YES(O(1),O(n^2)) 236.20/73.43 YES(O(1),O(n^2)) 236.20/73.43 236.20/73.43 We are left with following problem, upon which TcT provides the 236.20/73.43 certificate YES(O(1),O(n^2)). 236.20/73.43 236.20/73.43 Strict Trs: 236.20/73.43 { active(f(a(), X, X)) -> mark(f(X, b(), b())) 236.20/73.43 , active(b()) -> mark(a()) 236.20/73.43 , f(X1, X2, active(X3)) -> f(X1, X2, X3) 236.20/73.43 , f(X1, X2, mark(X3)) -> f(X1, X2, X3) 236.20/73.43 , f(X1, active(X2), X3) -> f(X1, X2, X3) 236.20/73.43 , f(X1, mark(X2), X3) -> f(X1, X2, X3) 236.20/73.43 , f(active(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.43 , f(mark(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.43 , mark(f(X1, X2, X3)) -> active(f(X1, mark(X2), X3)) 236.20/73.43 , mark(a()) -> active(a()) 236.20/73.43 , mark(b()) -> active(b()) } 236.20/73.43 Obligation: 236.20/73.43 derivational complexity 236.20/73.43 Answer: 236.20/73.43 YES(O(1),O(n^2)) 236.20/73.43 236.20/73.43 The weightgap principle applies (using the following nonconstant 236.20/73.43 growth matrix-interpretation) 236.20/73.43 236.20/73.43 TcT has computed the following triangular matrix interpretation. 236.20/73.43 Note that the diagonal of the component-wise maxima of 236.20/73.43 interpretation-entries contains no more than 1 non-zero entries. 236.20/73.43 236.20/73.43 [active](x1) = [1] x1 + [0] 236.20/73.43 236.20/73.43 [f](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 236.20/73.43 236.20/73.43 [a] = [0] 236.20/73.43 236.20/73.43 [mark](x1) = [1] x1 + [1] 236.20/73.43 236.20/73.43 [b] = [0] 236.20/73.43 236.20/73.43 The order satisfies the following ordering constraints: 236.20/73.43 236.20/73.43 [active(f(a(), X, X))] = [2] X + [0] 236.20/73.43 ? [1] X + [1] 236.20/73.43 = [mark(f(X, b(), b()))] 236.20/73.43 236.20/73.43 [active(b())] = [0] 236.20/73.43 ? [1] 236.20/73.43 = [mark(a())] 236.20/73.43 236.20/73.43 [f(X1, X2, active(X3))] = [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, X2, mark(X3))] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 > [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, active(X2), X3)] = [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, mark(X2), X3)] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 > [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(active(X1), X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(mark(X1), X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 > [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [mark(f(X1, X2, X3))] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 = [active(f(X1, mark(X2), X3))] 236.20/73.43 236.20/73.43 [mark(a())] = [1] 236.20/73.43 > [0] 236.20/73.43 = [active(a())] 236.20/73.43 236.20/73.43 [mark(b())] = [1] 236.20/73.43 > [0] 236.20/73.43 = [active(b())] 236.20/73.43 236.20/73.43 236.20/73.43 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 236.20/73.43 236.20/73.43 We are left with following problem, upon which TcT provides the 236.20/73.43 certificate YES(O(1),O(n^2)). 