YES(O(1),O(n^1)) 789.34/297.04 YES(O(1),O(n^1)) 789.34/297.04 789.34/297.04 We are left with following problem, upon which TcT provides the 789.34/297.04 certificate YES(O(1),O(n^1)). 789.34/297.04 789.34/297.04 Strict Trs: 789.34/297.04 { a(a(x1)) -> b(x1) 789.34/297.04 , a(a(a(x1))) -> a(b(a(x1))) 789.34/297.04 , a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) 789.34/297.04 , a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) 789.34/297.04 , a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) 789.34/297.04 , a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) 789.34/297.04 , a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) 789.34/297.04 , a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) 789.34/297.04 , a(b(a(x1))) -> b(b(b(x1))) 789.34/297.04 , a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) 789.34/297.04 , a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) 789.34/297.04 , a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) 789.34/297.04 , a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) 789.34/297.04 , a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) 789.34/297.04 , a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) } 789.34/297.04 Obligation: 789.34/297.04 derivational complexity 789.34/297.04 Answer: 789.34/297.04 YES(O(1),O(n^1)) 789.34/297.04 789.34/297.04 The weightgap principle applies (using the following nonconstant 789.34/297.04 growth matrix-interpretation) 789.34/297.04 789.34/297.04 TcT has computed the following triangular matrix interpretation. 789.34/297.04 Note that the diagonal of the component-wise maxima of 789.34/297.04 interpretation-entries contains no more than 1 non-zero entries. 789.34/297.04 789.34/297.04 [a](x1) = [1] x1 + [1] 789.34/297.04 789.34/297.04 [b](x1) = [1] x1 + [1] 789.34/297.04 789.34/297.04 The order satisfies the following ordering constraints: 789.34/297.04 789.34/297.04 [a(a(x1))] = [1] x1 + [2] 789.34/297.04 > [1] x1 + [1] 789.34/297.04 = [b(x1)] 789.34/297.04 789.34/297.04 [a(a(a(x1)))] = [1] x1 + [3] 789.34/297.04 >= [1] x1 + [3] 789.34/297.04 = [a(b(a(x1)))] 789.34/297.04 789.34/297.04 [a(a(a(a(x1))))] = [1] x1 + [4] 789.34/297.04 ? [1] x1 + [5] 789.34/297.04 = [a(a(b(a(a(x1)))))] 789.34/297.04 789.34/297.04 [a(a(a(a(a(x1)))))] = [1] x1 + [5] 789.34/297.04 ? [1] x1 + [7] 789.34/297.04 = [a(a(a(b(a(a(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(a(b(a(x1)))))] = [1] x1 + [5] 789.34/297.04 ? [1] x1 + [7] 789.34/297.04 = [a(a(b(b(a(a(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(b(a(x1))))] = [1] x1 + [4] 789.34/297.04 ? [1] x1 + [5] 789.34/297.04 = [a(b(b(a(b(x1)))))] 789.34/297.04 789.34/297.04 [a(a(b(a(a(x1)))))] = [1] x1 + [5] 789.34/297.04 ? [1] x1 + [7] 789.34/297.04 = [a(b(a(b(a(b(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(b(b(a(x1)))))] = [1] x1 + [5] 789.34/297.04 ? [1] x1 + [7] 789.34/297.04 = [a(b(b(b(a(b(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(a(x1)))] = [1] x1 + [3] 789.34/297.04 >= [1] x1 + [3] 789.34/297.04 = [b(b(b(x1)))] 789.34/297.04 789.34/297.04 [a(b(a(a(x1))))] = [1] x1 + [4] 789.34/297.04 ? [1] x1 + [5] 789.34/297.04 = [b(a(b(b(a(x1)))))] 789.34/297.04 789.34/297.04 [a(b(a(a(a(x1)))))] = [1] x1 + [5] 789.34/297.04 ? [1] x1 + [7] 789.34/297.04 = [b(a(a(b(b(a(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(a(b(a(x1)))))] = [1] x1 + [5] 789.34/297.04 ? [1] x1 + [7] 789.34/297.04 = [b(a(b(b(b(a(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(b(a(x1))))] = [1] x1 + [4] 789.34/297.04 ? [1] x1 + [5] 789.34/297.04 = [b(b(b(b(b(x1)))))] 789.34/297.04 789.34/297.04 [a(b(b(a(a(x1)))))] = [1] x1 + [5] 789.34/297.04 ? [1] x1 + [7] 789.34/297.04 = [b(b(a(b(b(b(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(b(b(a(x1)))))] = [1] x1 + [5] 789.34/297.04 ? [1] x1 + [7] 789.34/297.04 = [b(b(b(b(b(b(b(x1)))))))] 789.34/297.04 789.34/297.04 789.34/297.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 789.34/297.04 789.34/297.04 We are left with following problem, upon which TcT provides the 789.34/297.04 certificate YES(O(1),O(n^1)). 789.34/297.04 789.34/297.04 Strict Trs: 789.34/297.04 { a(a(a(x1))) -> a(b(a(x1))) 789.34/297.04 , a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) 789.34/297.04 , a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) 789.34/297.04 , a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) 789.34/297.04 , a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) 789.34/297.04 , a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) 789.34/297.04 , a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) 789.34/297.04 , a(b(a(x1))) -> b(b(b(x1))) 789.34/297.04 , a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) 789.34/297.04 , a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) 789.34/297.04 , a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) 789.34/297.04 , a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) 789.34/297.04 , a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) 789.34/297.04 , a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) } 789.34/297.04 Weak Trs: { a(a(x1)) -> b(x1) } 789.34/297.04 Obligation: 789.34/297.04 derivational complexity 789.34/297.04 Answer: 789.34/297.04 YES(O(1),O(n^1)) 789.34/297.04 789.34/297.04 The weightgap principle applies (using the following nonconstant 789.34/297.04 growth matrix-interpretation) 789.34/297.04 789.34/297.04 TcT has computed the following triangular matrix interpretation. 