YES(O(1),O(n^2)) 184.50/60.06 YES(O(1),O(n^2)) 184.50/60.06 184.50/60.06 We are left with following problem, upon which TcT provides the 184.50/60.06 certificate YES(O(1),O(n^2)). 184.50/60.06 184.50/60.06 Strict Trs: 184.50/60.06 { f(X) -> n__f(X) 184.50/60.06 , f(f(a())) -> c(n__f(n__g(n__f(n__a())))) 184.50/60.06 , a() -> n__a() 184.50/60.06 , g(X) -> n__g(X) 184.50/60.06 , activate(X) -> X 184.50/60.06 , activate(n__f(X)) -> f(activate(X)) 184.50/60.06 , activate(n__g(X)) -> g(activate(X)) 184.50/60.06 , activate(n__a()) -> a() } 184.50/60.06 Obligation: 184.50/60.06 derivational complexity 184.50/60.06 Answer: 184.50/60.06 YES(O(1),O(n^2)) 184.50/60.06 184.50/60.06 We use the processor 'matrix interpretation of dimension 1' to 184.50/60.06 orient following rules strictly. 184.50/60.06 184.50/60.06 Trs: 184.50/60.06 { activate(X) -> X 184.50/60.06 , activate(n__a()) -> a() } 184.50/60.06 184.50/60.06 The induced complexity on above rules (modulo remaining rules) is 184.50/60.06 YES(?,O(n^1)) . These rules are moved into the corresponding weak 184.50/60.06 component(s). 184.50/60.06 184.50/60.06 Sub-proof: 184.50/60.06 ---------- 184.50/60.06 TcT has computed the following triangular matrix interpretation. 184.50/60.06 184.50/60.06 [f](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [a] = [0] 184.50/60.06 184.50/60.06 [c](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__f](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__g](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__a] = [0] 184.50/60.06 184.50/60.06 [g](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [activate](x1) = [1] x1 + [1] 184.50/60.06 184.50/60.06 The order satisfies the following ordering constraints: 184.50/60.06 184.50/60.06 [f(X)] = [1] X + [0] 184.50/60.06 >= [1] X + [0] 184.50/60.06 = [n__f(X)] 184.50/60.06 184.50/60.06 [f(f(a()))] = [0] 184.50/60.06 >= [0] 184.50/60.06 = [c(n__f(n__g(n__f(n__a()))))] 184.50/60.06 184.50/60.06 [a()] = [0] 184.50/60.06 >= [0] 184.50/60.06 = [n__a()] 184.50/60.06 184.50/60.06 [g(X)] = [1] X + [0] 184.50/60.06 >= [1] X + [0] 184.50/60.06 = [n__g(X)] 184.50/60.06 184.50/60.06 [activate(X)] = [1] X + [1] 184.50/60.06 > [1] X + [0] 184.50/60.06 = [X] 184.50/60.06 184.50/60.06 [activate(n__f(X))] = [1] X + [1] 184.50/60.06 >= [1] X + [1] 184.50/60.06 = [f(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__g(X))] = [1] X + [1] 184.50/60.06 >= [1] X + [1] 184.50/60.06 = [g(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__a())] = [1] 184.50/60.06 > [0] 184.50/60.06 = [a()] 184.50/60.06 184.50/60.06 184.50/60.06 We return to the main proof. 184.50/60.06 184.50/60.06 We are left with following problem, upon which TcT provides the 184.50/60.06 certificate YES(O(1),O(n^2)). 184.50/60.06 184.50/60.06 Strict Trs: 184.50/60.06 { f(X) -> n__f(X) 184.50/60.06 , f(f(a())) -> c(n__f(n__g(n__f(n__a())))) 184.50/60.06 , a() -> n__a() 184.50/60.06 , g(X) -> n__g(X) 184.50/60.06 , activate(n__f(X)) -> f(activate(X)) 184.50/60.06 , activate(n__g(X)) -> g(activate(X)) } 184.50/60.06 Weak Trs: 184.50/60.06 { activate(X) -> X 184.50/60.06 , activate(n__a()) -> a() } 184.50/60.06 Obligation: 184.50/60.06 derivational complexity 184.50/60.06 Answer: 184.50/60.06 YES(O(1),O(n^2)) 184.50/60.06 184.50/60.06 We use the processor 'matrix interpretation of dimension 1' to 184.50/60.06 orient following rules strictly. 