YES(O(1),O(n^1)) 167.14/60.21 YES(O(1),O(n^1)) 167.14/60.21 167.14/60.21 We are left with following problem, upon which TcT provides the 167.14/60.21 certificate YES(O(1),O(n^1)). 167.14/60.21 167.14/60.21 Strict Trs: 167.14/60.21 { f(a(), f(b(), f(a(), x))) -> f(a(), f(b(), f(b(), f(a(), x)))) 167.14/60.21 , f(b(), f(b(), f(b(), x))) -> f(b(), f(b(), x)) } 167.14/60.21 Obligation: 167.14/60.21 derivational complexity 167.14/60.21 Answer: 167.14/60.21 YES(O(1),O(n^1)) 167.14/60.21 167.14/60.21 We uncurry the input using the following uncurry rules. 167.14/60.21 167.14/60.21 { f(a(), x_1) -> a_1(x_1) 167.14/60.21 , f(b(), x_1) -> b_1(x_1) } 167.14/60.21 167.14/60.21 We are left with following problem, upon which TcT provides the 167.14/60.21 certificate YES(O(1),O(n^1)). 167.14/60.21 167.14/60.21 Strict Trs: 167.14/60.21 { a_1(b_1(a_1(x))) -> a_1(b_1(b_1(a_1(x)))) 167.14/60.21 , b_1(b_1(b_1(x))) -> b_1(b_1(x)) } 167.14/60.21 Weak Trs: 167.14/60.21 { f(a(), x_1) -> a_1(x_1) 167.14/60.21 , f(b(), x_1) -> b_1(x_1) } 167.14/60.21 Obligation: 167.14/60.21 derivational complexity 167.14/60.21 Answer: 167.14/60.21 YES(O(1),O(n^1)) 167.14/60.21 167.14/60.21 The weightgap principle applies (using the following nonconstant 167.14/60.21 growth matrix-interpretation) 167.14/60.21 167.14/60.21 TcT has computed the following triangular matrix interpretation. 167.14/60.21 Note that the diagonal of the component-wise maxima of 167.14/60.21 interpretation-entries contains no more than 1 non-zero entries. 167.14/60.21 167.14/60.21 [f](x1, x2) = [1] x1 + [1] x2 + [2] 167.14/60.21 167.14/60.21 [a_1](x1) = [1] x1 + [1] 167.14/60.21 167.14/60.21 [b_1](x1) = [1] x1 + [2] 167.14/60.21 167.14/60.21 [a] = [1] 167.14/60.21 167.14/60.21 [b] = [1] 167.14/60.21 167.14/60.21 The order satisfies the following ordering constraints: 167.14/60.21 167.14/60.21 [f(a(), x_1)] = [1] x_1 + [3] 167.14/60.21 > [1] x_1 + [1] 167.14/60.21 = [a_1(x_1)] 167.14/60.21 167.14/60.21 [f(b(), x_1)] = [1] x_1 + [3] 167.14/60.21 > [1] x_1 + [2] 167.14/60.21 = [b_1(x_1)] 167.14/60.21 167.14/60.21 [a_1(b_1(a_1(x)))] = [1] x + [4] 167.14/60.21 ? [1] x + [6] 167.14/60.21 = [a_1(b_1(b_1(a_1(x))))] 167.14/60.21 167.14/60.21 [b_1(b_1(b_1(x)))] = [1] x + [6] 167.14/60.21 > [1] x + [4] 167.14/60.21 = [b_1(b_1(x))] 167.14/60.21 167.14/60.21 167.14/60.21 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 167.14/60.21 167.14/60.21 We are left with following problem, upon which TcT provides the 167.14/60.21 certificate YES(O(1),O(n^1)). 167.14/60.21 167.14/60.21 Strict Trs: { a_1(b_1(a_1(x))) -> a_1(b_1(b_1(a_1(x)))) } 167.14/60.21 Weak Trs: 167.14/60.21 { f(a(), x_1) -> a_1(x_1) 167.14/60.21 , f(b(), x_1) -> b_1(x_1) 167.14/60.21 , b_1(b_1(b_1(x))) -> b_1(b_1(x)) } 167.14/60.21 Obligation: 167.14/60.21 derivational complexity 167.14/60.21 Answer: 167.14/60.21 YES(O(1),O(n^1)) 167.14/60.21 167.14/60.21 We use the processor 'matrix interpretation of dimension 3' to 167.14/60.21 orient following rules strictly. 