YES(O(1),O(n^2)) 157.87/60.02 YES(O(1),O(n^2)) 157.87/60.02 157.87/60.02 We are left with following problem, upon which TcT provides the 157.87/60.02 certificate YES(O(1),O(n^2)). 157.87/60.02 157.87/60.02 Strict Trs: 157.87/60.02 { f(X) -> n__f(X) 157.87/60.02 , f(f(a())) -> f(g(n__f(n__a()))) 157.87/60.02 , a() -> n__a() 157.87/60.02 , activate(X) -> X 157.87/60.02 , activate(n__f(X)) -> f(activate(X)) 157.87/60.02 , activate(n__a()) -> a() } 157.87/60.02 Obligation: 157.87/60.02 derivational complexity 157.87/60.02 Answer: 157.87/60.02 YES(O(1),O(n^2)) 157.87/60.02 157.87/60.02 We use the processor 'matrix interpretation of dimension 1' to 157.87/60.02 orient following rules strictly. 157.87/60.02 157.87/60.02 Trs: 157.87/60.02 { activate(X) -> X 157.87/60.02 , activate(n__a()) -> a() } 157.87/60.02 157.87/60.02 The induced complexity on above rules (modulo remaining rules) is 157.87/60.02 YES(?,O(n^1)) . These rules are moved into the corresponding weak 157.87/60.02 component(s). 157.87/60.02 157.87/60.02 Sub-proof: 157.87/60.02 ---------- 157.87/60.02 TcT has computed the following triangular matrix interpretation. 157.87/60.02 157.87/60.02 [f](x1) = [1] x1 + [0] 157.87/60.02 157.87/60.02 [a] = [0] 157.87/60.02 157.87/60.02 [g](x1) = [1] x1 + [0] 157.87/60.02 157.87/60.02 [n__f](x1) = [1] x1 + [0] 157.87/60.02 157.87/60.02 [n__a] = [0] 157.87/60.02 157.87/60.02 [activate](x1) = [1] x1 + [1] 157.87/60.02 157.87/60.02 The order satisfies the following ordering constraints: 157.87/60.02 157.87/60.02 [f(X)] = [1] X + [0] 157.87/60.02 >= [1] X + [0] 157.87/60.02 = [n__f(X)] 157.87/60.02 157.87/60.02 [f(f(a()))] = [0] 157.87/60.02 >= [0] 157.87/60.02 = [f(g(n__f(n__a())))] 157.87/60.02 157.87/60.02 [a()] = [0] 157.87/60.02 >= [0] 157.87/60.02 = [n__a()] 157.87/60.02 157.87/60.02 [activate(X)] = [1] X + [1] 157.87/60.02 > [1] X + [0] 157.87/60.02 = [X] 157.87/60.02 157.87/60.02 [activate(n__f(X))] = [1] X + [1] 157.87/60.02 >= [1] X + [1] 157.87/60.02 = [f(activate(X))] 157.87/60.02 157.87/60.02 [activate(n__a())] = [1] 157.87/60.02 > [0] 157.87/60.02 = [a()] 157.87/60.02 157.87/60.02 157.87/60.02 We return to the main proof. 157.87/60.02 157.87/60.02 We are left with following problem, upon which TcT provides the 157.87/60.02 certificate YES(O(1),O(n^2)). 157.87/60.02 157.87/60.02 Strict Trs: 157.87/60.02 { f(X) -> n__f(X) 157.87/60.02 , f(f(a())) -> f(g(n__f(n__a()))) 157.87/60.02 , a() -> n__a() 157.87/60.02 , activate(n__f(X)) -> f(activate(X)) } 157.87/60.02 Weak Trs: 157.87/60.02 { activate(X) -> X 157.87/60.02 , activate(n__a()) -> a() } 157.87/60.02 Obligation: 157.87/60.02 derivational complexity 157.87/60.02 Answer: 157.87/60.02 YES(O(1),O(n^2)) 157.87/60.02 157.87/60.02 We use the processor 'matrix interpretation of dimension 1' to 157.87/60.02 orient following rules strictly. 157.87/60.02 157.87/60.02 Trs: 157.87/60.02 { f(f(a())) -> f(g(n__f(n__a()))) 157.87/60.02 , a() -> n__a() } 157.87/60.02 157.87/60.02 The induced complexity on above rules (modulo remaining rules) is 157.87/60.02 YES(?,O(n^1)) . These rules are moved into the corresponding weak 157.87/60.02 component(s). 