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