236.20/73.43 236.20/73.43 Strict Trs: 236.20/73.43 { active(f(a(), X, X)) -> mark(f(X, b(), b())) 236.20/73.43 , active(b()) -> mark(a()) 236.20/73.43 , f(X1, X2, active(X3)) -> f(X1, X2, X3) 236.20/73.43 , f(X1, active(X2), X3) -> f(X1, X2, X3) 236.20/73.43 , f(active(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.43 , mark(f(X1, X2, X3)) -> active(f(X1, mark(X2), X3)) } 236.20/73.43 Weak Trs: 236.20/73.43 { f(X1, X2, mark(X3)) -> f(X1, X2, X3) 236.20/73.43 , f(X1, mark(X2), X3) -> f(X1, X2, X3) 236.20/73.43 , f(mark(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.43 , mark(a()) -> active(a()) 236.20/73.43 , mark(b()) -> active(b()) } 236.20/73.43 Obligation: 236.20/73.43 derivational complexity 236.20/73.43 Answer: 236.20/73.43 YES(O(1),O(n^2)) 236.20/73.43 236.20/73.43 The weightgap principle applies (using the following nonconstant 236.20/73.43 growth matrix-interpretation) 236.20/73.43 236.20/73.43 TcT has computed the following triangular matrix interpretation. 236.20/73.43 Note that the diagonal of the component-wise maxima of 236.20/73.43 interpretation-entries contains no more than 1 non-zero entries. 236.20/73.43 236.20/73.43 [active](x1) = [1] x1 + [0] 236.20/73.43 236.20/73.43 [f](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 236.20/73.43 236.20/73.43 [a] = [0] 236.20/73.43 236.20/73.43 [mark](x1) = [1] x1 + [0] 236.20/73.43 236.20/73.43 [b] = [2] 236.20/73.43 236.20/73.43 The order satisfies the following ordering constraints: 236.20/73.43 236.20/73.43 [active(f(a(), X, X))] = [2] X + [0] 236.20/73.43 ? [1] X + [4] 236.20/73.43 = [mark(f(X, b(), b()))] 236.20/73.43 236.20/73.43 [active(b())] = [2] 236.20/73.43 > [0] 236.20/73.43 = [mark(a())] 236.20/73.43 236.20/73.43 [f(X1, X2, active(X3))] = [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, X2, mark(X3))] = [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, active(X2), X3)] = [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, mark(X2), X3)] = [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(active(X1), X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(mark(X1), X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [mark(f(X1, X2, X3))] = [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 >= [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [active(f(X1, mark(X2), X3))] 236.20/73.43 236.20/73.43 [mark(a())] = [0] 236.20/73.43 >= [0] 236.20/73.43 = [active(a())] 236.20/73.43 236.20/73.43 [mark(b())] = [2] 236.20/73.43 >= [2] 236.20/73.43 = [active(b())] 236.20/73.43 236.20/73.43 236.20/73.43 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 236.20/73.43 236.20/73.43 We are left with following problem, upon which TcT provides the 236.20/73.43 certificate YES(O(1),O(n^2)). 236.20/73.43 236.20/73.43 Strict Trs: 236.20/73.43 { active(f(a(), X, X)) -> mark(f(X, b(), b())) 236.20/73.43 , f(X1, X2, active(X3)) -> f(X1, X2, X3) 236.20/73.43 , f(X1, active(X2), X3) -> f(X1, X2, X3) 236.20/73.43 , f(active(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.43 , mark(f(X1, X2, X3)) -> active(f(X1, mark(X2), X3)) } 236.20/73.43 Weak Trs: 236.20/73.43 { active(b()) -> mark(a()) 236.20/73.43 , f(X1, X2, mark(X3)) -> f(X1, X2, X3) 236.20/73.43 , f(X1, mark(X2), X3) -> f(X1, X2, X3) 236.20/73.43 , f(mark(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.43 , mark(a()) -> active(a()) 236.20/73.43 , mark(b()) -> active(b()) } 236.20/73.43 Obligation: 236.20/73.43 derivational complexity 236.20/73.43 Answer: 236.20/73.43 YES(O(1),O(n^2)) 236.20/73.43 236.20/73.43 The weightgap principle applies (using the following nonconstant 236.20/73.43 growth matrix-interpretation) 236.20/73.43 236.20/73.43 TcT has computed the following triangular matrix interpretation. 