184.50/60.06 184.50/60.06 Trs: 184.50/60.06 { f(f(a())) -> c(n__f(n__g(n__f(n__a())))) 184.50/60.06 , a() -> n__a() } 184.50/60.06 184.50/60.06 The induced complexity on above rules (modulo remaining rules) is 184.50/60.06 YES(?,O(n^1)) . These rules are moved into the corresponding weak 184.50/60.06 component(s). 184.50/60.06 184.50/60.06 Sub-proof: 184.50/60.06 ---------- 184.50/60.06 TcT has computed the following triangular matrix interpretation. 184.50/60.06 184.50/60.06 [f](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [a] = [1] 184.50/60.06 184.50/60.06 [c](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__f](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__g](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__a] = [0] 184.50/60.06 184.50/60.06 [g](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [activate](x1) = [1] x1 + [1] 184.50/60.06 184.50/60.06 The order satisfies the following ordering constraints: 184.50/60.06 184.50/60.06 [f(X)] = [1] X + [0] 184.50/60.06 >= [1] X + [0] 184.50/60.06 = [n__f(X)] 184.50/60.06 184.50/60.06 [f(f(a()))] = [1] 184.50/60.06 > [0] 184.50/60.06 = [c(n__f(n__g(n__f(n__a()))))] 184.50/60.06 184.50/60.06 [a()] = [1] 184.50/60.06 > [0] 184.50/60.06 = [n__a()] 184.50/60.06 184.50/60.06 [g(X)] = [1] X + [0] 184.50/60.06 >= [1] X + [0] 184.50/60.06 = [n__g(X)] 184.50/60.06 184.50/60.06 [activate(X)] = [1] X + [1] 184.50/60.06 > [1] X + [0] 184.50/60.06 = [X] 184.50/60.06 184.50/60.06 [activate(n__f(X))] = [1] X + [1] 184.50/60.06 >= [1] X + [1] 184.50/60.06 = [f(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__g(X))] = [1] X + [1] 184.50/60.06 >= [1] X + [1] 184.50/60.06 = [g(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__a())] = [1] 184.50/60.06 >= [1] 184.50/60.06 = [a()] 184.50/60.06 184.50/60.06 184.50/60.06 We return to the main proof. 184.50/60.06 184.50/60.06 We are left with following problem, upon which TcT provides the 184.50/60.06 certificate YES(O(1),O(n^2)). 184.50/60.06 184.50/60.06 Strict Trs: 184.50/60.06 { f(X) -> n__f(X) 184.50/60.06 , g(X) -> n__g(X) 184.50/60.06 , activate(n__f(X)) -> f(activate(X)) 184.50/60.06 , activate(n__g(X)) -> g(activate(X)) } 184.50/60.06 Weak Trs: 184.50/60.06 { f(f(a())) -> c(n__f(n__g(n__f(n__a())))) 184.50/60.06 , a() -> n__a() 184.50/60.06 , activate(X) -> X 184.50/60.06 , activate(n__a()) -> a() } 184.50/60.06 Obligation: 184.50/60.06 derivational complexity 184.50/60.06 Answer: 184.50/60.06 YES(O(1),O(n^2)) 184.50/60.06 184.50/60.06 The weightgap principle applies (using the following nonconstant 184.50/60.06 growth matrix-interpretation) 184.50/60.06 184.50/60.06 TcT has computed the following triangular matrix interpretation. 184.50/60.06 Note that the diagonal of the component-wise maxima of 184.50/60.06 interpretation-entries contains no more than 1 non-zero entries. 184.50/60.06 184.50/60.06 [f](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [a] = [0] 184.50/60.06 184.50/60.06 [c](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__f](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__g](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__a] = [0] 184.50/60.06 184.50/60.06 [g](x1) = [1] x1 + [1] 184.50/60.06 184.50/60.06 [activate](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 The order satisfies the following ordering constraints: 184.50/60.06 184.50/60.06 [f(X)] = [1] X + [0] 184.50/60.06 >= [1] X + [0] 184.50/60.06 = [n__f(X)] 