789.34/297.04 Note that the diagonal of the component-wise maxima of 789.34/297.04 interpretation-entries contains no more than 1 non-zero entries. 789.34/297.04 789.34/297.04 [a](x1) = [1] x1 + [1] 789.34/297.04 789.34/297.04 [b](x1) = [1] x1 + [0] 789.34/297.04 789.34/297.04 The order satisfies the following ordering constraints: 789.34/297.04 789.34/297.04 [a(a(x1))] = [1] x1 + [2] 789.34/297.04 > [1] x1 + [0] 789.34/297.04 = [b(x1)] 789.34/297.04 789.34/297.04 [a(a(a(x1)))] = [1] x1 + [3] 789.34/297.04 > [1] x1 + [2] 789.34/297.04 = [a(b(a(x1)))] 789.34/297.04 789.34/297.04 [a(a(a(a(x1))))] = [1] x1 + [4] 789.34/297.04 >= [1] x1 + [4] 789.34/297.04 = [a(a(b(a(a(x1)))))] 789.34/297.04 789.34/297.04 [a(a(a(a(a(x1)))))] = [1] x1 + [5] 789.34/297.04 ? [1] x1 + [6] 789.34/297.04 = [a(a(a(b(a(a(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(a(b(a(x1)))))] = [1] x1 + [4] 789.34/297.04 >= [1] x1 + [4] 789.34/297.04 = [a(a(b(b(a(a(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(b(a(x1))))] = [1] x1 + [3] 789.34/297.04 > [1] x1 + [2] 789.34/297.04 = [a(b(b(a(b(x1)))))] 789.34/297.04 789.34/297.04 [a(a(b(a(a(x1)))))] = [1] x1 + [4] 789.34/297.04 >= [1] x1 + [4] 789.34/297.04 = [a(b(a(b(a(b(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(b(b(a(x1)))))] = [1] x1 + [3] 789.34/297.04 > [1] x1 + [2] 789.34/297.04 = [a(b(b(b(a(b(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(a(x1)))] = [1] x1 + [2] 789.34/297.04 > [1] x1 + [0] 789.34/297.04 = [b(b(b(x1)))] 789.34/297.04 789.34/297.04 [a(b(a(a(x1))))] = [1] x1 + [3] 789.34/297.04 > [1] x1 + [2] 789.34/297.04 = [b(a(b(b(a(x1)))))] 789.34/297.04 789.34/297.04 [a(b(a(a(a(x1)))))] = [1] x1 + [4] 789.34/297.04 >= [1] x1 + [4] 789.34/297.04 = [b(a(a(b(b(a(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(a(b(a(x1)))))] = [1] x1 + [3] 789.34/297.04 > [1] x1 + [2] 789.34/297.04 = [b(a(b(b(b(a(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(b(a(x1))))] = [1] x1 + [2] 789.34/297.04 > [1] x1 + [0] 789.34/297.04 = [b(b(b(b(b(x1)))))] 789.34/297.04 789.34/297.04 [a(b(b(a(a(x1)))))] = [1] x1 + [3] 789.34/297.04 > [1] x1 + [2] 789.34/297.04 = [b(b(a(b(b(b(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(b(b(a(x1)))))] = [1] x1 + [2] 789.34/297.04 > [1] x1 + [0] 789.34/297.04 = [b(b(b(b(b(b(b(x1)))))))] 789.34/297.04 789.34/297.04 789.34/297.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 789.34/297.04 789.34/297.04 We are left with following problem, upon which TcT provides the 789.34/297.04 certificate YES(O(1),O(n^1)). 789.34/297.04 789.34/297.04 Strict Trs: 789.34/297.04 { a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) 789.34/297.04 , a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) 789.34/297.04 , a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) 789.34/297.04 , a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) 789.34/297.04 , a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) } 789.34/297.04 Weak Trs: 789.34/297.04 { a(a(x1)) -> b(x1) 789.34/297.04 , a(a(a(x1))) -> a(b(a(x1))) 789.34/297.04 , a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) 789.34/297.04 , a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) 789.34/297.04 , a(b(a(x1))) -> b(b(b(x1))) 789.34/297.04 , a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) 789.34/297.04 , a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) 789.34/297.04 , a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) 789.34/297.04 , a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) 789.34/297.04 , a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) } 789.34/297.04 Obligation: 789.34/297.04 derivational complexity 789.34/297.04 Answer: 789.34/297.04 YES(O(1),O(n^1)) 789.34/297.04 789.34/297.04 We use the processor 'matrix interpretation of dimension 3' to 789.34/297.04 orient following rules strictly. 789.34/297.04 789.34/297.04 Trs: 789.34/297.04 { a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) 789.34/297.04 , a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) } 789.34/297.04 789.34/297.04 The induced complexity on above rules (modulo remaining rules) is 789.34/297.04 YES(?,O(n^1)) . These rules are moved into the corresponding weak 789.34/297.04 component(s). 789.34/297.04 789.34/297.04 Sub-proof: 789.34/297.04 ---------- 789.34/297.04 TcT has computed the following triangular matrix interpretation. 