YES(O(1),O(n^2)) 163.56/60.08 YES(O(1),O(n^2)) 163.56/60.08 163.56/60.08 We are left with following problem, upon which TcT provides the 163.56/60.08 certificate YES(O(1),O(n^2)). 163.56/60.08 163.56/60.08 Strict Trs: 163.56/60.08 { f(X) -> n__f(X) 163.56/60.08 , f(f(X)) -> c(n__f(n__g(n__f(X)))) 163.56/60.08 , c(X) -> d(activate(X)) 163.56/60.08 , d(X) -> n__d(X) 163.56/60.08 , activate(X) -> X 163.56/60.08 , activate(n__f(X)) -> f(activate(X)) 163.56/60.08 , activate(n__g(X)) -> g(X) 163.56/60.08 , activate(n__d(X)) -> d(X) 163.56/60.08 , h(X) -> c(n__d(X)) 163.56/60.08 , g(X) -> n__g(X) } 163.56/60.08 Obligation: 163.56/60.08 derivational complexity 163.56/60.08 Answer: 163.56/60.08 YES(O(1),O(n^2)) 163.56/60.08 163.56/60.08 We use the processor 'matrix interpretation of dimension 1' to 163.56/60.08 orient following rules strictly. 163.56/60.08 163.56/60.08 Trs: { h(X) -> c(n__d(X)) } 163.56/60.08 163.56/60.08 The induced complexity on above rules (modulo remaining rules) is 163.56/60.08 YES(?,O(n^1)) . These rules are moved into the corresponding weak 163.56/60.08 component(s). 163.56/60.08 163.56/60.08 Sub-proof: 163.56/60.08 ---------- 163.56/60.08 TcT has computed the following triangular matrix interpretation. 163.56/60.08 163.56/60.08 [f](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 [c](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 [n__f](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 [n__g](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 [d](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 [activate](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 [h](x1) = [1] x1 + [1] 163.56/60.08 163.56/60.08 [n__d](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 [g](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 The order satisfies the following ordering constraints: 163.56/60.08 163.56/60.08 [f(X)] = [1] X + [0] 163.56/60.08 >= [1] X + [0] 163.56/60.08 = [n__f(X)] 163.56/60.08 163.56/60.08 [f(f(X))] = [1] X + [0] 163.56/60.08 >= [1] X + [0] 163.56/60.08 = [c(n__f(n__g(n__f(X))))] 163.56/60.08 163.56/60.08 [c(X)] = [1] X + [0] 163.56/60.08 >= [1] X + [0] 163.56/60.08 = [d(activate(X))] 163.56/60.08 163.56/60.08 [d(X)] = [1] X + [0] 163.56/60.08 >= [1] X + [0] 163.56/60.08 = [n__d(X)] 163.56/60.08 163.56/60.08 [activate(X)] = [1] X + [0] 163.56/60.08 >= [1] X + [0] 163.56/60.08 = [X] 163.56/60.08 163.56/60.08 [activate(n__f(X))] = [1] X + [0] 163.56/60.08 >= [1] X + [0] 163.56/60.08 = [f(activate(X))] 163.56/60.08 163.56/60.08 [activate(n__g(X))] = [1] X + [0] 163.56/60.08 >= [1] X + [0] 163.56/60.08 = [g(X)] 163.56/60.08 163.56/60.08 [activate(n__d(X))] = [1] X + [0] 163.56/60.08 >= [1] X + [0] 163.56/60.08 = [d(X)] 163.56/60.08 163.56/60.08 [h(X)] = [1] X + [1] 163.56/60.08 > [1] X + [0] 163.56/60.08 = [c(n__d(X))] 163.56/60.08 163.56/60.08 [g(X)] = [1] X + [0] 163.56/60.08 >= [1] X + [0] 163.56/60.08 = [n__g(X)] 163.56/60.08 163.56/60.08 163.56/60.08 We return to the main proof. 