YES(O(1),O(n^2)) 197.68/60.04 YES(O(1),O(n^2)) 197.68/60.04 197.68/60.04 We are left with following problem, upon which TcT provides the 197.68/60.04 certificate YES(O(1),O(n^2)). 197.68/60.04 197.68/60.04 Strict Trs: 197.68/60.04 { active(f(f(a()))) -> mark(f(g(f(a())))) 197.68/60.04 , f(active(X)) -> f(X) 197.68/60.04 , f(mark(X)) -> f(X) 197.68/60.04 , mark(f(X)) -> active(f(X)) 197.68/60.04 , mark(a()) -> active(a()) 197.68/60.04 , mark(g(X)) -> active(g(mark(X))) 197.68/60.04 , g(active(X)) -> g(X) 197.68/60.04 , g(mark(X)) -> g(X) } 197.68/60.04 Obligation: 197.68/60.04 derivational complexity 197.68/60.04 Answer: 197.68/60.04 YES(O(1),O(n^2)) 197.68/60.04 197.68/60.04 The weightgap principle applies (using the following nonconstant 197.68/60.04 growth matrix-interpretation) 197.68/60.04 197.68/60.04 TcT has computed the following triangular matrix interpretation. 197.68/60.04 Note that the diagonal of the component-wise maxima of 197.68/60.04 interpretation-entries contains no more than 1 non-zero entries. 197.68/60.04 197.68/60.04 [active](x1) = [1] x1 + [0] 197.68/60.04 197.68/60.04 [f](x1) = [1] x1 + [0] 197.68/60.04 197.68/60.04 [a] = [0] 197.68/60.04 197.68/60.04 [mark](x1) = [1] x1 + [1] 197.68/60.04 197.68/60.04 [g](x1) = [1] x1 + [0] 197.68/60.04 197.68/60.04 The order satisfies the following ordering constraints: 197.68/60.04 197.68/60.04 [active(f(f(a())))] = [0] 197.68/60.04 ? [1] 197.68/60.04 = [mark(f(g(f(a()))))] 197.68/60.04 197.68/60.04 [f(active(X))] = [1] X + [0] 197.68/60.04 >= [1] X + [0] 197.68/60.04 = [f(X)] 197.68/60.04 197.68/60.04 [f(mark(X))] = [1] X + [1] 197.68/60.04 > [1] X + [0] 197.68/60.04 = [f(X)] 197.68/60.04 197.68/60.04 [mark(f(X))] = [1] X + [1] 197.68/60.04 > [1] X + [0] 197.68/60.04 = [active(f(X))] 197.68/60.04 197.68/60.04 [mark(a())] = [1] 197.68/60.04 > [0] 197.68/60.04 = [active(a())] 197.68/60.04 197.68/60.04 [mark(g(X))] = [1] X + [1] 197.68/60.04 >= [1] X + [1] 197.68/60.04 = [active(g(mark(X)))] 197.68/60.04 197.68/60.04 [g(active(X))] = [1] X + [0] 197.68/60.04 >= [1] X + [0] 197.68/60.04 = [g(X)] 197.68/60.04 197.68/60.04 [g(mark(X))] = [1] X + [1] 197.68/60.04 > [1] X + [0] 197.68/60.04 = [g(X)] 197.68/60.04 197.68/60.04 197.68/60.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 197.68/60.04 197.68/60.04 We are left with following problem, upon which TcT provides the 197.68/60.04 certificate YES(O(1),O(n^2)). 