YES(O(1),O(n^2)) 164.17/60.07 YES(O(1),O(n^2)) 164.17/60.07 164.17/60.07 We are left with following problem, upon which TcT provides the 164.17/60.07 certificate YES(O(1),O(n^2)). 164.17/60.07 164.17/60.07 Strict Trs: 164.17/60.07 { +(x, 0()) -> x 164.17/60.07 , +(x, s(y)) -> s(+(x, y)) 164.17/60.07 , +(0(), s(y)) -> s(y) 164.17/60.07 , s(+(0(), y)) -> s(y) } 164.17/60.07 Obligation: 164.17/60.07 derivational complexity 164.17/60.07 Answer: 164.17/60.07 YES(O(1),O(n^2)) 164.17/60.07 164.17/60.07 We use the processor 'matrix interpretation of dimension 1' to 164.17/60.07 orient following rules strictly. 164.17/60.07 164.17/60.07 Trs: 164.17/60.07 { +(x, 0()) -> x 164.17/60.07 , +(0(), s(y)) -> s(y) 164.17/60.07 , s(+(0(), y)) -> s(y) } 164.17/60.07 164.17/60.07 The induced complexity on above rules (modulo remaining rules) is 164.17/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 164.17/60.07 component(s). 164.17/60.07 164.17/60.07 Sub-proof: 164.17/60.07 ---------- 164.17/60.07 TcT has computed the following triangular matrix interpretation. 164.17/60.07 164.17/60.07 [+](x1, x2) = [1] x1 + [1] x2 + [0] 164.17/60.07 164.17/60.07 [0] = [1] 164.17/60.07 164.17/60.07 [s](x1) = [1] x1 + [0] 164.17/60.07 164.17/60.07 The order satisfies the following ordering constraints: 164.17/60.07 164.17/60.07 [+(x, 0())] = [1] x + [1] 164.17/60.07 > [1] x + [0] 164.17/60.07 = [x] 164.17/60.07 164.17/60.07 [+(x, s(y))] = [1] x + [1] y + [0] 164.17/60.07 >= [1] x + [1] y + [0] 164.17/60.07 = [s(+(x, y))] 164.17/60.07 164.17/60.07 [+(0(), s(y))] = [1] y + [1] 164.17/60.07 > [1] y + [0] 164.17/60.07 = [s(y)] 164.17/60.07 164.17/60.07 [s(+(0(), y))] = [1] y + [1] 164.17/60.07 > [1] y + [0] 164.17/60.07 = [s(y)] 164.17/60.07 164.17/60.07 164.17/60.07 We return to the main proof. 164.17/60.07 164.17/60.07 We are left with following problem, upon which TcT provides the 164.17/60.07 certificate YES(O(1),O(n^2)). 164.17/60.07 164.17/60.07 Strict Trs: { +(x, s(y)) -> s(+(x, y)) } 164.17/60.07 Weak Trs: 164.17/60.07 { +(x, 0()) -> x 164.17/60.07 , +(0(), s(y)) -> s(y) 164.17/60.07 , s(+(0(), y)) -> s(y) } 164.17/60.07 Obligation: 164.17/60.07 derivational complexity 164.17/60.07 Answer: 164.17/60.07 YES(O(1),O(n^2)) 164.17/60.07 164.17/60.07 We use the processor 'matrix interpretation of dimension 2' to 164.17/60.07 orient following rules strictly. 164.17/60.07 164.17/60.07 Trs: { +(x, s(y)) -> s(+(x, y)) } 164.17/60.07 164.17/60.07 The induced complexity on above rules (modulo remaining rules) is 164.17/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 164.17/60.07 component(s). 164.17/60.07 164.17/60.07 Sub-proof: 164.17/60.07 ---------- 164.17/60.07 TcT has computed the following triangular matrix interpretation. 164.17/60.07 164.17/60.07 [+](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 164.17/60.07 [0 1] [0 1] [0] 164.17/60.07 164.17/60.07 [0] = [0] 164.17/60.07 [2] 164.17/60.07 164.17/60.07 [s](x1) = [1 0] x1 + [0] 164.17/60.07 [0 1] [2] 164.17/60.07 164.17/60.07 The order satisfies the following ordering constraints: 164.17/60.07 164.17/60.07 [+(x, 0())] = [1 0] x + [2] 164.17/60.07 [0 1] [2] 164.17/60.07 > [1 0] x + [0] 164.17/60.07 [0 1] [0] 164.17/60.07 = [x] 164.17/60.07 164.17/60.07 [+(x, s(y))] = [1 0] x + [1 1] y + [2] 164.17/60.07 [0 1] [0 1] [2] 164.17/60.07 > [1 0] x + [1 1] y + [0] 164.17/60.07 [0 1] [0 1] [2] 164.17/60.07 = [s(+(x, y))] 164.17/60.07 164.17/60.07 [+(0(), s(y))] = [1 1] y + [2] 164.17/60.07 [0 1] [4] 164.17/60.07 > [1 0] y + [0] 164.17/60.07 [0 1] [2] 164.17/60.07 = [s(y)] 164.17/60.07 164.17/60.07 [s(+(0(), y))] = [1 1] y + [0] 164.17/60.07 [0 1] [4] 164.17/60.07 >= [1 0] y + [0] 164.17/60.07 [0 1] [2] 164.17/60.07 = [s(y)] 164.17/60.07 164.17/60.07 164.17/60.07 We return to the main proof. 164.17/60.07 164.17/60.07 We are left with following problem, upon which TcT provides the 164.17/60.07 certificate YES(O(1),O(1)). 164.17/60.07 164.17/60.07 Weak Trs: 164.17/60.07 { +(x, 0()) -> x 164.17/60.07 , +(x, s(y)) -> s(+(x, y)) 164.17/60.07 , +(0(), s(y)) -> s(y) 164.17/60.07 , s(+(0(), y)) -> s(y) } 164.17/60.07 Obligation: 164.17/60.07 derivational complexity 164.17/60.07 Answer: 164.17/60.07 YES(O(1),O(1)) 164.17/60.07 164.17/60.07 Empty rules are trivially bounded 164.17/60.07 164.17/60.07 Hurray, we answered YES(O(1),O(n^2)) 164.17/60.08 EOF