236.20/73.43 Note that the diagonal of the component-wise maxima of 236.20/73.43 interpretation-entries contains no more than 1 non-zero entries. 236.20/73.43 236.20/73.43 [active](x1) = [1] x1 + [1] 236.20/73.43 236.20/73.43 [f](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 236.20/73.43 236.20/73.43 [a] = [0] 236.20/73.43 236.20/73.43 [mark](x1) = [1] x1 + [1] 236.20/73.43 236.20/73.43 [b] = [0] 236.20/73.43 236.20/73.43 The order satisfies the following ordering constraints: 236.20/73.43 236.20/73.43 [active(f(a(), X, X))] = [2] X + [1] 236.20/73.43 >= [1] X + [1] 236.20/73.43 = [mark(f(X, b(), b()))] 236.20/73.43 236.20/73.43 [active(b())] = [1] 236.20/73.43 >= [1] 236.20/73.43 = [mark(a())] 236.20/73.43 236.20/73.43 [f(X1, X2, active(X3))] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 > [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, X2, mark(X3))] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 > [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, active(X2), X3)] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 > [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, mark(X2), X3)] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 > [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(active(X1), X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 > [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(mark(X1), X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 > [1] X1 + [1] X2 + [1] X3 + [0] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [mark(f(X1, X2, X3))] = [1] X1 + [1] X2 + [1] X3 + [1] 236.20/73.43 ? [1] X1 + [1] X2 + [1] X3 + [2] 236.20/73.43 = [active(f(X1, mark(X2), X3))] 236.20/73.43 236.20/73.43 [mark(a())] = [1] 236.20/73.43 >= [1] 236.20/73.43 = [active(a())] 236.20/73.43 236.20/73.43 [mark(b())] = [1] 236.20/73.43 >= [1] 236.20/73.43 = [active(b())] 236.20/73.43 236.20/73.43 236.20/73.43 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 236.20/73.43 236.20/73.43 We are left with following problem, upon which TcT provides the 236.20/73.43 certificate YES(O(1),O(n^2)). 236.20/73.43 236.20/73.43 Strict Trs: 236.20/73.43 { active(f(a(), X, X)) -> mark(f(X, b(), b())) 236.20/73.43 , mark(f(X1, X2, X3)) -> active(f(X1, mark(X2), X3)) } 236.20/73.43 Weak Trs: 236.20/73.43 { active(b()) -> mark(a()) 236.20/73.43 , f(X1, X2, active(X3)) -> f(X1, X2, X3) 236.20/73.43 , f(X1, X2, mark(X3)) -> f(X1, X2, X3) 236.20/73.43 , f(X1, active(X2), X3) -> f(X1, X2, X3) 236.20/73.43 , f(X1, mark(X2), X3) -> f(X1, X2, X3) 236.20/73.43 , f(active(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.43 , f(mark(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.43 , mark(a()) -> active(a()) 236.20/73.43 , mark(b()) -> active(b()) } 236.20/73.43 Obligation: 236.20/73.43 derivational complexity 236.20/73.43 Answer: 236.20/73.43 YES(O(1),O(n^2)) 236.20/73.43 236.20/73.43 The weightgap principle applies (using the following nonconstant 236.20/73.43 growth matrix-interpretation) 236.20/73.43 236.20/73.43 TcT has computed the following triangular matrix interpretation. 236.20/73.43 Note that the diagonal of the component-wise maxima of 236.20/73.43 interpretation-entries contains no more than 1 non-zero entries. 