YES
Problem:
strict:
f(b(x),y) -> f(x,b(y))
f(x,a(y)) -> f(a(x),y)
weak:
f(x,y) -> f(a(x),y)
f(x,y) -> f(x,b(y))
Proof:
Matrix Interpretation Processor: dim=5
interpretation:
[1 0 1 0 1] [1 0 0 0 0] [0]
[0 0 0 0 0] [0 0 1 1 0] [0]
[f](x0, x1) = [0 0 0 0 1]x0 + [0 0 0 0 0]x1 + [1]
[0 0 0 0 0] [0 0 0 0 0] [0]
[0 0 0 0 0] [0 0 0 0 0] [0],
[1 0 0 0 0] [0]
[0 0 0 0 0] [0]
[b](x0) = [0 0 1 1 0]x0 + [0]
[0 0 0 0 0] [0]
[0 0 1 0 1] [1],
[1 0 0 0 0]
[0 0 0 0 0]
[a](x0) = [0 0 0 0 0]x0
[0 1 1 1 1]
[0 0 0 0 0]
orientation:
[1 0 2 1 1] [1 0 0 0 0] [1] [1 0 1 0 1] [1 0 0 0 0] [0]
[0 0 0 0 0] [0 0 1 1 0] [0] [0 0 0 0 0] [0 0 1 1 0] [0]
f(b(x),y) = [0 0 1 0 1]x + [0 0 0 0 0]y + [2] >= [0 0 0 0 1]x + [0 0 0 0 0]y + [1] = f(x,b(y))
[0 0 0 0 0] [0 0 0 0 0] [0] [0 0 0 0 0] [0 0 0 0 0] [0]
[0 0 0 0 0] [0 0 0 0 0] [0] [0 0 0 0 0] [0 0 0 0 0] [0]
[1 0 1 0 1] [1 0 0 0 0] [0] [1 0 0 0 0] [1 0 0 0 0] [0]
[0 0 0 0 0] [0 1 1 1 1] [0] [0 0 0 0 0] [0 0 1 1 0] [0]
f(x,a(y)) = [0 0 0 0 1]x + [0 0 0 0 0]y + [1] >= [0 0 0 0 0]x + [0 0 0 0 0]y + [1] = f(a(x),y)
[0 0 0 0 0] [0 0 0 0 0] [0] [0 0 0 0 0] [0 0 0 0 0] [0]
[0 0 0 0 0] [0 0 0 0 0] [0] [0 0 0 0 0] [0 0 0 0 0] [0]
[1 0 1 0 1] [1 0 0 0 0] [0] [1 0 0 0 0] [1 0 0 0 0] [0]
[0 0 0 0 0] [0 0 1 1 0] [0] [0 0 0 0 0] [0 0 1 1 0] [0]
f(x,y) = [0 0 0 0 1]x + [0 0 0 0 0]y + [1] >= [0 0 0 0 0]x + [0 0 0 0 0]y + [1] = f(a(x),y)
[0 0 0 0 0] [0 0 0 0 0] [0] [0 0 0 0 0] [0 0 0 0 0] [0]
[0 0 0 0 0] [0 0 0 0 0] [0] [0 0 0 0 0] [0 0 0 0 0] [0]
[1 0 1 0 1] [1 0 0 0 0] [0] [1 0 1 0 1] [1 0 0 0 0] [0]
[0 0 0 0 0] [0 0 1 1 0] [0] [0 0 0 0 0] [0 0 1 1 0] [0]
f(x,y) = [0 0 0 0 1]x + [0 0 0 0 0]y + [1] >= [0 0 0 0 1]x + [0 0 0 0 0]y + [1] = f(x,b(y))
[0 0 0 0 0] [0 0 0 0 0] [0] [0 0 0 0 0] [0 0 0 0 0] [0]
[0 0 0 0 0] [0 0 0 0 0] [0] [0 0 0 0 0] [0 0 0 0 0] [0]
problem:
strict:
f(x,a(y)) -> f(a(x),y)
weak:
f(x,y) -> f(a(x),y)
f(x,y) -> f(x,b(y))
Matrix Interpretation Processor: dim=5
interpretation:
[1 0 0 1 0] [1 1 1 0 1]
[1 0 0 1 0] [0 0 0 0 0]
[f](x0, x1) = [0 0 0 0 0]x0 + [1 0 0 0 0]x1
[0 0 0 0 0] [0 0 0 0 0]
[0 0 0 1 0] [0 1 0 1 1] ,
[1 0 0 0 0]
[0 0 0 0 0]
[b](x0) = [0 1 0 0 0]x0
[0 1 0 1 0]
[0 0 0 0 0] ,
[1 0 0 1 0] [0]
[0 0 0 1 1] [0]
[a](x0) = [0 0 0 0 0]x0 + [1]
[0 0 0 0 0] [0]
[0 1 1 1 0] [0]
orientation:
[1 0 0 1 0] [1 1 1 3 1] [1] [1 0 0 1 0] [1 1 1 0 1]
[1 0 0 1 0] [0 0 0 0 0] [0] [1 0 0 1 0] [0 0 0 0 0]
f(x,a(y)) = [0 0 0 0 0]x + [1 0 0 1 0]y + [0] >= [0 0 0 0 0]x + [1 0 0 0 0]y = f(a(x),y)
[0 0 0 0 0] [0 0 0 0 0] [0] [0 0 0 0 0] [0 0 0 0 0]
[0 0 0 1 0] [0 1 1 2 1] [0] [0 0 0 0 0] [0 1 0 1 1]
[1 0 0 1 0] [1 1 1 0 1] [1 0 0 1 0] [1 1 1 0 1]
[1 0 0 1 0] [0 0 0 0 0] [1 0 0 1 0] [0 0 0 0 0]
f(x,y) = [0 0 0 0 0]x + [1 0 0 0 0]y >= [0 0 0 0 0]x + [1 0 0 0 0]y = f(a(x),y)
[0 0 0 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 0 0 0 0] [0 1 0 1 1]
[1 0 0 1 0] [1 1 1 0 1] [1 0 0 1 0] [1 1 0 0 0]
[1 0 0 1 0] [0 0 0 0 0] [1 0 0 1 0] [0 0 0 0 0]
f(x,y) = [0 0 0 0 0]x + [1 0 0 0 0]y >= [0 0 0 0 0]x + [1 0 0 0 0]y = f(x,b(y))
[0 0 0 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 0 0 1 0] [0 1 0 1 0]
problem:
strict:
weak:
f(x,y) -> f(a(x),y)
f(x,y) -> f(x,b(y))
Qed