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