157.87/60.02 157.87/60.02 Sub-proof: 157.87/60.02 ---------- 157.87/60.02 TcT has computed the following triangular matrix interpretation. 157.87/60.02 157.87/60.02 [f](x1) = [1] x1 + [0] 157.87/60.02 157.87/60.02 [a] = [1] 157.87/60.02 157.87/60.02 [g](x1) = [1] x1 + [0] 157.87/60.02 157.87/60.02 [n__f](x1) = [1] x1 + [0] 157.87/60.02 157.87/60.02 [n__a] = [0] 157.87/60.02 157.87/60.02 [activate](x1) = [1] x1 + [1] 157.87/60.02 157.87/60.02 The order satisfies the following ordering constraints: 157.87/60.02 157.87/60.02 [f(X)] = [1] X + [0] 157.87/60.02 >= [1] X + [0] 157.87/60.02 = [n__f(X)] 157.87/60.02 157.87/60.02 [f(f(a()))] = [1] 157.87/60.02 > [0] 157.87/60.02 = [f(g(n__f(n__a())))] 157.87/60.02 157.87/60.02 [a()] = [1] 157.87/60.02 > [0] 157.87/60.02 = [n__a()] 157.87/60.02 157.87/60.02 [activate(X)] = [1] X + [1] 157.87/60.02 > [1] X + [0] 157.87/60.02 = [X] 157.87/60.02 157.87/60.02 [activate(n__f(X))] = [1] X + [1] 157.87/60.02 >= [1] X + [1] 157.87/60.02 = [f(activate(X))] 157.87/60.02 157.87/60.02 [activate(n__a())] = [1] 157.87/60.02 >= [1] 157.87/60.02 = [a()] 157.87/60.02 157.87/60.02 157.87/60.02 We return to the main proof. 157.87/60.02 157.87/60.02 We are left with following problem, upon which TcT provides the 157.87/60.02 certificate YES(O(1),O(n^2)). 157.87/60.02 157.87/60.02 Strict Trs: 157.87/60.02 { f(X) -> n__f(X) 157.87/60.02 , activate(n__f(X)) -> f(activate(X)) } 157.87/60.02 Weak Trs: 157.87/60.02 { f(f(a())) -> f(g(n__f(n__a()))) 157.87/60.02 , a() -> n__a() 157.87/60.02 , activate(X) -> X 157.87/60.02 , activate(n__a()) -> a() } 157.87/60.02 Obligation: 157.87/60.02 derivational complexity 157.87/60.02 Answer: 157.87/60.02 YES(O(1),O(n^2)) 157.87/60.02 157.87/60.02 The weightgap principle applies (using the following nonconstant 157.87/60.02 growth matrix-interpretation) 157.87/60.02 157.87/60.02 TcT has computed the following triangular matrix interpretation. 157.87/60.02 Note that the diagonal of the component-wise maxima of 157.87/60.02 interpretation-entries contains no more than 1 non-zero entries. 157.87/60.02 157.87/60.02 [f](x1) = [1] x1 + [1] 157.87/60.02 157.87/60.02 [a] = [2] 157.87/60.02 157.87/60.02 [g](x1) = [1] x1 + [1] 157.87/60.02 157.87/60.02 [n__f](x1) = [1] x1 + [0] 157.87/60.02 157.87/60.02 [n__a] = [2] 157.87/60.02 157.87/60.02 [activate](x1) = [1] x1 + [0] 157.87/60.02 157.87/60.02 The order satisfies the following ordering constraints: 157.87/60.02 157.87/60.02 [f(X)] = [1] X + [1] 157.87/60.02 > [1] X + [0] 157.87/60.02 = [n__f(X)] 157.87/60.02 157.87/60.02 [f(f(a()))] = [4] 157.87/60.02 >= [4] 157.87/60.02 = [f(g(n__f(n__a())))] 157.87/60.02 157.87/60.02 [a()] = [2] 157.87/60.02 >= [2] 157.87/60.02 = [n__a()] 157.87/60.02 157.87/60.02 [activate(X)] = [1] X + [0] 157.87/60.02 >= [1] X + [0] 157.87/60.02 = [X] 157.87/60.02 157.87/60.02 [activate(n__f(X))] = [1] X + [0] 157.87/60.02 ? [1] X + [1] 157.87/60.02 = [f(activate(X))] 157.87/60.02 157.87/60.02 [activate(n__a())] = [2] 157.87/60.02 >= [2] 157.87/60.02 = [a()] 157.87/60.02 157.87/60.02 157.87/60.02 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 157.87/60.02 157.87/60.02 We are left with following problem, upon which TcT provides the 157.87/60.02 certificate YES(O(1),O(n^2)). 