236.20/73.43 236.20/73.43 [active](x1) = [1 1] x1 + [0] 236.20/73.43 [0 0] [0] 236.20/73.43 236.20/73.43 [f](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 236.20/73.43 [a] = [0] 236.20/73.43 [0] 236.20/73.43 236.20/73.43 [mark](x1) = [1 0] x1 + [0] 236.20/73.43 [0 0] [0] 236.20/73.43 236.20/73.43 [b] = [0] 236.20/73.43 [0] 236.20/73.43 236.20/73.43 The order satisfies the following ordering constraints: 236.20/73.43 236.20/73.43 [active(f(a(), X, X))] = [2 0] X + [1] 236.20/73.43 [0 0] [0] 236.20/73.43 > [1 0] X + [0] 236.20/73.43 [0 0] [0] 236.20/73.43 = [mark(f(X, b(), b()))] 236.20/73.43 236.20/73.43 [active(b())] = [0] 236.20/73.43 [0] 236.20/73.43 >= [0] 236.20/73.43 [0] 236.20/73.43 = [mark(a())] 236.20/73.43 236.20/73.43 [f(X1, X2, active(X3))] = [1 0] X1 + [1 0] X2 + [1 1] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 >= [1 0] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, X2, mark(X3))] = [1 0] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 >= [1 0] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, active(X2), X3)] = [1 0] X1 + [1 1] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 >= [1 0] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, mark(X2), X3)] = [1 0] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 >= [1 0] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(active(X1), X2, X3)] = [1 1] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 >= [1 0] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(mark(X1), X2, X3)] = [1 0] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 >= [1 0] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [1] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [mark(f(X1, X2, X3))] = [1 0] X1 + [1 0] X2 + [1 0] X3 + [0] 236.20/73.43 [0 0] [0 0] [0 0] [0] 236.20/73.43 ? [1 0] X1 + [1 0] X2 + [1 0] X3 + [1] 236.20/73.43 [0 0] [0 0] [0 0] [0] 236.20/73.43 = [active(f(X1, mark(X2), X3))] 236.20/73.43 236.20/73.43 [mark(a())] = [0] 236.20/73.43 [0] 236.20/73.43 >= [0] 236.20/73.43 [0] 236.20/73.43 = [active(a())] 236.20/73.43 236.20/73.43 [mark(b())] = [0] 236.20/73.43 [0] 236.20/73.43 >= [0] 236.20/73.43 [0] 236.20/73.43 = [active(b())] 236.20/73.43 236.20/73.43 236.20/73.43 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 236.20/73.43 236.20/73.43 We are left with following problem, upon which TcT provides the 236.20/73.43 certificate YES(O(1),O(n^2)). 236.20/73.43 236.20/73.43 Strict Trs: { mark(f(X1, X2, X3)) -> active(f(X1, mark(X2), X3)) } 236.20/73.43 Weak Trs: 236.20/73.43 { active(f(a(), X, X)) -> mark(f(X, b(), b())) 236.20/73.43 , active(b()) -> mark(a()) 236.20/73.43 , f(X1, X2, active(X3)) -> f(X1, X2, X3) 236.20/73.43 , f(X1, X2, mark(X3)) -> f(X1, X2, X3) 236.20/73.43 , f(X1, active(X2), X3) -> f(X1, X2, X3) 236.20/73.43 , f(X1, mark(X2), X3) -> f(X1, X2, X3) 236.20/73.43 , f(active(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.43 , f(mark(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.43 , mark(a()) -> active(a()) 236.20/73.43 , mark(b()) -> active(b()) } 236.20/73.43 Obligation: 236.20/73.43 derivational complexity 236.20/73.43 Answer: 236.20/73.43 YES(O(1),O(n^2)) 236.20/73.43 236.20/73.43 We use the processor 'matrix interpretation of dimension 4' to 236.20/73.43 orient following rules strictly. 236.20/73.43 236.20/73.43 Trs: { mark(f(X1, X2, X3)) -> active(f(X1, mark(X2), X3)) } 236.20/73.43 236.20/73.43 The induced complexity on above rules (modulo remaining rules) is 236.20/73.43 YES(?,O(n^2)) . These rules are moved into the corresponding weak 236.20/73.43 component(s). 