YES(O(1),O(n^2)) 178.53/60.02 YES(O(1),O(n^2)) 178.53/60.02 178.53/60.02 We are left with following problem, upon which TcT provides the 178.53/60.02 certificate YES(O(1),O(n^2)). 178.53/60.02 178.53/60.02 Strict Trs: 178.53/60.02 { f(+(x, 0())) -> f(x) 178.53/60.02 , +(x, +(y, z)) -> +(+(x, y), z) } 178.53/60.02 Obligation: 178.53/60.02 derivational complexity 178.53/60.02 Answer: 178.53/60.02 YES(O(1),O(n^2)) 178.53/60.02 178.53/60.02 We use the processor 'matrix interpretation of dimension 1' to 178.53/60.02 orient following rules strictly. 178.53/60.02 178.53/60.02 Trs: { f(+(x, 0())) -> f(x) } 178.53/60.02 178.53/60.02 The induced complexity on above rules (modulo remaining rules) is 178.53/60.02 YES(?,O(n^1)) . These rules are moved into the corresponding weak 178.53/60.02 component(s). 178.53/60.02 178.53/60.02 Sub-proof: 178.53/60.02 ---------- 178.53/60.02 TcT has computed the following triangular matrix interpretation. 178.53/60.02 178.53/60.02 [f](x1) = [1] x1 + [0] 178.53/60.02 178.53/60.02 [+](x1, x2) = [1] x1 + [1] x2 + [1] 178.53/60.02 178.53/60.02 [0] = [2] 178.53/60.02 178.53/60.02 The order satisfies the following ordering constraints: 178.53/60.02 178.53/60.02 [f(+(x, 0()))] = [1] x + [3] 178.53/60.02 > [1] x + [0] 178.53/60.02 = [f(x)] 178.53/60.02 178.53/60.02 [+(x, +(y, z))] = [1] x + [1] y + [1] z + [2] 178.53/60.02 >= [1] x + [1] y + [1] z + [2] 178.53/60.02 = [+(+(x, y), z)] 178.53/60.02 178.53/60.02 178.53/60.02 We return to the main proof. 178.53/60.02 178.53/60.02 We are left with following problem, upon which TcT provides the 178.53/60.02 certificate YES(O(1),O(n^2)). 178.53/60.02 178.53/60.02 Strict Trs: { +(x, +(y, z)) -> +(+(x, y), z) } 178.53/60.02 Weak Trs: { f(+(x, 0())) -> f(x) } 178.53/60.02 Obligation: 178.53/60.02 derivational complexity 178.53/60.02 Answer: 178.53/60.02 YES(O(1),O(n^2)) 178.53/60.02 178.53/60.02 We use the processor 'matrix interpretation of dimension 2' to 178.53/60.02 orient following rules strictly. 178.53/60.02 178.53/60.02 Trs: { +(x, +(y, z)) -> +(+(x, y), z) } 178.53/60.02 178.53/60.02 The induced complexity on above rules (modulo remaining rules) is 178.53/60.02 YES(?,O(n^2)) . These rules are moved into the corresponding weak 178.53/60.02 component(s). 178.53/60.02 178.53/60.02 Sub-proof: 178.53/60.02 ---------- 178.53/60.02 TcT has computed the following triangular matrix interpretation. 178.53/60.02 178.53/60.02 [f](x1) = [1 0] x1 + [2] 178.53/60.02 [0 1] [1] 178.53/60.02 178.53/60.02 [+](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 178.53/60.02 [0 1] [0 1] [1] 178.53/60.02 178.53/60.02 [0] = [2] 178.53/60.02 [0] 178.53/60.02 178.53/60.02 The order satisfies the following ordering constraints: 178.53/60.02 178.53/60.02 [f(+(x, 0()))] = [1 0] x + [4] 178.53/60.02 [0 1] [2] 178.53/60.02 > [1 0] x + [2] 178.53/60.02 [0 1] [1] 178.53/60.02 = [f(x)] 178.53/60.02 178.53/60.02 [+(x, +(y, z))] = [1 0] x + [1 1] y + [1 2] z + [1] 178.53/60.02 [0 1] [0 1] [0 1] [2] 178.53/60.02 > [1 0] x + [1 1] y + [1 1] z + [0] 178.53/60.02 [0 1] [0 1] [0 1] [2] 178.53/60.02 = [+(+(x, y), z)] 178.53/60.02 178.53/60.02 178.53/60.02 We return to the main proof. 178.53/60.02 178.53/60.02 We are left with following problem, upon which TcT provides the 178.53/60.02 certificate YES(O(1),O(1)). 178.53/60.02 178.53/60.02 Weak Trs: 178.53/60.02 { f(+(x, 0())) -> f(x) 178.53/60.02 , +(x, +(y, z)) -> +(+(x, y), z) } 178.53/60.02 Obligation: 178.53/60.02 derivational complexity 178.53/60.02 Answer: 178.53/60.02 YES(O(1),O(1)) 178.53/60.02 178.53/60.02 Empty rules are trivially bounded 178.53/60.02 178.53/60.02 Hurray, we answered YES(O(1),O(n^2)) 178.67/60.19 EOF