163.56/60.08 163.56/60.08 We are left with following problem, upon which TcT provides the 163.56/60.08 certificate YES(O(1),O(n^2)). 163.56/60.08 163.56/60.08 Strict Trs: 163.56/60.08 { f(X) -> n__f(X) 163.56/60.08 , f(f(X)) -> c(n__f(n__g(n__f(X)))) 163.56/60.08 , c(X) -> d(activate(X)) 163.56/60.08 , d(X) -> n__d(X) 163.56/60.08 , activate(X) -> X 163.56/60.08 , activate(n__f(X)) -> f(activate(X)) 163.56/60.08 , activate(n__g(X)) -> g(X) 163.56/60.08 , activate(n__d(X)) -> d(X) 163.56/60.08 , g(X) -> n__g(X) } 163.56/60.08 Weak Trs: { h(X) -> c(n__d(X)) } 163.56/60.08 Obligation: 163.56/60.08 derivational complexity 163.56/60.08 Answer: 163.56/60.08 YES(O(1),O(n^2)) 163.56/60.08 163.56/60.08 The weightgap principle applies (using the following nonconstant 163.56/60.08 growth matrix-interpretation) 163.56/60.08 163.56/60.08 TcT has computed the following triangular matrix interpretation. 163.56/60.08 Note that the diagonal of the component-wise maxima of 163.56/60.08 interpretation-entries contains no more than 1 non-zero entries. 163.56/60.08 163.56/60.08 [f](x1) = [1] x1 + [1] 163.56/60.08 163.56/60.08 [c](x1) = [1] x1 + [1] 163.56/60.08 163.56/60.08 [n__f](x1) = [1] x1 + [2] 163.56/60.08 163.56/60.08 [n__g](x1) = [1] x1 + [1] 163.56/60.08 163.56/60.08 [d](x1) = [1] x1 + [1] 163.56/60.08 163.56/60.08 [activate](x1) = [1] x1 + [1] 163.56/60.08 163.56/60.08 [h](x1) = [1] x1 + [2] 163.56/60.08 163.56/60.08 [n__d](x1) = [1] x1 + [1] 163.56/60.08 163.56/60.08 [g](x1) = [1] x1 + [1] 163.56/60.08 163.56/60.08 The order satisfies the following ordering constraints: 163.56/60.08 163.56/60.08 [f(X)] = [1] X + [1] 163.56/60.08 ? [1] X + [2] 163.56/60.08 = [n__f(X)] 163.56/60.08 163.56/60.08 [f(f(X))] = [1] X + [2] 163.56/60.08 ? [1] X + [6] 163.56/60.08 = [c(n__f(n__g(n__f(X))))] 163.56/60.08 163.56/60.08 [c(X)] = [1] X + [1] 163.56/60.08 ? [1] X + [2] 163.56/60.08 = [d(activate(X))] 163.56/60.08 163.56/60.08 [d(X)] = [1] X + [1] 163.56/60.08 >= [1] X + [1] 163.56/60.08 = [n__d(X)] 163.56/60.08 163.56/60.08 [activate(X)] = [1] X + [1] 163.56/60.08 > [1] X + [0] 163.56/60.08 = [X] 163.56/60.08 163.56/60.08 [activate(n__f(X))] = [1] X + [3] 163.56/60.08 > [1] X + [2] 163.56/60.08 = [f(activate(X))] 163.56/60.08 163.56/60.08 [activate(n__g(X))] = [1] X + [2] 163.56/60.08 > [1] X + [1] 163.56/60.08 = [g(X)] 163.56/60.08 163.56/60.08 [activate(n__d(X))] = [1] X + [2] 163.56/60.08 > [1] X + [1] 163.56/60.08 = [d(X)] 163.56/60.08 163.56/60.08 [h(X)] = [1] X + [2] 163.56/60.08 >= [1] X + [2] 163.56/60.08 = [c(n__d(X))] 163.56/60.08 163.56/60.08 [g(X)] = [1] X + [1] 163.56/60.08 >= [1] X + [1] 163.56/60.08 = [n__g(X)] 163.56/60.08 163.56/60.08 163.56/60.