197.68/60.04 197.68/60.04 Strict Trs: 197.68/60.04 { active(f(f(a()))) -> mark(f(g(f(a())))) 197.68/60.04 , f(active(X)) -> f(X) 197.68/60.04 , mark(g(X)) -> active(g(mark(X))) 197.68/60.04 , g(active(X)) -> g(X) } 197.68/60.04 Weak Trs: 197.68/60.04 { f(mark(X)) -> f(X) 197.68/60.04 , mark(f(X)) -> active(f(X)) 197.68/60.04 , mark(a()) -> active(a()) 197.68/60.04 , g(mark(X)) -> g(X) } 197.68/60.04 Obligation: 197.68/60.04 derivational complexity 197.68/60.04 Answer: 197.68/60.04 YES(O(1),O(n^2)) 197.68/60.04 197.68/60.04 The weightgap principle applies (using the following nonconstant 197.68/60.04 growth matrix-interpretation) 197.68/60.04 197.68/60.04 TcT has computed the following triangular matrix interpretation. 197.68/60.04 Note that the diagonal of the component-wise maxima of 197.68/60.04 interpretation-entries contains no more than 1 non-zero entries. 197.68/60.04 197.68/60.04 [active](x1) = [1] x1 + [1] 197.68/60.04 197.68/60.04 [f](x1) = [1] x1 + [0] 197.68/60.04 197.68/60.04 [a] = [0] 197.68/60.04 197.68/60.04 [mark](x1) = [1] x1 + [1] 197.68/60.04 197.68/60.04 [g](x1) = [1] x1 + [0] 197.68/60.04 197.68/60.04 The order satisfies the following ordering constraints: 197.68/60.04 197.68/60.04 [active(f(f(a())))] = [1] 197.68/60.04 >= [1] 197.68/60.04 = [mark(f(g(f(a()))))] 197.68/60.04 197.68/60.04 [f(active(X))] = [1] X + [1] 197.68/60.04 > [1] X + [0] 197.68/60.04 = [f(X)] 197.68/60.04 197.68/60.04 [f(mark(X))] = [1] X + [1] 197.68/60.04 > [1] X + [0] 197.68/60.04 = [f(X)] 197.68/60.04 197.68/60.04 [mark(f(X))] = [1] X + [1] 197.68/60.04 >= [1] X + [1] 197.68/60.04 = [active(f(X))] 197.68/60.04 197.68/60.04 [mark(a())] = [1] 197.68/60.04 >= [1] 197.68/60.04 = [active(a())] 197.68/60.04 197.68/60.04 [mark(g(X))] = [1] X + [1] 197.68/60.04 ? [1] X + [2] 197.68/60.04 = [active(g(mark(X)))] 197.68/60.04 197.68/60.04 [g(active(X))] = [1] X + [1] 197.68/60.04 > [1] X + [0] 197.68/60.04 = [g(X)] 197.68/60.04 197.68/60.04 [g(mark(X))] = [1] X + [1] 197.68/60.04 > [1] X + [0] 197.68/60.04 = [g(X)] 197.68/60.04 197.68/60.04 197.68/60.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 197.68/60.04 197.68/60.04 We are left with following problem, upon which TcT provides the 197.68/60.04 certificate YES(O(1),O(n^2)). 197.68/60.04 197.68/60.04 Strict Trs: 197.68/60.04 { active(f(f(a()))) -> mark(f(g(f(a())))) 197.68/60.04 , mark(g(X)) -> active(g(mark(X))) } 197.68/60.04 Weak Trs: 197.68/60.04 { f(active(X)) -> f(X) 197.68/60.04 , f(mark(X)) -> f(X) 197.68/60.04 , mark(f(X)) -> active(f(X)) 197.68/60.04 , mark(a()) -> active(a()) 197.68/60.04 , g(active(X)) -> g(X) 197.68/60.04 , g(mark(X)) -> g(X) } 197.68/60.04 Obligation: 197.68/60.04 derivational complexity 197.68/60.04 Answer: 197.68/60.04 YES(O(1),O(n^2)) 197.68/60.04 197.68/60.04 We use the processor 'matrix interpretation of dimension 2' to 197.68/60.04 orient following rules strictly. 197.68/60.04 197.68/60.04 Trs: { active(f(f(a()))) -> mark(f(g(f(a())))) } 197.68/60.04 197.68/60.04 The induced complexity on above rules (modulo remaining rules) is 197.68/60.04 YES(?,O(n^1)) . These rules are moved into the corresponding weak 197.68/60.04 component(s). 