184.50/60.06 184.50/60.06 [f(f(a()))] = [0] 184.50/60.06 >= [0] 184.50/60.06 = [c(n__f(n__g(n__f(n__a()))))] 184.50/60.06 184.50/60.06 [a()] = [0] 184.50/60.06 >= [0] 184.50/60.06 = [n__a()] 184.50/60.06 184.50/60.06 [g(X)] = [1] X + [1] 184.50/60.06 > [1] X + [0] 184.50/60.06 = [n__g(X)] 184.50/60.06 184.50/60.06 [activate(X)] = [1] X + [0] 184.50/60.06 >= [1] X + [0] 184.50/60.06 = [X] 184.50/60.06 184.50/60.06 [activate(n__f(X))] = [1] X + [0] 184.50/60.06 >= [1] X + [0] 184.50/60.06 = [f(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__g(X))] = [1] X + [0] 184.50/60.06 ? [1] X + [1] 184.50/60.06 = [g(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__a())] = [0] 184.50/60.06 >= [0] 184.50/60.06 = [a()] 184.50/60.06 184.50/60.06 184.50/60.06 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 184.50/60.06 184.50/60.06 We are left with following problem, upon which TcT provides the 184.50/60.06 certificate YES(O(1),O(n^2)). 184.50/60.06 184.50/60.06 Strict Trs: 184.50/60.06 { f(X) -> n__f(X) 184.50/60.06 , activate(n__f(X)) -> f(activate(X)) 184.50/60.06 , activate(n__g(X)) -> g(activate(X)) } 184.50/60.06 Weak Trs: 184.50/60.06 { f(f(a())) -> c(n__f(n__g(n__f(n__a())))) 184.50/60.06 , a() -> n__a() 184.50/60.06 , g(X) -> n__g(X) 184.50/60.06 , activate(X) -> X 184.50/60.06 , activate(n__a()) -> a() } 184.50/60.06 Obligation: 184.50/60.06 derivational complexity 184.50/60.06 Answer: 184.50/60.06 YES(O(1),O(n^2)) 184.50/60.06 184.50/60.06 The weightgap principle applies (using the following nonconstant 184.50/60.06 growth matrix-interpretation) 184.50/60.06 184.50/60.06 TcT has computed the following triangular matrix interpretation. 184.50/60.06 Note that the diagonal of the component-wise maxima of 184.50/60.06 interpretation-entries contains no more than 1 non-zero entries. 184.50/60.06 184.50/60.06 [f](x1) = [1] x1 + [1] 184.50/60.06 184.50/60.06 [a] = [2] 184.50/60.06 184.50/60.06 [c](x1) = [1] x1 + [2] 184.50/60.06 184.50/60.06 [n__f](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__g](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [n__a] = [2] 184.50/60.06 184.50/60.06 [g](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 [activate](x1) = [1] x1 + [0] 184.50/60.06 184.50/60.06 The order satisfies the following ordering constraints: 184.50/60.06 184.50/60.06 [f(X)] = [1] X + [1] 184.50/60.06 > [1] X + [0] 184.50/60.06 = [n__f(X)] 184.50/60.06 184.50/60.06 [f(f(a()))] = [4] 184.50/60.06 >= [4] 184.50/60.06 = [c(n__f(n__g(n__f(n__a()))))] 184.50/60.06 184.50/60.06 [a()] = [2] 184.50/60.06 >= [2] 184.50/60.06 = [n__a()] 184.50/60.06 184.50/60.06 [g(X)] = [1] X + [0] 184.50/60.06 >= [1] X + [0] 184.50/60.06 = [n__g(X)] 184.50/60.06 184.50/60.06 [activate(X)] = [1] X + [0] 184.50/60.06 >= [1] X + [0] 184.50/60.06 = [X] 184.50/60.06 184.50/60.06 [activate(n__f(X))] = [1] X + [0] 184.50/60.06 ? [1] X + [1] 184.50/60.06 = [f(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__g(X))] = [1] X + [0] 184.50/60.06 >= [1] X + [0] 184.50/60.06 = [g(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__a())] = [2] 184.50/60.06 >= [2] 184.50/60.06 = [a()] 184.50/60.06 184.50/60.06 184.50/60.06 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 184.50/60.06 184.50/60.06 We are left with following problem, upon which TcT provides the 184.50/60.06 certificate YES(O(1),O(n^2)). 