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 163.56/60.08 163.56/60.08 We are left with following problem, upon which TcT provides the 163.56/60.08 certificate YES(O(1),O(n^2)). 163.56/60.08 163.56/60.08 Strict Trs: 163.56/60.08 { f(X) -> n__f(X) 163.56/60.08 , f(f(X)) -> c(n__f(n__g(n__f(X)))) 163.56/60.08 , c(X) -> d(activate(X)) 163.56/60.08 , d(X) -> n__d(X) 163.56/60.08 , g(X) -> n__g(X) } 163.56/60.08 Weak Trs: 163.56/60.08 { activate(X) -> X 163.56/60.08 , activate(n__f(X)) -> f(activate(X)) 163.56/60.08 , activate(n__g(X)) -> g(X) 163.56/60.08 , activate(n__d(X)) -> d(X) 163.56/60.08 , h(X) -> c(n__d(X)) } 163.56/60.08 Obligation: 163.56/60.08 derivational complexity 163.56/60.08 Answer: 163.56/60.08 YES(O(1),O(n^2)) 163.56/60.08 163.56/60.08 The weightgap principle applies (using the following nonconstant 163.56/60.08 growth matrix-interpretation) 163.56/60.08 163.56/60.08 TcT has computed the following triangular matrix interpretation. 163.56/60.08 Note that the diagonal of the component-wise maxima of 163.56/60.08 interpretation-entries contains no more than 1 non-zero entries. 163.56/60.08 163.56/60.08 [f](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 [c](x1) = [1] x1 + [2] 163.56/60.08 163.56/60.08 [n__f](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 [n__g](x1) = [1] x1 + [2] 163.56/60.08 163.56/60.08 [d](x1) = [1] x1 + [0] 163.56/60.08 163.56/60.08 [activate](x1) = [1] x1 + [0] 163.56/60.09 163.56/60.09 [h](x1) = [1] x1 + [2] 163.56/60.09 163.56/60.09 [n__d](x1) = [1] x1 + [0] 163.56/60.09 163.56/60.09 [g](x1) = [1] x1 + [1] 163.56/60.09 163.56/60.09 The order satisfies the following ordering constraints: 163.56/60.09 163.56/60.09 [f(X)] = [1] X + [0] 163.56/60.09 >= [1] X + [0] 163.56/60.09 = [n__f(X)] 163.56/60.09 163.56/60.09 [f(f(X))] = [1] X + [0] 163.56/60.09 ? [1] X + [4] 163.56/60.09 = [c(n__f(n__g(n__f(X))))] 163.56/60.09 163.56/60.09 [c(X)] = [1] X + [2] 163.56/60.09 > [1] X + [0] 163.56/60.09 = [d(activate(X))] 163.56/60.09 163.56/60.09 [d(X)] = [1] X + [0] 163.56/60.09 >= [1] X + [0] 163.56/60.09 = [n__d(X)] 163.56/60.09 163.56/60.09 [activate(X)] = [1] X + [0] 163.56/60.09 >= [1] X + [0] 163.56/60.09 = [X] 163.56/60.09 163.56/60.09 [activate(n__f(X))] = [1] X + [0] 163.56/60.09 >= [1] X + [0] 163.56/60.09 = [f(activate(X))] 163.56/60.09 163.56/60.09 [activate(n__g(X))] = [1] X + [2] 163.56/60.09 > [1] X + [1] 163.56/60.09 = [g(X)] 163.56/60.09 163.56/60.09 [activate(n__d(X))] = [1] X + [0] 163.56/60.09 >= [1] X + [0] 163.56/60.09 = [d(X)] 163.56/60.09 163.56/60.09 [h(X)] = [1] X + [2] 163.56/60.09 >= [1] X + [2] 163.56/60.09 = [c(n__d(X))] 163.56/60.09 163.56/60.09 [g(X)] = [1] X + [1] 163.56/60.09 ? [1] X + [2] 163.56/60.09 = [n__g(X)] 163.56/60.09 163.56/60.09 163.56/60.