197.68/60.04 197.68/60.04 Sub-proof: 197.68/60.04 ---------- 197.68/60.04 TcT has computed the following triangular matrix interpretation. 197.68/60.04 197.68/60.04 [active](x1) = [1 0] x1 + [0] 197.68/60.04 [0 1] [0] 197.68/60.04 197.68/60.04 [f](x1) = [1 1] x1 + [0] 197.68/60.04 [0 0] [1] 197.68/60.04 197.68/60.04 [a] = [0] 197.68/60.04 [0] 197.68/60.04 197.68/60.04 [mark](x1) = [1 0] x1 + [0] 197.68/60.04 [0 1] [0] 197.68/60.04 197.68/60.04 [g](x1) = [1 0] x1 + [0] 197.68/60.04 [0 0] [0] 197.68/60.04 197.68/60.04 The order satisfies the following ordering constraints: 197.68/60.04 197.68/60.04 [active(f(f(a())))] = [1] 197.68/60.04 [1] 197.68/60.04 > [0] 197.68/60.04 [1] 197.68/60.04 = [mark(f(g(f(a()))))] 197.68/60.04 197.68/60.04 [f(active(X))] = [1 1] X + [0] 197.68/60.04 [0 0] [1] 197.68/60.04 >= [1 1] X + [0] 197.68/60.04 [0 0] [1] 197.68/60.04 = [f(X)] 197.68/60.04 197.68/60.04 [f(mark(X))] = [1 1] X + [0] 197.68/60.04 [0 0] [1] 197.68/60.04 >= [1 1] X + [0] 197.68/60.04 [0 0] [1] 197.68/60.04 = [f(X)] 197.68/60.04 197.68/60.04 [mark(f(X))] = [1 1] X + [0] 197.68/60.04 [0 0] [1] 197.68/60.04 >= [1 1] X + [0] 197.68/60.04 [0 0] [1] 197.68/60.04 = [active(f(X))] 197.68/60.04 197.68/60.04 [mark(a())] = [0] 197.68/60.04 [0] 197.68/60.04 >= [0] 197.68/60.04 [0] 197.68/60.04 = [active(a())] 197.68/60.04 197.68/60.04 [mark(g(X))] = [1 0] X + [0] 197.68/60.04 [0 0] [0] 197.68/60.04 >= [1 0] X + [0] 197.68/60.04 [0 0] [0] 197.68/60.04 = [active(g(mark(X)))] 197.68/60.04 197.68/60.04 [g(active(X))] = [1 0] X + [0] 197.68/60.04 [0 0] [0] 197.68/60.04 >= [1 0] X + [0] 197.68/60.04 [0 0] [0] 197.68/60.04 = [g(X)] 197.68/60.04 197.68/60.04 [g(mark(X))] = [1 0] X + [0] 197.68/60.04 [0 0] [0] 197.68/60.04 >= [1 0] X + [0] 197.68/60.04 [0 0] [0] 197.68/60.04 = [g(X)] 197.68/60.04 197.68/60.04 197.68/60.04 We return to the main proof. 197.68/60.04 197.68/60.04 We are left with following problem, upon which TcT provides the 197.68/60.04 certificate YES(O(1),O(n^2)). 197.68/60.04 197.68/60.04 Strict Trs: { mark(g(X)) -> active(g(mark(X))) } 197.68/60.04 Weak Trs: 197.68/60.04 { active(f(f(a()))) -> mark(f(g(f(a())))) 197.68/60.04 , f(active(X)) -> f(X) 197.68/60.04 , f(mark(X)) -> f(X) 197.68/60.04 , mark(f(X)) -> active(f(X)) 197.68/60.04 , mark(a()) -> active(a()) 197.68/60.04 , g(active(X)) -> g(X) 197.68/60.04 , g(mark(X)) -> g(X) } 197.68/60.04 Obligation: 197.68/60.04 derivational complexity 197.68/60.04 Answer: 197.68/60.04 YES(O(1),O(n^2)) 197.68/60.04 197.68/60.04 We use the processor 'matrix interpretation of dimension 2' to 197.68/60.04 orient following rules strictly. 