184.50/60.06 184.50/60.06 Strict Trs: 184.50/60.06 { activate(n__f(X)) -> f(activate(X)) 184.50/60.06 , activate(n__g(X)) -> g(activate(X)) } 184.50/60.06 Weak Trs: 184.50/60.06 { f(X) -> n__f(X) 184.50/60.06 , f(f(a())) -> c(n__f(n__g(n__f(n__a())))) 184.50/60.06 , a() -> n__a() 184.50/60.06 , g(X) -> n__g(X) 184.50/60.06 , activate(X) -> X 184.50/60.06 , activate(n__a()) -> a() } 184.50/60.06 Obligation: 184.50/60.06 derivational complexity 184.50/60.06 Answer: 184.50/60.06 YES(O(1),O(n^2)) 184.50/60.06 184.50/60.06 We use the processor 'matrix interpretation of dimension 2' to 184.50/60.06 orient following rules strictly. 184.50/60.06 184.50/60.06 Trs: { activate(n__f(X)) -> f(activate(X)) } 184.50/60.06 184.50/60.06 The induced complexity on above rules (modulo remaining rules) is 184.50/60.06 YES(?,O(n^2)) . These rules are moved into the corresponding weak 184.50/60.06 component(s). 184.50/60.06 184.50/60.06 Sub-proof: 184.50/60.06 ---------- 184.50/60.06 TcT has computed the following triangular matrix interpretation. 184.50/60.06 184.50/60.06 [f](x1) = [1 0] x1 + [0] 184.50/60.06 [0 1] [1] 184.50/60.06 184.50/60.06 [a] = [0] 184.50/60.06 [2] 184.50/60.06 184.50/60.06 [c](x1) = [1 0] x1 + [0] 184.50/60.06 [0 0] [0] 184.50/60.06 184.50/60.06 [n__f](x1) = [1 0] x1 + [0] 184.50/60.06 [0 1] [1] 184.50/60.06 184.50/60.06 [n__g](x1) = [1 0] x1 + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 184.50/60.06 [n__a] = [0] 184.50/60.06 [2] 184.50/60.06 184.50/60.06 [g](x1) = [1 0] x1 + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 184.50/60.06 [activate](x1) = [1 1] x1 + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 184.50/60.06 The order satisfies the following ordering constraints: 184.50/60.06 184.50/60.06 [f(X)] = [1 0] X + [0] 184.50/60.06 [0 1] [1] 184.50/60.06 >= [1 0] X + [0] 184.50/60.06 [0 1] [1] 184.50/60.06 = [n__f(X)] 184.50/60.06 184.50/60.06 [f(f(a()))] = [0] 184.50/60.06 [4] 184.50/60.06 >= [0] 184.50/60.06 [0] 184.50/60.06 = [c(n__f(n__g(n__f(n__a()))))] 184.50/60.06 184.50/60.06 [a()] = [0] 184.50/60.06 [2] 184.50/60.06 >= [0] 184.50/60.06 [2] 184.50/60.06 = [n__a()] 184.50/60.06 184.50/60.06 [g(X)] = [1 0] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 >= [1 0] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 = [n__g(X)] 184.50/60.06 184.50/60.06 [activate(X)] = [1 1] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 >= [1 0] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 = [X] 184.50/60.06 184.50/60.06 [activate(n__f(X))] = [1 1] X + [1] 184.50/60.06 [0 1] [1] 184.50/60.06 > [1 1] X + [0] 184.50/60.06 [0 1] [1] 184.50/60.06 = [f(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__g(X))] = [1 1] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 >= [1 1] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 = [g(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__a())] = [2] 184.50/60.06 [2] 184.50/60.06 > [0] 184.50/60.06 [2] 184.50/60.06 = [a()] 184.50/60.06 184.50/60.06 184.50/60.06 We return to the main proof. 184.50/60.06 184.50/60.06 We are left with following problem, upon which TcT provides the 184.50/60.06 certificate YES(O(1),O(n^2)). 184.50/60.06 184.50/60.06 Strict Trs: { activate(n__g(X)) -> g(activate(X)) } 184.50/60.06 Weak Trs: 184.50/60.06 { f(X) -> n__f(X) 184.50/60.06 , f(f(a())) -> c(n__f(n__g(n__f(n__a())))) 184.50/60.06 , a() -> n__a() 184.50/60.06 , g(X) -> n__g(X) 184.50/60.06 , activate(X) -> X 184.50/60.06 , activate(n__f(X)) -> f(activate(X)) 184.50/60.06 , activate(n__a()) -> a() } 184.50/60.06 Obligation: 184.50/60.06 derivational complexity 184.50/60.06 Answer: 184.50/60.06 YES(O(1),O(n^2)) 184.50/60.06 184.50/60.06 We use the processor 'matrix interpretation of dimension 2' to 184.50/60.06 orient following rules strictly. 