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 163.56/60.09 163.56/60.09 We are left with following problem, upon which TcT provides the 163.56/60.09 certificate YES(O(1),O(n^2)). 163.56/60.09 163.56/60.09 Strict Trs: 163.56/60.09 { f(X) -> n__f(X) 163.56/60.09 , f(f(X)) -> c(n__f(n__g(n__f(X)))) 163.56/60.09 , d(X) -> n__d(X) 163.56/60.09 , g(X) -> n__g(X) } 163.56/60.09 Weak Trs: 163.56/60.09 { c(X) -> d(activate(X)) 163.56/60.09 , activate(X) -> X 163.56/60.09 , activate(n__f(X)) -> f(activate(X)) 163.56/60.09 , activate(n__g(X)) -> g(X) 163.56/60.09 , activate(n__d(X)) -> d(X) 163.56/60.09 , h(X) -> c(n__d(X)) } 163.56/60.09 Obligation: 163.56/60.09 derivational complexity 163.56/60.09 Answer: 163.56/60.09 YES(O(1),O(n^2)) 163.56/60.09 163.56/60.09 The weightgap principle applies (using the following nonconstant 163.56/60.09 growth matrix-interpretation) 163.56/60.09 163.56/60.09 TcT has computed the following triangular matrix interpretation. 163.56/60.09 Note that the diagonal of the component-wise maxima of 163.56/60.09 interpretation-entries contains no more than 1 non-zero entries. 163.56/60.09 163.56/60.09 [f](x1) = [1] x1 + [0] 163.56/60.09 163.56/60.09 [c](x1) = [1] x1 + [1] 163.56/60.09 163.56/60.09 [n__f](x1) = [1] x1 + [0] 163.56/60.09 163.56/60.09 [n__g](x1) = [1] x1 + [0] 163.56/60.09 163.56/60.09 [d](x1) = [1] x1 + [0] 163.56/60.09 163.56/60.09 [activate](x1) = [1] x1 + [1] 163.56/60.09 163.56/60.09 [h](x1) = [1] x1 + [2] 163.56/60.09 163.56/60.09 [n__d](x1) = [1] x1 + [0] 163.56/60.09 163.56/60.09 [g](x1) = [1] x1 + [1] 163.56/60.09 163.56/60.09 The order satisfies the following ordering constraints: 163.56/60.09 163.56/60.09 [f(X)] = [1] X + [0] 163.56/60.09 >= [1] X + [0] 163.56/60.09 = [n__f(X)] 163.56/60.09 163.56/60.09 [f(f(X))] = [1] X + [0] 163.56/60.09 ? [1] X + [1] 163.56/60.09 = [c(n__f(n__g(n__f(X))))] 163.56/60.09 163.56/60.09 [c(X)] = [1] X + [1] 163.56/60.09 >= [1] X + [1] 163.56/60.09 = [d(activate(X))] 163.56/60.09 163.56/60.09 [d(X)] = [1] X + [0] 163.56/60.09 >= [1] X + [0] 163.56/60.09 = [n__d(X)] 163.56/60.09 163.56/60.09 [activate(X)] = [1] X + [1] 163.56/60.09 > [1] X + [0] 163.56/60.09 = [X] 163.56/60.09 163.56/60.09 [activate(n__f(X))] = [1] X + [1] 163.56/60.09 >= [1] X + [1] 163.56/60.09 = [f(activate(X))] 163.56/60.09 163.56/60.09 [activate(n__g(X))] = [1] X + [1] 163.56/60.09 >= [1] X + [1] 163.56/60.09 = [g(X)] 163.56/60.09 163.56/60.09 [activate(n__d(X))] = [1] X + [1] 163.56/60.09 > [1] X + [0] 163.56/60.09 = [d(X)] 163.56/60.09 163.56/60.09 [h(X)] = [1] X + [2] 163.56/60.09 > [1] X + [1] 163.56/60.09 = [c(n__d(X))] 163.56/60.09 163.56/60.09 [g(X)] = [1] X + [1] 163.56/60.09 > [1] X + [0] 163.56/60.09 = [n__g(X)] 163.56/60.09 163.56/60.09 163.56/60.