Problem: rev(a()) -> a() rev(b()) -> b() rev(++(x,y)) -> ++(rev(y),rev(x)) rev(++(x,x)) -> rev(x) Proof: Complexity Transformation Processor: strict: rev(a()) -> a() rev(b()) -> b() rev(++(x,y)) -> ++(rev(y),rev(x)) rev(++(x,x)) -> rev(x) weak: Matrix Interpretation Processor: dimension: 1 max_matrix: 1 interpretation: [++](x0, x1) = x0 + x1 + 1, [b] = 4, [rev](x0) = x0 + 6, [a] = 12 orientation: rev(a()) = 18 >= 12 = a() rev(b()) = 10 >= 4 = b() rev(++(x,y)) = x + y + 7 >= x + y + 13 = ++(rev(y),rev(x)) rev(++(x,x)) = 2x + 7 >= x + 6 = rev(x) problem: strict: rev(++(x,y)) -> ++(rev(y),rev(x)) weak: rev(a()) -> a() rev(b()) -> b() rev(++(x,x)) -> rev(x) Matrix Interpretation Processor: dimension: 3 max_matrix: [1 1 1] [0 1 1] [0 0 0] interpretation: [1 0 0] [1 0 1] [0] [++](x0, x1) = [0 1 1]x0 + [0 1 1]x1 + [1] [0 0 0] [0 0 0] [0], [0] [b] = [0] [0], [1 1 0] [rev](x0) = [0 1 1]x0 [0 0 0] , [0] [a] = [0] [0] orientation: [1 1 1] [1 1 2] [1] [1 1 0] [1 1 0] [0] rev(++(x,y)) = [0 1 1]x + [0 1 1]y + [1] >= [0 1 1]x + [0 1 1]y + [1] = ++(rev(y),rev(x)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] rev(a()) = [0] >= [0] = a() [0] [0] [0] [0] rev(b()) = [0] >= [0] = b() [0] [0] [2 2 3] [1] [1 1 0] rev(++(x,x)) = [0 2 2]x + [1] >= [0 1 1]x = rev(x) [0 0 0] [0] [0 0 0] problem: strict: weak: rev(++(x,y)) -> ++(rev(y),rev(x)) rev(a()) -> a() rev(b()) -> b() rev(++(x,x)) -> rev(x) Qed