167.14/60.21 167.14/60.21 Trs: { a_1(b_1(a_1(x))) -> a_1(b_1(b_1(a_1(x)))) } 167.14/60.21 167.14/60.21 The induced complexity on above rules (modulo remaining rules) is 167.14/60.21 YES(?,O(n^1)) . These rules are moved into the corresponding weak 167.14/60.21 component(s). 167.14/60.21 167.14/60.21 Sub-proof: 167.14/60.21 ---------- 167.14/60.21 TcT has computed the following triangular matrix interpretation. 167.14/60.21 Note that the diagonal of the component-wise maxima of 167.14/60.21 interpretation-entries contains no more than 1 non-zero entries. 167.14/60.21 167.14/60.21 [1 0 0] [1 2 2] [0] 167.14/60.21 [f](x1, x2) = [0 0 0] x1 + [0 0 2] x2 + [0] 167.14/60.21 [0 0 0] [0 0 0] [2] 167.14/60.21 167.14/60.21 [1 2 0] [0] 167.14/60.21 [a_1](x1) = [0 0 0] x1 + [0] 167.14/60.21 [0 0 0] [1] 167.14/60.21 167.14/60.21 [1 0 0] [0] 167.14/60.21 [b_1](x1) = [0 0 2] x1 + [0] 167.14/60.21 [0 0 0] [0] 167.14/60.21 167.14/60.21 [0] 167.14/60.21 [a] = [2] 167.14/60.21 [0] 167.14/60.21 167.14/60.21 [0] 167.14/60.21 [b] = [2] 167.14/60.21 [0] 167.14/60.21 167.14/60.21 The order satisfies the following ordering constraints: 167.14/60.21 167.14/60.21 [f(a(), x_1)] = [1 2 2] [0] 167.14/60.21 [0 0 2] x_1 + [0] 167.14/60.21 [0 0 0] [2] 167.14/60.21 >= [1 2 0] [0] 167.14/60.21 [0 0 0] x_1 + [0] 167.14/60.21 [0 0 0] [1] 167.14/60.21 = [a_1(x_1)] 167.14/60.21 167.14/60.21 [f(b(), x_1)] = [1 2 2] [0] 167.14/60.21 [0 0 2] x_1 + [0] 167.14/60.21 [0 0 0] [2] 167.14/60.21 >= [1 0 0] [0] 167.14/60.21 [0 0 2] x_1 + [0] 167.14/60.21 [0 0 0] [0] 167.14/60.21 = [b_1(x_1)] 167.14/60.21 167.14/60.21 [a_1(b_1(a_1(x)))] = [1 2 0] [4] 167.14/60.21 [0 0 0] x + [0] 167.14/60.21 [0 0 0] [1] 167.14/60.21 > [1 2 0] [0] 167.14/60.21 [0 0 0] x + [0] 167.14/60.21 [0 0 0] [1] 167.14/60.21 = [a_1(b_1(b_1(a_1(x))))] 167.14/60.21 167.14/60.21 [b_1(b_1(b_1(x)))] = [1 0 0] [0] 167.14/60.21 [0 0 0] x + [0] 167.14/60.21 [0 0 0] [0] 167.14/60.21 >= [1 0 0] [0] 167.14/60.21 [0 0 0] x + [0] 167.14/60.21 [0 0 0] [0] 167.14/60.21 = [b_1(b_1(x))] 167.14/60.21 167.14/60.21 167.14/60.21 We return to the main proof. 167.14/60.21 167.14/60.21 We are left with following problem, upon which TcT provides the 167.14/60.21 certificate YES(O(1),O(1)). 167.14/60.21 167.14/60.21 Weak Trs: 167.14/60.21 { f(a(), x_1) -> a_1(x_1) 167.14/60.21 , f(b(), x_1) -> b_1(x_1) 167.14/60.21 , a_1(b_1(a_1(x))) -> a_1(b_1(b_1(a_1(x)))) 167.14/60.21 , b_1(b_1(b_1(x))) -> b_1(b_1(x)) } 167.14/60.21 Obligation: 167.14/60.21 derivational complexity 167.14/60.21 Answer: 167.14/60.21 YES(O(1),O(1)) 167.14/60.21 167.14/60.21 Empty rules are trivially bounded 167.14/60.21 167.14/60.21 Hurray, we answered YES(O(1),O(n^1)) 167.14/60.27 EOF