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 163.56/60.09 163.56/60.09 We are left with following problem, upon which TcT provides the 163.56/60.09 certificate YES(O(1),O(n^2)). 163.56/60.09 163.56/60.09 Strict Trs: 163.56/60.09 { f(X) -> n__f(X) 163.56/60.09 , f(f(X)) -> c(n__f(n__g(n__f(X)))) 163.56/60.09 , d(X) -> n__d(X) } 163.56/60.09 Weak Trs: 163.56/60.09 { c(X) -> d(activate(X)) 163.56/60.09 , activate(X) -> X 163.56/60.09 , activate(n__f(X)) -> f(activate(X)) 163.56/60.09 , activate(n__g(X)) -> g(X) 163.56/60.09 , activate(n__d(X)) -> d(X) 163.56/60.09 , h(X) -> c(n__d(X)) 163.56/60.09 , g(X) -> n__g(X) } 163.56/60.09 Obligation: 163.56/60.09 derivational complexity 163.56/60.09 Answer: 163.56/60.09 YES(O(1),O(n^2)) 163.56/60.09 163.56/60.09 The weightgap principle applies (using the following nonconstant 163.56/60.09 growth matrix-interpretation) 163.56/60.09 163.56/60.09 TcT has computed the following triangular matrix interpretation. 163.56/60.09 Note that the diagonal of the component-wise maxima of 163.56/60.09 interpretation-entries contains no more than 1 non-zero entries. 163.56/60.09 163.56/60.09 [f](x1) = [1] x1 + [0] 163.56/60.09 163.56/60.09 [c](x1) = [1] x1 + [2] 163.56/60.09 163.56/60.09 [n__f](x1) = [1] x1 + [0] 163.56/60.09 163.56/60.09 [n__g](x1) = [1] x1 + [2] 163.56/60.09 163.56/60.09 [d](x1) = [1] x1 + [1] 163.56/60.09 163.56/60.09 [activate](x1) = [1] x1 + [1] 163.56/60.09 163.56/60.09 [h](x1) = [1] x1 + [2] 163.56/60.09 163.56/60.09 [n__d](x1) = [1] x1 + [0] 163.56/60.09 163.56/60.09 [g](x1) = [1] x1 + [2] 163.56/60.09 163.56/60.09 The order satisfies the following ordering constraints: 163.56/60.09 163.56/60.09 [f(X)] = [1] X + [0] 163.56/60.09 >= [1] X + [0] 163.56/60.09 = [n__f(X)] 163.56/60.09 163.56/60.09 [f(f(X))] = [1] X + [0] 163.56/60.09 ? [1] X + [4] 163.56/60.09 = [c(n__f(n__g(n__f(X))))] 163.56/60.09 163.56/60.09 [c(X)] = [1] X + [2] 163.56/60.09 >= [1] X + [2] 163.56/60.09 = [d(activate(X))] 163.56/60.09 163.56/60.09 [d(X)] = [1] X + [1] 163.56/60.09 > [1] X + [0] 163.56/60.09 = [n__d(X)] 163.56/60.09 163.56/60.09 [activate(X)] = [1] X + [1] 163.56/60.09 > [1] X + [0] 163.56/60.09 = [X] 163.56/60.09 163.56/60.09 [activate(n__f(X))] = [1] X + [1] 163.56/60.09 >= [1] X + [1] 163.56/60.09 = [f(activate(X))] 163.56/60.09 163.56/60.09 [activate(n__g(X))] = [1] X + [3] 163.56/60.09 > [1] X + [2] 163.56/60.09 = [g(X)] 163.56/60.09 163.56/60.09 [activate(n__d(X))] = [1] X + [1] 163.56/60.09 >= [1] X + [1] 163.56/60.09 = [d(X)] 163.56/60.09 163.56/60.09 [h(X)] = [1] X + [2] 163.56/60.09 >= [1] X + [2] 163.56/60.09 = [c(n__d(X))] 163.56/60.09 163.56/60.09 [g(X)] = [1] X + [2] 163.56/60.09 >= [1] X + [2] 163.56/60.09 = [n__g(X)] 163.56/60.09 163.56/60.09 163.56/60.