YES(?,O(n^2)) Problem: f(+(x,0())) -> f(x) +(x,+(y,z)) -> +(+(x,y),z) Proof: Complexity Transformation Processor: strict: f(+(x,0())) -> f(x) +(x,+(y,z)) -> +(+(x,y),z) weak: Matrix Interpretation Processor: dimension: 1 max_matrix: 1 interpretation: [f](x0) = x0 + 1, [+](x0, x1) = x0 + x1, [0] = 1 orientation: f(+(x,0())) = x + 2 >= x + 1 = f(x) +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) problem: strict: +(x,+(y,z)) -> +(+(x,y),z) weak: f(+(x,0())) -> f(x) Matrix Interpretation Processor: dimension: 2 max_matrix: [1 1] [0 1] interpretation: [1 0] [f](x0) = [0 0]x0, [1 1] [0] [+](x0, x1) = x0 + [0 1]x1 + [1], [0] [0] = [0] orientation: [1 1] [1 2] [1] [1 1] [1 1] [0] +(x,+(y,z)) = x + [0 1]y + [0 1]z + [2] >= x + [0 1]y + [0 1]z + [2] = +(+(x,y),z) [1 0] [1 0] f(+(x,0())) = [0 0]x >= [0 0]x = f(x) problem: strict: weak: +(x,+(y,z)) -> +(+(x,y),z) f(+(x,0())) -> f(x) Qed