YES(O(1),O(n^2)) 181.73/60.03 YES(O(1),O(n^2)) 181.73/60.03 181.73/60.03 We are left with following problem, upon which TcT provides the 181.73/60.03 certificate YES(O(1),O(n^2)). 181.73/60.03 181.73/60.03 Strict Trs: 181.73/60.03 { f(a(), f(b(), x)) -> f(b(), f(a(), x)) 181.73/60.03 , f(b(), f(c(), x)) -> f(c(), f(b(), x)) 181.73/60.03 , f(c(), f(a(), x)) -> f(a(), f(c(), x)) } 181.73/60.03 Obligation: 181.73/60.03 derivational complexity 181.73/60.03 Answer: 181.73/60.03 YES(O(1),O(n^2)) 181.73/60.03 181.73/60.03 We uncurry the input using the following uncurry rules. 181.73/60.03 181.73/60.03 { f(a(), x_1) -> a_1(x_1) 181.73/60.03 , f(b(), x_1) -> b_1(x_1) 181.73/60.03 , f(c(), x_1) -> c_1(x_1) } 181.73/60.03 181.73/60.03 We are left with following problem, upon which TcT provides the 181.73/60.03 certificate YES(O(1),O(n^2)). 181.73/60.03 181.73/60.03 Strict Trs: 181.73/60.03 { a_1(b_1(x)) -> b_1(a_1(x)) 181.73/60.03 , b_1(c_1(x)) -> c_1(b_1(x)) 181.73/60.03 , c_1(a_1(x)) -> a_1(c_1(x)) } 181.73/60.03 Weak Trs: 181.73/60.03 { f(a(), x_1) -> a_1(x_1) 181.73/60.03 , f(b(), x_1) -> b_1(x_1) 181.73/60.03 , f(c(), x_1) -> c_1(x_1) } 181.73/60.03 Obligation: 181.73/60.03 derivational complexity 181.73/60.03 Answer: 181.73/60.03 YES(O(1),O(n^2)) 181.73/60.03 181.73/60.03 We use the processor 'matrix interpretation of dimension 2' to 181.73/60.03 orient following rules strictly. 181.73/60.03 181.73/60.03 Trs: { b_1(c_1(x)) -> c_1(b_1(x)) } 181.73/60.03 181.73/60.03 The induced complexity on above rules (modulo remaining rules) is 181.73/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak 181.73/60.03 component(s). 181.73/60.03 181.73/60.03 Sub-proof: 181.73/60.03 ---------- 181.73/60.03 TcT has computed the following triangular matrix interpretation. 181.73/60.03 181.73/60.03 [f](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 181.73/60.03 [0 0] [0 1] [1] 181.73/60.03 181.73/60.03 [a_1](x1) = [1 0] x1 + [0] 181.73/60.03 [0 1] [0] 181.73/60.03 181.73/60.03 [b_1](x1) = [1 2] x1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 181.73/60.03 [c_1](x1) = [1 1] x1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 181.73/60.03 [a] = [0] 181.73/60.03 [0] 181.73/60.03 181.73/60.03 [b] = [0] 181.73/60.03 [0] 181.73/60.03 181.73/60.03 [c] = [0] 181.73/60.03 [0] 181.73/60.03 181.73/60.03 The order satisfies the following ordering constraints: 181.73/60.03 181.73/60.03 [f(a(), x_1)] = [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 0] x_1 + [0] 181.73/60.03 [0 1] [0] 181.73/60.03 = [a_1(x_1)] 181.73/60.03 181.73/60.03 [f(b(), x_1)] = [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 = [b_1(x_1)] 181.73/60.03 181.73/60.03 [f(c(), x_1)] = [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 1] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 = [c_1(x_1)] 181.73/60.03 181.73/60.03 [a_1(b_1(x))] = [1 2] x + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 2] x + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 = [b_1(a_1(x))] 181.73/60.03 181.73/60.03 [b_1(c_1(x))] = [1 3] x + [2] 181.73/60.03 [0 1] [2] 181.73/60.03 > [1 3] x + [1] 181.73/60.03 [0 1] [2] 181.73/60.03 = [c_1(b_1(x))] 181.73/60.03 181.73/60.03 [c_1(a_1(x))] = [1 1] x + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 1] x + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 = [a_1(c_1(x))] 181.73/60.03 181.73/60.03 181.73/60.03 We return to the main proof. 