789.34/297.04 Note that the diagonal of the component-wise maxima of 789.34/297.04 interpretation-entries contains no more than 1 non-zero entries. 789.34/297.04 789.34/297.04 [1 1 0] [0] 789.34/297.04 [a](x1) = [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 789.34/297.04 [1 0 0] [0] 789.34/297.04 [b](x1) = [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 789.34/297.04 The order satisfies the following ordering constraints: 789.34/297.04 789.34/297.04 [a(a(x1))] = [1 1 0] [1] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 > [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(x1)] 789.34/297.04 789.34/297.04 [a(a(a(x1)))] = [1 1 0] [2] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 > [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [a(b(a(x1)))] 789.34/297.04 789.34/297.04 [a(a(a(a(x1))))] = [1 1 0] [3] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 > [1 1 0] [2] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [a(a(b(a(a(x1)))))] 789.34/297.04 789.34/297.04 [a(a(a(a(a(x1)))))] = [1 1 0] [4] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 >= [1 1 0] [4] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [a(a(a(b(a(a(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(a(b(a(x1)))))] = [1 1 0] [2] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 >= [1 0 0] [2] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [a(a(b(b(a(a(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(b(a(x1))))] = [1 1 0] [1] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 > [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [a(b(b(a(b(x1)))))] 789.34/297.04 789.34/297.04 [a(a(b(a(a(x1)))))] = [1 1 0] [2] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 > [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [a(b(a(b(a(b(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(b(b(a(x1)))))] = [1 1 0] [1] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 > [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [a(b(b(b(a(b(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(a(x1)))] = [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(b(b(x1)))] 789.34/297.04 789.34/297.04 [a(b(a(a(x1))))] = [1 1 0] [1] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 > [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(a(b(b(a(x1)))))] 789.34/297.04 789.34/297.04 [a(b(a(a(a(x1)))))] = [1 1 0] [2] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 >= [1 1 0] [2] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(a(a(b(b(a(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(a(b(a(x1)))))] = [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(a(b(b(b(a(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(b(a(x1))))] = [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(b(b(b(b(x1)))))] 789.34/297.04 789.34/297.04 [a(b(b(a(a(x1)))))] = [1 1 0] [1] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 > [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(b(a(b(b(b(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(b(b(a(x1)))))] = [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [1] 789.34/297.04 [0 0 0] [0] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(b(b(b(b(b(b(x1)))))))] 789.34/297.04 789.34/297.04 789.34/297.04 We return to the main proof. 789.34/297.04 789.34/297.04 We are left with following problem, upon which TcT provides the 789.34/297.04 certificate YES(O(1),O(n^1)). 789.34/297.04 789.34/297.04 Strict Trs: 789.34/297.04 { a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) 789.34/297.04 , a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) 789.34/297.04 , a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) } 789.34/297.04 Weak Trs: 789.34/297.04 { a(a(x1)) -> b(x1) 789.34/297.04 , a(a(a(x1))) -> a(b(a(x1))) 789.34/297.04 , a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) 789.34/297.04 , a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) 789.34/297.04 , a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) 789.34/297.04 , a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) 789.34/297.04 , a(b(a(x1))) -> b(b(b(x1))) 789.34/297.04 , a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) 789.34/297.04 , a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) 789.34/297.04 , a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) 789.34/297.04 , a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) 789.34/297.04 , a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) } 789.34/297.04 Obligation: 789.34/297.04 derivational complexity 789.34/297.04 Answer: 789.34/297.04 YES(O(1),O(n^1)) 789.34/297.04 789.34/297.04 We use the processor 'matrix interpretation of dimension 3' to 789.34/297.04 orient following rules strictly. 789.34/297.04 789.34/297.04 Trs: 789.34/297.04 { a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) 789.34/297.04 , a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) 789.34/297.04 , a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) } 789.34/297.04 789.34/297.04 The induced complexity on above rules (modulo remaining rules) is 789.34/297.04 YES(?,O(n^1)) . These rules are moved into the corresponding weak 789.34/297.04 component(s). 