236.20/73.43 236.20/73.43 Sub-proof: 236.20/73.43 ---------- 236.20/73.43 TcT has computed the following triangular matrix interpretation. 236.20/73.43 Note that the diagonal of the component-wise maxima of 236.20/73.43 interpretation-entries contains no more than 2 non-zero entries. 236.20/73.43 236.20/73.43 [1 1 0 0] [0] 236.20/73.43 [active](x1) = [0 0 0 0] x1 + [0] 236.20/73.43 [0 0 1 1] [0] 236.20/73.43 [0 0 0 0] [0] 236.20/73.43 236.20/73.43 [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 236.20/73.43 [f](x1, x2, x3) = [0 0 1 0] x1 + [0 0 0 0] x2 + [0 0 1 0] x3 + [0] 236.20/73.43 [0 0 1 0] [0 0 1 1] [0 0 1 0] [0] 236.20/73.43 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.43 236.20/73.43 [0] 236.20/73.43 [a] = [0] 236.20/73.43 [1] 236.20/73.43 [0] 236.20/73.43 236.20/73.43 [1 0 1 1] [0] 236.20/73.43 [mark](x1) = [0 0 0 0] x1 + [0] 236.20/73.43 [0 0 1 1] [0] 236.20/73.43 [0 0 0 0] [0] 236.20/73.43 236.20/73.43 [0] 236.20/73.43 [b] = [1] 236.20/73.43 [0] 236.20/73.43 [1] 236.20/73.43 236.20/73.43 The order satisfies the following ordering constraints: 236.20/73.43 236.20/73.43 [active(f(a(), X, X))] = [2 0 2 0] [2] 236.20/73.43 [0 0 0 0] X + [0] 236.20/73.43 [0 0 2 1] [2] 236.20/73.43 [0 0 0 0] [0] 236.20/73.43 >= [1 0 2 0] [2] 236.20/73.43 [0 0 0 0] X + [0] 236.20/73.43 [0 0 1 0] [2] 236.20/73.43 [0 0 0 0] [0] 236.20/73.43 = [mark(f(X, b(), b()))] 236.20/73.43 236.20/73.43 [active(b())] = [1] 236.20/73.43 [0] 236.20/73.43 [1] 236.20/73.43 [0] 236.20/73.43 >= [1] 236.20/73.43 [0] 236.20/73.43 [1] 236.20/73.43 [0] 236.20/73.43 = [mark(a())] 236.20/73.43 236.20/73.43 [f(X1, X2, active(X3))] = [1 0 1 0] [1 0 0 0] [1 1 1 1] [0] 236.20/73.43 [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 1] X3 + [0] 236.20/73.43 [0 0 1 0] [0 0 1 1] [0 0 1 1] [0] 236.20/73.43 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.43 >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 236.20/73.43 [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] 236.20/73.43 [0 0 1 0] [0 0 1 1] [0 0 1 0] [0] 236.20/73.43 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.43 = [f(X1, X2, X3)] 236.20/73.43 236.20/73.43 [f(X1, X2, mark(X3))] = [1 0 1 0] [1 0 0 0] [1 0 2 2] [0] 236.20/73.43 [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 1] X3 + [0] 236.20/73.43 [0 0 1 0] [0 0 1 1] [0 0 1 1] [0] 236.20/73.43 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.44 >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 236.20/73.44 [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] 236.20/73.44 [0 0 1 0] [0 0 1 1] [0 0 1 0] [0] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.44 = [f(X1, X2, X3)] 236.20/73.44 236.20/73.44 [f(X1, active(X2), X3)] = [1 0 1 0] [1 1 0 0] [1 0 1 0] [0] 236.20/73.44 [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] 236.20/73.44 [0 0 1 0] [0 0 1 1] [0 0 1 0] [0] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.44 >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 236.20/73.44 [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] 236.20/73.44 [0 0 1 0] [0 0 1 1] [0 0 1 0] [0] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.44 = [f(X1, X2, X3)] 236.20/73.44 236.20/73.44 [f(X1, mark(X2), X3)] = [1 0 1 0] [1 0 1 1] [1 0 1 0] [0] 236.20/73.44 [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] 