181.73/60.03 181.73/60.03 We are left with following problem, upon which TcT provides the 181.73/60.03 certificate YES(O(1),O(n^2)). 181.73/60.03 181.73/60.03 Strict Trs: 181.73/60.03 { a_1(b_1(x)) -> b_1(a_1(x)) 181.73/60.03 , c_1(a_1(x)) -> a_1(c_1(x)) } 181.73/60.03 Weak Trs: 181.73/60.03 { f(a(), x_1) -> a_1(x_1) 181.73/60.03 , f(b(), x_1) -> b_1(x_1) 181.73/60.03 , f(c(), x_1) -> c_1(x_1) 181.73/60.03 , b_1(c_1(x)) -> c_1(b_1(x)) } 181.73/60.03 Obligation: 181.73/60.03 derivational complexity 181.73/60.03 Answer: 181.73/60.03 YES(O(1),O(n^2)) 181.73/60.03 181.73/60.03 We use the processor 'matrix interpretation of dimension 2' to 181.73/60.03 orient following rules strictly. 181.73/60.03 181.73/60.03 Trs: { a_1(b_1(x)) -> b_1(a_1(x)) } 181.73/60.03 181.73/60.03 The induced complexity on above rules (modulo remaining rules) is 181.73/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak 181.73/60.03 component(s). 181.73/60.03 181.73/60.03 Sub-proof: 181.73/60.03 ---------- 181.73/60.03 TcT has computed the following triangular matrix interpretation. 181.73/60.03 181.73/60.03 [f](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 181.73/60.03 [0 0] [0 1] [1] 181.73/60.03 181.73/60.03 [a_1](x1) = [1 2] x1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 181.73/60.03 [b_1](x1) = [1 1] x1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 181.73/60.03 [c_1](x1) = [1 0] x1 + [0] 181.73/60.03 [0 1] [0] 181.73/60.03 181.73/60.03 [a] = [0] 181.73/60.03 [0] 181.73/60.03 181.73/60.03 [b] = [0] 181.73/60.03 [0] 181.73/60.03 181.73/60.03 [c] = [0] 181.73/60.03 [0] 181.73/60.03 181.73/60.03 The order satisfies the following ordering constraints: 181.73/60.03 181.73/60.03 [f(a(), x_1)] = [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 = [a_1(x_1)] 181.73/60.03 181.73/60.03 [f(b(), x_1)] = [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 1] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 = [b_1(x_1)] 181.73/60.03 181.73/60.03 [f(c(), x_1)] = [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 0] x_1 + [0] 181.73/60.03 [0 1] [0] 181.73/60.03 = [c_1(x_1)] 181.73/60.03 181.73/60.03 [a_1(b_1(x))] = [1 3] x + [2] 181.73/60.03 [0 1] [2] 181.73/60.03 > [1 3] x + [1] 181.73/60.03 [0 1] [2] 181.73/60.03 = [b_1(a_1(x))] 181.73/60.03 181.73/60.03 [b_1(c_1(x))] = [1 1] x + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 1] x + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 = [c_1(b_1(x))] 181.73/60.03 181.73/60.03 [c_1(a_1(x))] = [1 2] x + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 2] x + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 = [a_1(c_1(x))] 181.73/60.03 181.73/60.03 181.73/60.03 We return to the main proof. 181.73/60.03 181.73/60.03 We are left with following problem, upon which TcT provides the 181.73/60.03 certificate YES(O(1),O(n^2)). 