YES(?,O(n^2)) Problem: +(*(x,y),*(a(),y)) -> *(+(x,a()),y) *(*(x,y),z) -> *(x,*(y,z)) Proof: Complexity Transformation Processor: strict: +(*(x,y),*(a(),y)) -> *(+(x,a()),y) *(*(x,y),z) -> *(x,*(y,z)) weak: Matrix Interpretation Processor: dimension: 1 max_matrix: 1 interpretation: [+](x0, x1) = x0 + x1, [a] = 0, [*](x0, x1) = x0 + x1 + 1 orientation: +(*(x,y),*(a(),y)) = x + 2y + 2 >= x + y + 1 = *(+(x,a()),y) *(*(x,y),z) = x + y + z + 2 >= x + y + z + 2 = *(x,*(y,z)) problem: strict: *(*(x,y),z) -> *(x,*(y,z)) weak: +(*(x,y),*(a(),y)) -> *(+(x,a()),y) Matrix Interpretation Processor: dimension: 2 max_matrix: [1 1] [0 1] interpretation: [1 1] [0] [+](x0, x1) = [0 0]x0 + x1 + [1], [1] [a] = [1], [1 1] [0] [*](x0, x1) = [0 1]x0 + x1 + [1] orientation: [1 2] [1 1] [1] [1 1] [1 1] [0] *(*(x,y),z) = [0 1]x + [0 1]y + z + [2] >= [0 1]x + [0 1]y + z + [2] = *(x,*(y,z)) [1 2] [2 1] [3] [1 1] [3] +(*(x,y),*(a(),y)) = [0 0]x + [0 1]y + [3] >= [0 0]x + y + [3] = *(+(x,a()),y) problem: strict: weak: *(*(x,y),z) -> *(x,*(y,z)) +(*(x,y),*(a(),y)) -> *(+(x,a()),y) Qed