197.68/60.04 197.68/60.04 Trs: { mark(g(X)) -> active(g(mark(X))) } 197.68/60.04 197.68/60.04 The induced complexity on above rules (modulo remaining rules) is 197.68/60.04 YES(?,O(n^2)) . These rules are moved into the corresponding weak 197.68/60.04 component(s). 197.68/60.04 197.68/60.04 Sub-proof: 197.68/60.04 ---------- 197.68/60.04 TcT has computed the following triangular matrix interpretation. 197.68/60.04 197.68/60.04 [active](x1) = [1 0] x1 + [0] 197.68/60.04 [0 1] [0] 197.68/60.04 197.68/60.04 [f](x1) = [1 0] x1 + [0] 197.68/60.04 [0 0] [0] 197.68/60.04 197.68/60.04 [a] = [0] 197.68/60.04 [0] 197.68/60.04 197.68/60.04 [mark](x1) = [1 1] x1 + [0] 197.68/60.04 [0 1] [0] 197.68/60.04 197.68/60.04 [g](x1) = [1 0] x1 + [0] 197.68/60.04 [0 1] [1] 197.68/60.05 197.68/60.05 The order satisfies the following ordering constraints: 197.68/60.05 197.68/60.05 [active(f(f(a())))] = [0] 197.68/60.05 [0] 197.68/60.05 >= [0] 197.68/60.05 [0] 197.68/60.05 = [mark(f(g(f(a()))))] 197.68/60.05 197.68/60.05 [f(active(X))] = [1 0] X + [0] 197.68/60.05 [0 0] [0] 197.68/60.05 >= [1 0] X + [0] 197.68/60.05 [0 0] [0] 197.68/60.05 = [f(X)] 197.68/60.05 197.68/60.05 [f(mark(X))] = [1 1] X + [0] 197.68/60.05 [0 0] [0] 197.68/60.05 >= [1 0] X + [0] 197.68/60.05 [0 0] [0] 197.68/60.05 = [f(X)] 197.68/60.05 197.68/60.05 [mark(f(X))] = [1 0] X + [0] 197.68/60.05 [0 0] [0] 197.68/60.05 >= [1 0] X + [0] 197.68/60.05 [0 0] [0] 197.68/60.05 = [active(f(X))] 197.68/60.05 197.68/60.05 [mark(a())] = [0] 197.68/60.05 [0] 197.68/60.05 >= [0] 197.68/60.05 [0] 197.68/60.05 = [active(a())] 197.68/60.05 197.68/60.05 [mark(g(X))] = [1 1] X + [1] 197.68/60.05 [0 1] [1] 197.68/60.05 > [1 1] X + [0] 197.68/60.05 [0 1] [1] 197.68/60.05 = [active(g(mark(X)))] 197.68/60.05 197.68/60.05 [g(active(X))] = [1 0] X + [0] 197.68/60.05 [0 1] [1] 197.68/60.05 >= [1 0] X + [0] 197.68/60.05 [0 1] [1] 197.68/60.05 = [g(X)] 197.68/60.05 197.68/60.05 [g(mark(X))] = [1 1] X + [0] 197.68/60.05 [0 1] [1] 197.68/60.05 >= [1 0] X + [0] 197.68/60.05 [0 1] [1] 197.68/60.05 = [g(X)] 197.68/60.05 197.68/60.05 197.68/60.05 We return to the main proof. 197.68/60.05 197.68/60.05 We are left with following problem, upon which TcT provides the 197.68/60.05 certificate YES(O(1),O(1)). 197.68/60.05 197.68/60.05 Weak Trs: 197.68/60.05 { active(f(f(a()))) -> mark(f(g(f(a())))) 197.68/60.05 , f(active(X)) -> f(X) 197.68/60.05 , f(mark(X)) -> f(X) 197.68/60.05 , mark(f(X)) -> active(f(X)) 197.68/60.05 , mark(a()) -> active(a()) 197.68/60.05 , mark(g(X)) -> active(g(mark(X))) 197.68/60.05 , g(active(X)) -> g(X) 197.68/60.05 , g(mark(X)) -> g(X) } 197.68/60.05 Obligation: 197.68/60.05 derivational complexity 197.68/60.05 Answer: 197.68/60.05 YES(O(1),O(1)) 197.68/60.05 197.68/60.05 Empty rules are trivially bounded 197.68/60.05 197.68/60.05 Hurray, we answered YES(O(1),O(n^2)) 197.68/60.07 EOF