236.20/73.44 [0 0 1 0] [0 0 1 1] [0 0 1 0] [0] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.44 >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 236.20/73.44 [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] 236.20/73.44 [0 0 1 0] [0 0 1 1] [0 0 1 0] [0] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.44 = [f(X1, X2, X3)] 236.20/73.44 236.20/73.44 [f(active(X1), X2, X3)] = [1 1 1 1] [1 0 0 0] [1 0 1 0] [0] 236.20/73.44 [0 0 1 1] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] 236.20/73.44 [0 0 1 1] [0 0 1 1] [0 0 1 0] [0] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.44 >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 236.20/73.44 [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] 236.20/73.44 [0 0 1 0] [0 0 1 1] [0 0 1 0] [0] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.44 = [f(X1, X2, X3)] 236.20/73.44 236.20/73.44 [f(mark(X1), X2, X3)] = [1 0 2 2] [1 0 0 0] [1 0 1 0] [0] 236.20/73.44 [0 0 1 1] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] 236.20/73.44 [0 0 1 1] [0 0 1 1] [0 0 1 0] [0] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.44 >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 236.20/73.44 [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] 236.20/73.44 [0 0 1 0] [0 0 1 1] [0 0 1 0] [0] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 236.20/73.44 = [f(X1, X2, X3)] 236.20/73.44 236.20/73.44 [mark(f(X1, X2, X3))] = [1 0 2 0] [1 0 1 1] [1 0 2 0] [1] 236.20/73.44 [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] 236.20/73.44 [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] 236.20/73.44 > [1 0 2 0] [1 0 1 1] [1 0 2 0] [0] 236.20/73.44 [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] 236.20/73.44 [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] 236.20/73.44 [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] 236.20/73.44 = [active(f(X1, mark(X2), X3))] 236.20/73.44 236.20/73.44 [mark(a())] = [1] 236.20/73.44 [0] 236.20/73.44 [1] 236.20/73.44 [0] 236.20/73.44 > [0] 236.20/73.44 [0] 236.20/73.44 [1] 236.20/73.44 [0] 236.20/73.44 = [active(a())] 236.20/73.44 236.20/73.44 [mark(b())] = [1] 236.20/73.44 [0] 236.20/73.44 [1] 236.20/73.44 [0] 236.20/73.44 >= [1] 236.20/73.44 [0] 236.20/73.44 [1] 236.20/73.44 [0] 236.20/73.44 = [active(b())] 236.20/73.44 236.20/73.44 236.20/73.44 We return to the main proof. 236.20/73.44 236.20/73.44 We are left with following problem, upon which TcT provides the 236.20/73.44 certificate YES(O(1),O(1)). 236.20/73.44 236.20/73.44 Weak Trs: 236.20/73.44 { active(f(a(), X, X)) -> mark(f(X, b(), b())) 236.20/73.44 , active(b()) -> mark(a()) 236.20/73.44 , f(X1, X2, active(X3)) -> f(X1, X2, X3) 236.20/73.44 , f(X1, X2, mark(X3)) -> f(X1, X2, X3) 236.20/73.44 , f(X1, active(X2), X3) -> f(X1, X2, X3) 236.20/73.44 , f(X1, mark(X2), X3) -> f(X1, X2, X3) 236.20/73.44 , f(active(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.44 , f(mark(X1), X2, X3) -> f(X1, X2, X3) 236.20/73.44 , mark(f(X1, X2, X3)) -> active(f(X1, mark(X2), X3)) 236.20/73.44 , mark(a()) -> active(a()) 236.20/73.44 , mark(b()) -> active(b()) } 236.20/73.44 Obligation: 236.20/73.44 derivational complexity 236.20/73.44 Answer: 236.20/73.44 YES(O(1),O(1)) 236.20/73.44 236.20/73.44 Empty rules are trivially bounded 236.20/73.44 236.20/73.44 Hurray, we answered YES(O(1),O(n^2)) 236.20/73.45 EOF