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 163.56/60.09 163.56/60.09 We are left with following problem, upon which TcT provides the 163.56/60.09 certificate YES(O(1),O(n^2)). 163.56/60.09 163.56/60.09 Strict Trs: 163.56/60.09 { f(X) -> n__f(X) 163.56/60.09 , f(f(X)) -> c(n__f(n__g(n__f(X)))) } 163.56/60.09 Weak Trs: 163.56/60.09 { c(X) -> d(activate(X)) 163.56/60.09 , d(X) -> n__d(X) 163.56/60.09 , activate(X) -> X 163.56/60.09 , activate(n__f(X)) -> f(activate(X)) 163.56/60.09 , activate(n__g(X)) -> g(X) 163.56/60.09 , activate(n__d(X)) -> d(X) 163.56/60.09 , h(X) -> c(n__d(X)) 163.56/60.09 , g(X) -> n__g(X) } 163.56/60.09 Obligation: 163.56/60.09 derivational complexity 163.56/60.09 Answer: 163.56/60.09 YES(O(1),O(n^2)) 163.56/60.09 163.56/60.09 We use the processor 'matrix interpretation of dimension 2' to 163.56/60.09 orient following rules strictly. 163.56/60.09 163.56/60.09 Trs: { f(X) -> n__f(X) } 163.56/60.09 163.56/60.09 The induced complexity on above rules (modulo remaining rules) is 163.56/60.09 YES(?,O(n^2)) . These rules are moved into the corresponding weak 163.56/60.09 component(s). 163.56/60.09 163.56/60.09 Sub-proof: 163.56/60.09 ---------- 163.56/60.09 TcT has computed the following triangular matrix interpretation. 163.56/60.09 163.56/60.09 [f](x1) = [1 0] x1 + [2] 163.56/60.09 [0 1] [2] 163.56/60.09 163.56/60.09 [c](x1) = [1 2] x1 + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 163.56/60.09 [n__f](x1) = [1 0] x1 + [0] 163.56/60.09 [0 1] [2] 163.56/60.09 163.56/60.09 [n__g](x1) = [1 0] x1 + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 163.56/60.09 [d](x1) = [1 0] x1 + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 163.56/60.09 [activate](x1) = [1 1] x1 + [0] 163.56/60.09 [0 1] [0] 163.56/60.09 163.56/60.09 [h](x1) = [1 2] x1 + [2] 163.56/60.09 [0 1] [2] 163.56/60.09 163.56/60.09 [n__d](x1) = [1 0] x1 + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 163.56/60.09 [g](x1) = [1 0] x1 + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 163.56/60.09 The order satisfies the following ordering constraints: 163.56/60.09 163.56/60.09 [f(X)] = [1 0] X + [2] 163.56/60.09 [0 1] [2] 163.56/60.09 > [1 0] X + [0] 163.56/60.09 [0 1] [2] 163.56/60.09 = [n__f(X)] 163.56/60.09 163.56/60.09 [f(f(X))] = [1 0] X + [4] 163.56/60.09 [0 1] [4] 163.56/60.09 >= [1 0] X + [4] 163.56/60.09 [0 0] [0] 163.56/60.09 = [c(n__f(n__g(n__f(X))))] 163.56/60.09 163.56/60.09 [c(X)] = [1 2] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 >= [1 1] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 = [d(activate(X))] 163.56/60.09 163.56/60.09 [d(X)] = [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 >= [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 = [n__d(X)] 163.56/60.09 163.56/60.09 [activate(X)] = [1 1] X + [0] 163.56/60.09 [0 1] [0] 163.56/60.09 >= [1 0] X + [0] 163.56/60.09 [0 1] [0] 163.56/60.09 = [X] 163.56/60.09 163.56/60.09 [activate(n__f(X))] = [1 1] X + [2] 163.56/60.09 [0 1] [2] 163.56/60.09 >= [1 1] X + [2] 163.56/60.09 [0 1] [2] 163.56/60.09 = [f(activate(X))] 163.56/60.09 163.56/60.09 [activate(n__g(X))] = [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 >= [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 = [g(X)] 163.56/60.09 163.56/60.09 [activate(n__d(X))] = [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 >= [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 = [d(X)] 163.56/60.09 163.56/60.09 [h(X)] = [1 2] X + [2] 163.56/60.09 [0 1] [2] 163.56/60.09 > [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 = [c(n__d(X))] 163.56/60.09 163.56/60.09 [g(X)] = [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 >= [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 = [n__g(X)] 163.56/60.09 163.56/60.09 163.56/60.09 We return to the main proof. 163.56/60.09 163.56/60.09 We are left with following problem, upon which TcT provides the 163.56/60.09 certificate YES(O(1),O(n^2)). 163.56/60.09 163.56/60.09 Strict Trs: { f(f(X)) -> c(n__f(n__g(n__f(X)))) } 163.56/60.09 Weak Trs: 163.56/60.09 { f(X) -> n__f(X) 163.56/60.09 , c(X) -> d(activate(X)) 163.56/60.09 , d(X) -> n__d(X) 163.56/60.09 , activate(X) -> X 163.56/60.09 , activate(n__f(X)) -> f(activate(X)) 163.56/60.09 , activate(n__g(X)) -> g(X) 163.56/60.09 , activate(n__d(X)) -> d(X) 163.56/60.09 , h(X) -> c(n__d(X)) 163.56/60.09 , g(X) -> n__g(X) } 163.56/60.09 Obligation: 163.56/60.09 derivational complexity 163.56/60.09 Answer: 163.56/60.09 YES(O(1),O(n^2)) 163.56/60.09 163.56/60.09 We use the processor 'matrix interpretation of dimension 2' to 163.56/60.09 orient following rules strictly. 163.56/60.09 163.56/60.09 Trs: { f(f(X)) -> c(n__f(n__g(n__f(X)))) } 163.56/60.09 163.56/60.09 The induced complexity on above rules (modulo remaining rules) is 163.56/60.09 YES(?,O(n^2)) . These rules are moved into the corresponding weak 163.56/60.09 component(s). 163.56/60.09 163.56/60.09 Sub-proof: 163.56/60.09 ---------- 163.56/60.09 TcT has computed the following triangular matrix interpretation. 163.56/60.09 163.56/60.09 [f](x1) = [1 0] x1 + [2] 163.56/60.09 [0 1] [2] 163.56/60.09 163.56/60.09 [c](x1) = [1 1] x1 + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 163.56/60.09 [n__f](x1) = [1 0] x1 + [0] 163.56/60.09 [0 1] [2] 163.56/60.09 163.56/60.09 [n__g](x1) = [1 0] x1 + [1] 163.56/60.09 [0 0] [0] 163.56/60.09 163.56/60.09 [d](x1) = [1 0] x1 + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 163.56/60.09 [activate](x1) = [1 1] x1 + [0] 163.56/60.09 [0 1] [0] 163.56/60.09 