184.50/60.06 184.50/60.06 Trs: { activate(n__g(X)) -> g(activate(X)) } 184.50/60.06 184.50/60.06 The induced complexity on above rules (modulo remaining rules) is 184.50/60.06 YES(?,O(n^2)) . These rules are moved into the corresponding weak 184.50/60.06 component(s). 184.50/60.06 184.50/60.06 Sub-proof: 184.50/60.06 ---------- 184.50/60.06 TcT has computed the following triangular matrix interpretation. 184.50/60.06 184.50/60.06 [f](x1) = [1 0] x1 + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 184.50/60.06 [a] = [0] 184.50/60.06 [0] 184.50/60.06 184.50/60.06 [c](x1) = [1 0] x1 + [0] 184.50/60.06 [0 0] [0] 184.50/60.06 184.50/60.06 [n__f](x1) = [1 0] x1 + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 184.50/60.06 [n__g](x1) = [1 0] x1 + [0] 184.50/60.06 [0 1] [1] 184.50/60.06 184.50/60.06 [n__a] = [0] 184.50/60.06 [0] 184.50/60.06 184.50/60.06 [g](x1) = [1 0] x1 + [0] 184.50/60.06 [0 1] [1] 184.50/60.06 184.50/60.06 [activate](x1) = [1 1] x1 + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 184.50/60.06 The order satisfies the following ordering constraints: 184.50/60.06 184.50/60.06 [f(X)] = [1 0] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 >= [1 0] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 = [n__f(X)] 184.50/60.06 184.50/60.06 [f(f(a()))] = [0] 184.50/60.06 [0] 184.50/60.06 >= [0] 184.50/60.06 [0] 184.50/60.06 = [c(n__f(n__g(n__f(n__a()))))] 184.50/60.06 184.50/60.06 [a()] = [0] 184.50/60.06 [0] 184.50/60.06 >= [0] 184.50/60.06 [0] 184.50/60.06 = [n__a()] 184.50/60.06 184.50/60.06 [g(X)] = [1 0] X + [0] 184.50/60.06 [0 1] [1] 184.50/60.06 >= [1 0] X + [0] 184.50/60.06 [0 1] [1] 184.50/60.06 = [n__g(X)] 184.50/60.06 184.50/60.06 [activate(X)] = [1 1] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 >= [1 0] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 = [X] 184.50/60.06 184.50/60.06 [activate(n__f(X))] = [1 1] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 >= [1 1] X + [0] 184.50/60.06 [0 1] [0] 184.50/60.06 = [f(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__g(X))] = [1 1] X + [1] 184.50/60.06 [0 1] [1] 184.50/60.06 > [1 1] X + [0] 184.50/60.06 [0 1] [1] 184.50/60.06 = [g(activate(X))] 184.50/60.06 184.50/60.06 [activate(n__a())] = [0] 184.50/60.06 [0] 184.50/60.06 >= [0] 184.50/60.06 [0] 184.50/60.06 = [a()] 184.50/60.06 184.50/60.06 184.50/60.06 We return to the main proof. 184.50/60.06 184.50/60.06 We are left with following problem, upon which TcT provides the 184.50/60.06 certificate YES(O(1),O(1)). 184.50/60.06 184.50/60.06 Weak Trs: 184.50/60.06 { f(X) -> n__f(X) 184.50/60.06 , f(f(a())) -> c(n__f(n__g(n__f(n__a())))) 184.50/60.06 , a() -> n__a() 184.50/60.06 , g(X) -> n__g(X) 184.50/60.06 , activate(X) -> X 184.50/60.06 , activate(n__f(X)) -> f(activate(X)) 184.50/60.06 , activate(n__g(X)) -> g(activate(X)) 184.50/60.06 , activate(n__a()) -> a() } 184.50/60.06 Obligation: 184.50/60.06 derivational complexity 184.50/60.06 Answer: 184.50/60.06 YES(O(1),O(1)) 184.50/60.06 184.50/60.06 Empty rules are trivially bounded 184.50/60.06 184.50/60.06 Hurray, we answered YES(O(1),O(n^2)) 184.50/60.08 EOF