YES(O(1),O(n^2)) 121.02/60.07 YES(O(1),O(n^2)) 121.02/60.07 121.02/60.07 We are left with following problem, upon which TcT provides the 121.02/60.07 certificate YES(O(1),O(n^2)). 121.02/60.07 121.02/60.07 Strict Trs: 121.02/60.07 { +(*(x, y), *(a(), y)) -> *(+(x, a()), y) 121.02/60.07 , *(*(x, y), z) -> *(x, *(y, z)) } 121.02/60.07 Obligation: 121.02/60.07 derivational complexity 121.02/60.07 Answer: 121.02/60.07 YES(O(1),O(n^2)) 121.02/60.07 121.02/60.07 We use the processor 'matrix interpretation of dimension 1' to 121.02/60.07 orient following rules strictly. 121.02/60.07 121.02/60.07 Trs: { +(*(x, y), *(a(), y)) -> *(+(x, a()), y) } 121.02/60.07 121.02/60.07 The induced complexity on above rules (modulo remaining rules) is 121.02/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 121.02/60.07 component(s). 121.02/60.07 121.02/60.07 Sub-proof: 121.02/60.07 ---------- 121.02/60.07 TcT has computed the following triangular matrix interpretation. 121.02/60.07 121.02/60.07 [+](x1, x2) = [1] x1 + [1] x2 + [0] 121.02/60.07 121.02/60.07 [*](x1, x2) = [1] x1 + [1] x2 + [2] 121.02/60.07 121.02/60.07 [a] = [1] 121.02/60.07 121.02/60.07 The order satisfies the following ordering constraints: 121.02/60.07 121.02/60.07 [+(*(x, y), *(a(), y))] = [1] x + [2] y + [5] 121.02/60.07 > [1] x + [1] y + [3] 121.02/60.07 = [*(+(x, a()), y)] 121.02/60.07 121.02/60.07 [*(*(x, y), z)] = [1] x + [1] y + [1] z + [4] 121.02/60.07 >= [1] x + [1] y + [1] z + [4] 121.02/60.07 = [*(x, *(y, z))] 121.02/60.07 121.02/60.07 121.02/60.07 We return to the main proof. 121.02/60.07 121.02/60.07 We are left with following problem, upon which TcT provides the 121.02/60.07 certificate YES(O(1),O(n^2)). 121.02/60.07 121.02/60.07 Strict Trs: { *(*(x, y), z) -> *(x, *(y, z)) } 121.02/60.07 Weak Trs: { +(*(x, y), *(a(), y)) -> *(+(x, a()), y) } 121.02/60.07 Obligation: 121.02/60.07 derivational complexity 121.02/60.07 Answer: 121.02/60.07 YES(O(1),O(n^2)) 121.02/60.07 121.02/60.07 We use the processor 'matrix interpretation of dimension 2' to 121.02/60.07 orient following rules strictly. 121.02/60.07 121.02/60.07 Trs: { *(*(x, y), z) -> *(x, *(y, z)) } 121.02/60.07 121.02/60.07 The induced complexity on above rules (modulo remaining rules) is 121.02/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 121.02/60.07 component(s). 121.02/60.07 121.02/60.07 Sub-proof: 121.02/60.07 ---------- 121.02/60.07 TcT has computed the following triangular matrix interpretation. 121.02/60.07 121.02/60.07 [+](x1, x2) = [1 2] x1 + [1 0] x2 + [0] 121.02/60.07 [0 0] [0 1] [0] 121.02/60.07 121.02/60.07 [*](x1, x2) = [1 2] x1 + [1 0] x2 + [0] 121.02/60.07 [0 1] [0 1] [2] 121.02/60.07 121.02/60.07 [a] = [0] 121.02/60.07 [0] 121.02/60.07 121.02/60.07 The order satisfies the following ordering constraints: 121.02/60.07 121.02/60.07 [+(*(x, y), *(a(), y))] = [1 4] x + [2 2] y + [4] 121.02/60.07 [0 0] [0 1] [2] 121.02/60.07 > [1 2] x + [1 0] y + [0] 121.02/60.07 [0 0] [0 1] [2] 121.02/60.07 = [*(+(x, a()), y)] 121.02/60.07 121.02/60.07 [*(*(x, y), z)] = [1 4] x + [1 2] y + [1 0] z + [4] 121.02/60.07 [0 1] [0 1] [0 1] [4] 121.02/60.07 > [1 2] x + [1 2] y + [1 0] z + [0] 121.02/60.07 [0 1] [0 1] [0 1] [4] 121.02/60.07 = [*(x, *(y, z))] 121.02/60.07 121.02/60.07 121.02/60.07 We return to the main proof. 121.02/60.07 121.02/60.07 We are left with following problem, upon which TcT provides the 121.02/60.07 certificate YES(O(1),O(1)). 121.02/60.07 121.02/60.07 Weak Trs: 121.02/60.07 { +(*(x, y), *(a(), y)) -> *(+(x, a()), y) 121.02/60.07 , *(*(x, y), z) -> *(x, *(y, z)) } 121.02/60.07 Obligation: 121.02/60.07 derivational complexity 121.02/60.07 Answer: 121.02/60.07 YES(O(1),O(1)) 121.02/60.07 121.02/60.07 Empty rules are trivially bounded 121.02/60.07 121.02/60.07 Hurray, we answered YES(O(1),O(n^2)) 121.14/60.11 EOF