YES(O(1),O(n^2)) 161.39/60.07 YES(O(1),O(n^2)) 161.39/60.07 161.39/60.07 We are left with following problem, upon which TcT provides the 161.39/60.07 certificate YES(O(1),O(n^2)). 161.39/60.07 161.39/60.07 Strict Trs: 161.39/60.07 { *(x, 0()) -> 0() 161.39/60.07 , *(*(x, y), z) -> *(x, *(y, z)) 161.39/60.07 , *(i(x), x) -> 1() 161.39/60.07 , *(1(), y) -> y } 161.39/60.07 Obligation: 161.39/60.07 derivational complexity 161.39/60.07 Answer: 161.39/60.07 YES(O(1),O(n^2)) 161.39/60.07 161.39/60.07 We use the processor 'matrix interpretation of dimension 1' to 161.39/60.07 orient following rules strictly. 161.39/60.07 161.39/60.07 Trs: 161.39/60.07 { *(x, 0()) -> 0() 161.39/60.07 , *(i(x), x) -> 1() 161.39/60.07 , *(1(), y) -> y } 161.39/60.07 161.39/60.07 The induced complexity on above rules (modulo remaining rules) is 161.39/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 161.39/60.07 component(s). 161.39/60.07 161.39/60.07 Sub-proof: 161.39/60.07 ---------- 161.39/60.07 TcT has computed the following triangular matrix interpretation. 161.39/60.07 161.39/60.07 [*](x1, x2) = [1] x1 + [1] x2 + [1] 161.39/60.07 161.39/60.07 [i](x1) = [1] x1 + [1] 161.39/60.07 161.39/60.07 [1] = [1] 161.39/60.07 161.39/60.07 [0] = [1] 161.39/60.07 161.39/60.07 The order satisfies the following ordering constraints: 161.39/60.07 161.39/60.07 [*(x, 0())] = [1] x + [2] 161.39/60.07 > [1] 161.39/60.07 = [0()] 161.39/60.07 161.39/60.07 [*(*(x, y), z)] = [1] x + [1] y + [1] z + [2] 161.39/60.07 >= [1] x + [1] y + [1] z + [2] 161.39/60.07 = [*(x, *(y, z))] 161.39/60.07 161.39/60.07 [*(i(x), x)] = [2] x + [2] 161.39/60.07 > [1] 161.39/60.07 = [1()] 161.39/60.07 161.39/60.07 [*(1(), y)] = [1] y + [2] 161.39/60.07 > [1] y + [0] 161.39/60.07 = [y] 161.39/60.07 161.39/60.07 161.39/60.07 We return to the main proof. 161.39/60.07 161.39/60.07 We are left with following problem, upon which TcT provides the 161.39/60.07 certificate YES(O(1),O(n^2)). 161.39/60.07 161.39/60.07 Strict Trs: { *(*(x, y), z) -> *(x, *(y, z)) } 161.39/60.07 Weak Trs: 161.39/60.07 { *(x, 0()) -> 0() 161.39/60.07 , *(i(x), x) -> 1() 161.39/60.07 , *(1(), y) -> y } 161.39/60.07 Obligation: 161.39/60.07 derivational complexity 161.39/60.07 Answer: 161.39/60.07 YES(O(1),O(n^2)) 161.39/60.07 161.39/60.07 We use the processor 'matrix interpretation of dimension 2' to 161.39/60.07 orient following rules strictly. 161.39/60.07 161.39/60.07 Trs: { *(*(x, y), z) -> *(x, *(y, z)) } 161.39/60.07 161.39/60.07 The induced complexity on above rules (modulo remaining rules) is 161.39/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 161.39/60.07 component(s). 161.39/60.07 161.39/60.07 Sub-proof: 161.39/60.07 ---------- 161.39/60.07 TcT has computed the following triangular matrix interpretation. 161.39/60.07 161.39/60.07 [*](x1, x2) = [1 1] x1 + [1 0] x2 + [0] 161.39/60.07 [0 1] [0 1] [1] 161.39/60.07 161.39/60.07 [i](x1) = [1 0] x1 + [0] 161.39/60.07 [0 0] [0] 161.39/60.07 161.39/60.07 [1] = [0] 161.39/60.07 [0] 161.39/60.07 161.39/60.07 [0] = [2] 161.39/60.07 [1] 161.39/60.07 161.39/60.07 The order satisfies the following ordering constraints: 161.39/60.07 161.39/60.07 [*(x, 0())] = [1 1] x + [2] 161.39/60.07 [0 1] [2] 161.39/60.07 >= [2] 161.39/60.07 [1] 161.39/60.07 = [0()] 161.39/60.07 161.39/60.07 [*(*(x, y), z)] = [1 2] x + [1 1] y + [1 0] z + [1] 161.39/60.07 [0 1] [0 1] [0 1] [2] 161.39/60.07 > [1 1] x + [1 1] y + [1 0] z + [0] 161.39/60.07 [0 1] [0 1] [0 1] [2] 161.39/60.07 = [*(x, *(y, z))] 161.39/60.07 161.39/60.07 [*(i(x), x)] = [2 0] x + [0] 161.39/60.07 [0 1] [1] 161.39/60.07 >= [0] 161.39/60.07 [0] 161.39/60.07 = [1()] 161.39/60.07 161.39/60.07 [*(1(), y)] = [1 0] y + [0] 161.39/60.07 [0 1] [1] 161.39/60.07 >= [1 0] y + [0] 161.39/60.07 [0 1] [0] 161.39/60.07 = [y] 161.39/60.07 161.39/60.07 161.39/60.07 We return to the main proof. 161.39/60.07 161.39/60.07 We are left with following problem, upon which TcT provides the 161.39/60.07 certificate YES(O(1),O(1)). 161.39/60.07 161.39/60.07 Weak Trs: 161.39/60.07 { *(x, 0()) -> 0() 161.39/60.07 , *(*(x, y), z) -> *(x, *(y, z)) 161.39/60.07 , *(i(x), x) -> 1() 161.39/60.07 , *(1(), y) -> y } 161.39/60.07 Obligation: 161.39/60.07 derivational complexity 161.39/60.07 Answer: 161.39/60.07 YES(O(1),O(1)) 161.39/60.07 161.39/60.07 Empty rules are trivially bounded 161.39/60.07 161.39/60.07 Hurray, we answered YES(O(1),O(n^2)) 161.59/60.10 EOF