2, [0] [a] = [2] orientation: [3 2] [4] [2 2] [0] active(f(a(),X,X)) = [0 0]X + [2] >= [0 0]X + [2] = mark(f(X,b(),b())) [0] [0] active(b()) = [2] >= [2] = mark(a()) [2 2] [2 0] [1 2] [0] [2 2] [2 0] [1 2] [0] mark(f(X1,X2,X3)) = [0 0]X1 + [0 0]X2 + [0 0]X3 + [2] >= [0 0]X1 + [0 0]X2 + [0 0]X3 + [2] = active(f(X1,mark(X2),X3)) [0] [0] mark(a()) = [2] >= [2] = active(a()) [0] [0] mark(b()) = [2] >= [2] = active(b()) [2 2] [2 0] [1 2] [2 2] [2 0] [1 2] f(X1,mark(X2),X3) = [0 0]X1 + [0 0]X2 + [0 0]X3 >= [0 0]X1 + [0 0]X2 + [0 0]X3 = f(X1,X2,X3) [2 2] [2 0] [1 2] [2 2] [2 0] [1 2] f(X1,active(X2),X3) = [0 0]X1 + [0 0]X2 + [0 0]X3 >= [0 0]X1 + [0 0]X2 + [0 0]X3 = f(X1,X2,X3) problem: active(b()) -> mark(a()) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) mark(a()) -> active(a()) mark(b()) -> active(b()) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [b] = 4, [active](x0) = x0, [f](x0, x1, x2) = 4x0 + x1 + 4x2, [a] = 0 orientation: active(b()) = 4 >= 0 = mark(a()) mark(f(X1,X2,X3)) = 4X1 + X2 + 4X3 >= 4X1 + X2 + 4X3 = active(f(X1,mark(X2),X3)) mark(a()) = 0 >= 0 = active(a()) mark(b()) = 4 >= 4 = active(b()) f(X1,mark(X2),X3) = 4X1 + X2 + 4X3 >= 4X1 + X2 + 4X3 = f(X1,X2,X3) f(X1,active(X2),X3) = 4X1 + X2 + 4X3 >= 4X1 + X2 + 4X3 = f(X1,X2,X3) problem: mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) mark(a()) -> active(a()) mark(b()) -> active(b()) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [mark](x0) = [0 0 1]x0 [0 1 0] , [0] [b] = [0] [0], [1 0 0] [active](x0) = [0 0 1]x0 [0 1 0] , [1 0 0] [1 0 0] [1 0 1] [0] [f](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [1] [0 0 0] [0 1 1] [0 0 0] [0], [0] [a] = [0] [0] orientation: [1 0 0] [1 1 1] [1 0 1] [1] [1 0 0] [1 1 1] [1 0 1] [0] mark(f(X1,X2,X3)) = [0 0 0]X1 + [0 1 1]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 1 1]X2 + [0 0 0]X3 + [0] = active(f(X1,mark(X2),X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [0] [0] mark(a()) = [0] >= [0] = active(a()) [0] [0] [0] [0] mark(b()) = [0] >= [0] = active(b()) [0] [0] [1 0 0] [1 1 1] [1 0 1] [0] [1 0 0] [1 0 0] [1 0 1] [0] f(X1,mark(X2),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] = f(X1,X2,X3) [0 0 0] [0 1 1] [0 0 0] [0] [0 0 0] [0 1 1] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [1 0 1] [0] f(X1,active(X2),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] = f(X1,X2,X3) [0 0 0] [0 1 1] [0 0 0] [0] [0 0 0] [0 1 1] [0 0 0] [0] problem: mark(a()) -> active(a()) mark(b()) -> active(b()) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [mark](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [0] [b] = [0] [0], [1 0 1] [active](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1 1 0] [1 0 0] [f](x0, x1, x2) = [0 0 0]x0 + [1 0 0]x1 + [0 0 0]x2 [0 0 0] [0 1 0] [0 0 0] , [0] [a] = [0] [1] orientation: [1] [1] mark(a()) = [0] >= [0] = active(a()) [1] [0] [1] [0] mark(b()) = [0] >= [0] = active(b()) [1] [0] [1 0 0] [1 1 0] [1 0 0] [1] [1 0 0] [1 1 0] [1 0 0] f(X1,mark(X2),X3) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 = f(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 1 1] [1 0 0] [1 0 0] [1 1 0] [1 0 0] f(X1,active(X2),X3) = [0 0 0]X1 + [1 0 1]X2 + [0 0 0]X3 >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 = f(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [0 0 0] problem: mark(a()) -> active(a()) f(X1,active(X2),X3) -> f(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [mark](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [1 0 0] [active](x0) = [0 1 0]x0 [0 1 0] , [1 0 0] [1 0 0] [1 0 0] [f](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [0] [a] = [1] [1] orientation: [1] [0] mark(a()) = [1] >= [1] = active(a()) [1] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] f(X1,active(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = f(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: f(X1,active(X2),X3) -> f(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [active](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 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 0] [0 0 0] [0 0 0] orientation: [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] f(X1,active(X2),X3) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 = f(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] problem: Qed