YES(O(1),O(n^2)) 174.14/60.02 YES(O(1),O(n^2)) 174.14/60.02 174.14/60.02 We are left with following problem, upon which TcT provides the 174.14/60.02 certificate YES(O(1),O(n^2)). 174.14/60.02 174.14/60.02 Strict Trs: 174.14/60.02 { f(x, a()) -> x 174.14/60.02 , f(x, g(y)) -> f(g(x), y) } 174.14/60.02 Obligation: 174.14/60.02 derivational complexity 174.14/60.02 Answer: 174.14/60.02 YES(O(1),O(n^2)) 174.14/60.02 174.14/60.02 We use the processor 'matrix interpretation of dimension 1' to 174.14/60.02 orient following rules strictly. 174.14/60.02 174.14/60.02 Trs: { f(x, a()) -> x } 174.14/60.02 174.14/60.02 The induced complexity on above rules (modulo remaining rules) is 174.14/60.02 YES(?,O(n^1)) . These rules are moved into the corresponding weak 174.14/60.02 component(s). 174.14/60.02 174.14/60.02 Sub-proof: 174.14/60.02 ---------- 174.14/60.02 TcT has computed the following triangular matrix interpretation. 174.14/60.02 174.14/60.02 [f](x1, x2) = [1] x1 + [1] x2 + [0] 174.14/60.02 174.14/60.02 [a] = [2] 174.14/60.02 174.14/60.02 [g](x1) = [1] x1 + [0] 174.14/60.02 174.14/60.02 The order satisfies the following ordering constraints: 174.14/60.02 174.14/60.02 [f(x, a())] = [1] x + [2] 174.14/60.02 > [1] x + [0] 174.14/60.02 = [x] 174.14/60.02 174.14/60.02 [f(x, g(y))] = [1] x + [1] y + [0] 174.14/60.02 >= [1] x + [1] y + [0] 174.14/60.02 = [f(g(x), y)] 174.14/60.02 174.14/60.02 174.14/60.02 We return to the main proof. 174.14/60.02 174.14/60.02 We are left with following problem, upon which TcT provides the 174.14/60.02 certificate YES(O(1),O(n^2)). 174.14/60.02 174.14/60.02 Strict Trs: { f(x, g(y)) -> f(g(x), y) } 174.14/60.02 Weak Trs: { f(x, a()) -> x } 174.14/60.02 Obligation: 174.14/60.02 derivational complexity 174.14/60.02 Answer: 174.14/60.02 YES(O(1),O(n^2)) 174.14/60.02 174.14/60.02 We use the processor 'matrix interpretation of dimension 2' to 174.14/60.02 orient following rules strictly. 174.14/60.02 174.14/60.02 Trs: { f(x, g(y)) -> f(g(x), y) } 174.14/60.02 174.14/60.02 The induced complexity on above rules (modulo remaining rules) is 174.14/60.02 YES(?,O(n^2)) . These rules are moved into the corresponding weak 174.14/60.02 component(s). 174.14/60.02 174.14/60.02 Sub-proof: 174.14/60.02 ---------- 174.14/60.02 TcT has computed the following triangular matrix interpretation. 174.14/60.02 174.14/60.02 [f](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 174.14/60.02 [0 1] [0 1] [2] 174.14/60.02 174.14/60.02 [a] = [1] 174.14/60.02 [1] 174.14/60.02 174.14/60.02 [g](x1) = [1 0] x1 + [0] 174.14/60.02 [0 1] [2] 174.14/60.02 174.14/60.02 The order satisfies the following ordering constraints: 174.14/60.02 174.14/60.02 [f(x, a())] = [1 0] x + [3] 174.14/60.02 [0 1] [3] 174.14/60.02 > [1 0] x + [0] 174.14/60.02 [0 1] [0] 174.14/60.02 = [x] 174.14/60.02 174.14/60.02 [f(x, g(y))] = [1 0] x + [1 2] y + [4] 174.14/60.02 [0 1] [0 1] [4] 174.14/60.02 > [1 0] x + [1 2] y + [0] 174.14/60.02 [0 1] [0 1] [4] 174.14/60.02 = [f(g(x), y)] 174.14/60.02 174.14/60.02 174.14/60.02 We return to the main proof. 174.14/60.02 174.14/60.02 We are left with following problem, upon which TcT provides the 174.14/60.02 certificate YES(O(1),O(1)). 174.14/60.02 174.14/60.02 Weak Trs: 174.14/60.02 { f(x, a()) -> x 174.14/60.02 , f(x, g(y)) -> f(g(x), y) } 174.14/60.02 Obligation: 174.14/60.02 derivational complexity 174.14/60.02 Answer: 174.14/60.02 YES(O(1),O(1)) 174.14/60.02 174.14/60.02 Empty rules are trivially bounded 174.14/60.02 174.14/60.02 Hurray, we answered YES(O(1),O(n^2)) 174.14/60.03 EOF