157.87/60.02 157.87/60.02 Strict Trs: { activate(n__f(X)) -> f(activate(X)) } 157.87/60.02 Weak Trs: 157.87/60.02 { f(X) -> n__f(X) 157.87/60.02 , f(f(a())) -> f(g(n__f(n__a()))) 157.87/60.02 , a() -> n__a() 157.87/60.02 , activate(X) -> X 157.87/60.02 , activate(n__a()) -> a() } 157.87/60.02 Obligation: 157.87/60.02 derivational complexity 157.87/60.02 Answer: 157.87/60.02 YES(O(1),O(n^2)) 157.87/60.02 157.87/60.02 We use the processor 'matrix interpretation of dimension 2' to 157.87/60.02 orient following rules strictly. 157.87/60.02 157.87/60.02 Trs: { activate(n__f(X)) -> f(activate(X)) } 157.87/60.02 157.87/60.02 The induced complexity on above rules (modulo remaining rules) is 157.87/60.02 YES(?,O(n^2)) . These rules are moved into the corresponding weak 157.87/60.02 component(s). 157.87/60.02 157.87/60.02 Sub-proof: 157.87/60.02 ---------- 157.87/60.02 TcT has computed the following triangular matrix interpretation. 157.87/60.02 157.87/60.02 [f](x1) = [1 0] x1 + [0] 157.87/60.02 [0 1] [1] 157.87/60.02 157.87/60.02 [a] = [0] 157.87/60.02 [2] 157.87/60.02 157.87/60.02 [g](x1) = [1 0] x1 + [0] 157.87/60.02 [0 0] [0] 157.87/60.02 157.87/60.02 [n__f](x1) = [1 0] x1 + [0] 157.87/60.02 [0 1] [1] 157.87/60.02 157.87/60.02 [n__a] = [0] 157.87/60.02 [2] 157.87/60.02 157.87/60.02 [activate](x1) = [1 1] x1 + [0] 157.87/60.02 [0 1] [0] 157.87/60.02 157.87/60.02 The order satisfies the following ordering constraints: 157.87/60.02 157.87/60.02 [f(X)] = [1 0] X + [0] 157.87/60.02 [0 1] [1] 157.87/60.02 >= [1 0] X + [0] 157.87/60.02 [0 1] [1] 157.87/60.02 = [n__f(X)] 157.87/60.02 157.87/60.02 [f(f(a()))] = [0] 157.87/60.02 [4] 157.87/60.02 >= [0] 157.87/60.02 [1] 157.87/60.02 = [f(g(n__f(n__a())))] 157.87/60.02 157.87/60.02 [a()] = [0] 157.87/60.02 [2] 157.87/60.02 >= [0] 157.87/60.02 [2] 157.87/60.02 = [n__a()] 157.87/60.02 157.87/60.02 [activate(X)] = [1 1] X + [0] 157.87/60.02 [0 1] [0] 157.87/60.02 >= [1 0] X + [0] 157.87/60.02 [0 1] [0] 157.87/60.02 = [X] 157.87/60.02 157.87/60.02 [activate(n__f(X))] = [1 1] X + [1] 157.87/60.02 [0 1] [1] 157.87/60.02 > [1 1] X + [0] 157.87/60.02 [0 1] [1] 157.87/60.02 = [f(activate(X))] 157.87/60.02 157.87/60.02 [activate(n__a())] = [2] 157.87/60.02 [2] 157.87/60.02 > [0] 157.87/60.02 [2] 157.87/60.02 = [a()] 157.87/60.02 157.87/60.02 157.87/60.02 We return to the main proof. 157.87/60.02 157.87/60.02 We are left with following problem, upon which TcT provides the 157.87/60.02 certificate YES(O(1),O(1)). 157.87/60.02 157.87/60.02 Weak Trs: 157.87/60.02 { f(X) -> n__f(X) 157.87/60.02 , f(f(a())) -> f(g(n__f(n__a()))) 157.87/60.02 , a() -> n__a() 157.87/60.02 , activate(X) -> X 157.87/60.02 , activate(n__f(X)) -> f(activate(X)) 157.87/60.02 , activate(n__a()) -> a() } 157.87/60.02 Obligation: 157.87/60.02 derivational complexity 157.87/60.02 Answer: 157.87/60.02 YES(O(1),O(1)) 157.87/60.02 157.87/60.02 Empty rules are trivially bounded 157.87/60.02 157.87/60.02 Hurray, we answered YES(O(1),O(n^2)) 157.87/60.03 EOF