163.56/60.09 [h](x1) = [1 1] x1 + [1] 163.56/60.09 [0 1] [1] 163.56/60.09 163.56/60.09 [n__d](x1) = [1 0] x1 + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 163.56/60.09 [g](x1) = [1 0] x1 + [1] 163.56/60.09 [0 0] [0] 163.56/60.09 163.56/60.09 The order satisfies the following ordering constraints: 163.56/60.09 163.56/60.09 [f(X)] = [1 0] X + [2] 163.56/60.09 [0 1] [2] 163.56/60.09 > [1 0] X + [0] 163.56/60.09 [0 1] [2] 163.56/60.09 = [n__f(X)] 163.56/60.09 163.56/60.09 [f(f(X))] = [1 0] X + [4] 163.56/60.09 [0 1] [4] 163.56/60.09 > [1 0] X + [3] 163.56/60.09 [0 0] [0] 163.56/60.09 = [c(n__f(n__g(n__f(X))))] 163.56/60.09 163.56/60.09 [c(X)] = [1 1] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 >= [1 1] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 = [d(activate(X))] 163.56/60.09 163.56/60.09 [d(X)] = [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 >= [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 = [n__d(X)] 163.56/60.09 163.56/60.09 [activate(X)] = [1 1] X + [0] 163.56/60.09 [0 1] [0] 163.56/60.09 >= [1 0] X + [0] 163.56/60.09 [0 1] [0] 163.56/60.09 = [X] 163.56/60.09 163.56/60.09 [activate(n__f(X))] = [1 1] X + [2] 163.56/60.09 [0 1] [2] 163.56/60.09 >= [1 1] X + [2] 163.56/60.09 [0 1] [2] 163.56/60.09 = [f(activate(X))] 163.56/60.09 163.56/60.09 [activate(n__g(X))] = [1 0] X + [1] 163.56/60.09 [0 0] [0] 163.56/60.09 >= [1 0] X + [1] 163.56/60.09 [0 0] [0] 163.56/60.09 = [g(X)] 163.56/60.09 163.56/60.09 [activate(n__d(X))] = [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 >= [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 = [d(X)] 163.56/60.09 163.56/60.09 [h(X)] = [1 1] X + [1] 163.56/60.09 [0 1] [1] 163.56/60.09 > [1 0] X + [0] 163.56/60.09 [0 0] [0] 163.56/60.09 = [c(n__d(X))] 163.56/60.09 163.56/60.09 [g(X)] = [1 0] X + [1] 163.56/60.09 [0 0] [0] 163.56/60.09 >= [1 0] X + [1] 163.56/60.09 [0 0] [0] 163.56/60.09 = [n__g(X)] 163.56/60.09 163.56/60.09 163.56/60.09 We return to the main proof. 163.56/60.09 163.56/60.09 We are left with following problem, upon which TcT provides the 163.56/60.09 certificate YES(O(1),O(1)). 163.56/60.09 163.56/60.09 Weak Trs: 163.56/60.09 { f(X) -> n__f(X) 163.56/60.09 , f(f(X)) -> c(n__f(n__g(n__f(X)))) 163.56/60.09 , c(X) -> d(activate(X)) 163.56/60.09 , d(X) -> n__d(X) 163.56/60.09 , activate(X) -> X 163.56/60.09 , activate(n__f(X)) -> f(activate(X)) 163.56/60.09 , activate(n__g(X)) -> g(X) 163.56/60.09 , activate(n__d(X)) -> d(X) 163.56/60.09 , h(X) -> c(n__d(X)) 163.56/60.09 , g(X) -> n__g(X) } 163.56/60.09 Obligation: 163.56/60.09 derivational complexity 163.56/60.09 Answer: 163.56/60.09 YES(O(1),O(1)) 163.56/60.09 163.56/60.09 Empty rules are trivially bounded 163.56/60.09 163.56/60.09 Hurray, we answered YES(O(1),O(n^2)) 163.81/60.10 EOF