789.34/297.04 789.34/297.04 Sub-proof: 789.34/297.04 ---------- 789.34/297.04 TcT has computed the following triangular matrix interpretation. 789.34/297.04 Note that the diagonal of the component-wise maxima of 789.34/297.04 interpretation-entries contains no more than 1 non-zero entries. 789.34/297.04 789.34/297.04 [1 1 0] [0] 789.34/297.04 [a](x1) = [0 0 1] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 789.34/297.04 [1 0 0] [0] 789.34/297.04 [b](x1) = [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 789.34/297.04 The order satisfies the following ordering constraints: 789.34/297.04 789.34/297.04 [a(a(x1))] = [1 1 1] [0] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(x1)] 789.34/297.04 789.34/297.04 [a(a(a(x1)))] = [1 1 1] [2] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 > [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 = [a(b(a(x1)))] 789.34/297.04 789.34/297.04 [a(a(a(a(x1))))] = [1 1 1] [4] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 > [1 1 1] [0] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 = [a(a(b(a(a(x1)))))] 789.34/297.04 789.34/297.04 [a(a(a(a(a(x1)))))] = [1 1 1] [6] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 > [1 1 1] [4] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 = [a(a(a(b(a(a(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(a(b(a(x1)))))] = [1 1 0] [2] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 > [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 = [a(a(b(b(a(a(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(b(a(x1))))] = [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 = [a(b(b(a(b(x1)))))] 789.34/297.04 789.34/297.04 [a(a(b(a(a(x1)))))] = [1 1 1] [0] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 >= [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 = [a(b(a(b(a(b(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(a(b(b(a(x1)))))] = [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [2] 789.34/297.04 [0 0 0] [2] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 = [a(b(b(b(a(b(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(a(x1)))] = [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(b(b(x1)))] 789.34/297.04 789.34/297.04 [a(b(a(a(x1))))] = [1 1 1] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 >= [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(a(b(b(a(x1)))))] 789.34/297.04 789.34/297.04 [a(b(a(a(a(x1)))))] = [1 1 1] [2] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 > [1 1 1] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(a(a(b(b(a(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(a(b(a(x1)))))] = [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(a(b(b(b(a(b(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(b(a(x1))))] = [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(b(b(b(b(x1)))))] 789.34/297.04 789.34/297.04 [a(b(b(a(a(x1)))))] = [1 1 1] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 >= [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(b(a(b(b(b(a(x1)))))))] 789.34/297.04 789.34/297.04 [a(b(b(b(a(x1)))))] = [1 1 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [2] 789.34/297.04 >= [1 0 0] [0] 789.34/297.04 [0 0 0] x1 + [0] 789.34/297.04 [0 0 0] [0] 789.34/297.04 = [b(b(b(b(b(b(b(x1)))))))] 789.34/297.04 789.34/297.04 789.34/297.04 We return to the main proof. 789.34/297.04 789.34/297.04 We are left with following problem, upon which TcT provides the 789.34/297.04 certificate YES(O(1),O(1)). 789.34/297.04 789.34/297.04 Weak Trs: 789.34/297.04 { a(a(x1)) -> b(x1) 789.34/297.04 , a(a(a(x1))) -> a(b(a(x1))) 789.34/297.04 , a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) 789.34/297.04 , a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) 789.34/297.04 , a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) 789.34/297.04 , a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) 789.34/297.04 , a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) 789.34/297.04 , a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) 789.34/297.04 , a(b(a(x1))) -> b(b(b(x1))) 789.34/297.04 , a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) 789.34/297.04 , a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) 789.34/297.04 , a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) 789.34/297.04 , a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) 789.34/297.04 , a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) 789.34/297.04 , a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) } 789.34/297.04 Obligation: 789.34/297.04 derivational complexity 789.34/297.04 Answer: 789.34/297.04 YES(O(1),O(1)) 789.34/297.04 789.34/297.04 Empty rules are trivially bounded 789.34/297.04 789.34/297.04 Hurray, we answered YES(O(1),O(n^1)) 789.50/297.14 EOF