181.73/60.03 181.73/60.03 Strict Trs: { c_1(a_1(x)) -> a_1(c_1(x)) } 181.73/60.03 Weak Trs: 181.73/60.03 { f(a(), x_1) -> a_1(x_1) 181.73/60.03 , f(b(), x_1) -> b_1(x_1) 181.73/60.03 , f(c(), x_1) -> c_1(x_1) 181.73/60.03 , a_1(b_1(x)) -> b_1(a_1(x)) 181.73/60.03 , b_1(c_1(x)) -> c_1(b_1(x)) } 181.73/60.03 Obligation: 181.73/60.03 derivational complexity 181.73/60.03 Answer: 181.73/60.03 YES(O(1),O(n^2)) 181.73/60.03 181.73/60.03 We use the processor 'matrix interpretation of dimension 2' to 181.73/60.03 orient following rules strictly. 181.73/60.03 181.73/60.03 Trs: { c_1(a_1(x)) -> a_1(c_1(x)) } 181.73/60.03 181.73/60.03 The induced complexity on above rules (modulo remaining rules) is 181.73/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak 181.73/60.03 component(s). 181.73/60.03 181.73/60.03 Sub-proof: 181.73/60.03 ---------- 181.73/60.03 TcT has computed the following triangular matrix interpretation. 181.73/60.03 181.73/60.03 [f](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 181.73/60.03 [0 0] [0 1] [1] 181.73/60.03 181.73/60.03 [a_1](x1) = [1 0] x1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 181.73/60.03 [b_1](x1) = [1 0] x1 + [0] 181.73/60.03 [0 0] [0] 181.73/60.03 181.73/60.03 [c_1](x1) = [1 2] x1 + [0] 181.73/60.03 [0 1] [0] 181.73/60.03 181.73/60.03 [a] = [0] 181.73/60.03 [0] 181.73/60.03 181.73/60.03 [b] = [0] 181.73/60.03 [0] 181.73/60.03 181.73/60.03 [c] = [0] 181.73/60.03 [0] 181.73/60.03 181.73/60.03 The order satisfies the following ordering constraints: 181.73/60.03 181.73/60.03 [f(a(), x_1)] = [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 0] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 = [a_1(x_1)] 181.73/60.03 181.73/60.03 [f(b(), x_1)] = [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 0] x_1 + [0] 181.73/60.03 [0 0] [0] 181.73/60.03 = [b_1(x_1)] 181.73/60.03 181.73/60.03 [f(c(), x_1)] = [1 2] x_1 + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 >= [1 2] x_1 + [0] 181.73/60.03 [0 1] [0] 181.73/60.03 = [c_1(x_1)] 181.73/60.03 181.73/60.03 [a_1(b_1(x))] = [1 0] x + [0] 181.73/60.03 [0 0] [1] 181.73/60.03 >= [1 0] x + [0] 181.73/60.03 [0 0] [0] 181.73/60.03 = [b_1(a_1(x))] 181.73/60.03 181.73/60.03 [b_1(c_1(x))] = [1 2] x + [0] 181.73/60.03 [0 0] [0] 181.73/60.03 >= [1 0] x + [0] 181.73/60.03 [0 0] [0] 181.73/60.03 = [c_1(b_1(x))] 181.73/60.03 181.73/60.03 [c_1(a_1(x))] = [1 2] x + [2] 181.73/60.03 [0 1] [1] 181.73/60.03 > [1 2] x + [0] 181.73/60.03 [0 1] [1] 181.73/60.03 = [a_1(c_1(x))] 181.73/60.03 181.73/60.03 181.73/60.03 We return to the main proof. 181.73/60.03 181.73/60.03 We are left with following problem, upon which TcT provides the 181.73/60.03 certificate YES(O(1),O(1)). 181.73/60.03 181.73/60.03 Weak Trs: 181.73/60.03 { f(a(), x_1) -> a_1(x_1) 181.73/60.03 , f(b(), x_1) -> b_1(x_1) 181.73/60.03 , f(c(), x_1) -> c_1(x_1) 181.73/60.03 , a_1(b_1(x)) -> b_1(a_1(x)) 181.73/60.03 , b_1(c_1(x)) -> c_1(b_1(x)) 181.73/60.03 , c_1(a_1(x)) -> a_1(c_1(x)) } 181.73/60.03 Obligation: 181.73/60.03 derivational complexity 181.73/60.03 Answer: 181.73/60.03 YES(O(1),O(1)) 181.73/60.03 181.73/60.03 Empty rules are trivially bounded 181.73/60.03 181.73/60.03 Hurray, we answered YES(O(1),O(n^2)) 181.73/60.03 EOF