YES(O(1),O(1)) 0.00/0.36 YES(O(1),O(1)) 0.00/0.36 0.00/0.36 We are left with following problem, upon which TcT provides the 0.00/0.36 certificate YES(O(1),O(1)). 0.00/0.36 0.00/0.36 Strict Trs: 0.00/0.36 { f(X) -> n__f(X) 0.00/0.36 , f(f(a())) -> f(g(n__f(a()))) 0.00/0.36 , activate(X) -> X 0.00/0.36 , activate(n__f(X)) -> f(X) } 0.00/0.36 Obligation: 0.00/0.36 runtime complexity 0.00/0.36 Answer: 0.00/0.36 YES(O(1),O(1)) 0.00/0.36 0.00/0.36 We add the following weak dependency pairs: 0.00/0.36 0.00/0.36 Strict DPs: 0.00/0.36 { f^#(X) -> c_1(X) 0.00/0.36 , f^#(f(a())) -> c_2(f^#(g(n__f(a())))) 0.00/0.36 , activate^#(X) -> c_3(X) 0.00/0.36 , activate^#(n__f(X)) -> c_4(f^#(X)) } 0.00/0.36 0.00/0.36 and mark the set of starting terms. 0.00/0.36 0.00/0.36 We are left with following problem, upon which TcT provides the 0.00/0.36 certificate YES(O(1),O(1)). 0.00/0.36 0.00/0.36 Strict DPs: 0.00/0.36 { f^#(X) -> c_1(X) 0.00/0.36 , f^#(f(a())) -> c_2(f^#(g(n__f(a())))) 0.00/0.36 , activate^#(X) -> c_3(X) 0.00/0.36 , activate^#(n__f(X)) -> c_4(f^#(X)) } 0.00/0.36 Strict Trs: 0.00/0.36 { f(X) -> n__f(X) 0.00/0.36 , f(f(a())) -> f(g(n__f(a()))) 0.00/0.36 , activate(X) -> X 0.00/0.36 , activate(n__f(X)) -> f(X) } 0.00/0.36 Obligation: 0.00/0.36 runtime complexity 0.00/0.36 Answer: 0.00/0.36 YES(O(1),O(1)) 0.00/0.36 0.00/0.36 No rule is usable, rules are removed from the input problem. 0.00/0.36 0.00/0.36 We are left with following problem, upon which TcT provides the 0.00/0.36 certificate YES(O(1),O(1)). 0.00/0.36 0.00/0.36 Strict DPs: 0.00/0.36 { f^#(X) -> c_1(X) 0.00/0.36 , f^#(f(a())) -> c_2(f^#(g(n__f(a())))) 0.00/0.36 , activate^#(X) -> c_3(X) 0.00/0.36 , activate^#(n__f(X)) -> c_4(f^#(X)) } 0.00/0.36 Obligation: 0.00/0.36 runtime complexity 0.00/0.36 Answer: 0.00/0.36 YES(O(1),O(1)) 0.00/0.36 0.00/0.36 The weightgap principle applies (using the following constant 0.00/0.36 growth matrix-interpretation) 0.00/0.36 0.00/0.36 The following argument positions are usable: 0.00/0.36 Uargs(c_2) = {1}, Uargs(c_4) = {1} 0.00/0.36 0.00/0.36 TcT has computed the following constructor-restricted matrix 0.00/0.36 interpretation. 0.00/0.36 0.00/0.36 [f](x1) = [0] 0.00/0.36 [0] 0.00/0.36 0.00/0.36 [a] = [0] 0.00/0.36 [0] 0.00/0.36 0.00/0.36 [g](x1) = [0] 0.00/0.36 [0] 0.00/0.36 0.00/0.36 [n__f](x1) = [0] 0.00/0.36 [0] 0.00/0.36 0.00/0.36 [f^#](x1) = [0] 0.00/0.36 [0] 0.00/0.36 0.00/0.36 [c_1](x1) = [1] 0.00/0.36 [0] 0.00/0.36 0.00/0.36 [c_2](x1) = [1 0] x1 + [1] 0.00/0.36 [0 1] [2] 0.00/0.36 0.00/0.36 [activate^#](x1) = [0 0] x1 + [1] 0.00/0.36 [1 2] [2] 0.00/0.36 0.00/0.36 [c_3](x1) = [0 0] x1 + [0] 0.00/0.36 [1 1] [1] 0.00/0.36 0.00/0.36 [c_4](x1) = [1 0] x1 + [0] 0.00/0.36 [0 1] [2] 0.00/0.36 0.00/0.36 The order satisfies the following ordering constraints: 0.00/0.36 0.00/0.36 [f^#(X)] = [0] 0.00/0.36 [0] 0.00/0.36 ? [1] 0.00/0.36 [0] 0.00/0.36 = [c_1(X)] 0.00/0.36 0.00/0.36 [f^#(f(a()))] = [0] 0.00/0.36 [0] 0.00/0.36 ? [1] 0.00/0.36 [2] 0.00/0.36 = [c_2(f^#(g(n__f(a()))))] 0.00/0.36 0.00/0.36 [activate^#(X)] = [0 0] X + [1] 0.00/0.36 [1 2] [2] 0.00/0.36 > [0 0] X + [0] 0.00/0.36 [1 1] [1] 0.00/0.36 = [c_3(X)] 0.00/0.36 0.00/0.36 [activate^#(n__f(X))] = [1] 0.00/0.36 [2] 0.00/0.36 > [0] 0.00/0.36 [2] 0.00/0.36 = [c_4(f^#(X))] 0.00/0.36 0.00/0.36 0.00/0.36 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.36 0.00/0.36 We are left with following problem, upon which TcT provides the 0.00/0.36 certificate YES(O(1),O(1)). 0.00/0.36 0.00/0.36 Strict DPs: 0.00/0.36 { f^#(X) -> c_1(X) 0.00/0.36 , f^#(f(a())) -> c_2(f^#(g(n__f(a())))) } 0.00/0.36 Weak DPs: 0.00/0.36 { activate^#(X) -> c_3(X) 0.00/0.36 , activate^#(n__f(X)) -> c_4(f^#(X)) } 0.00/0.36 Obligation: 0.00/0.36 runtime complexity 0.00/0.36 Answer: 0.00/0.36 YES(O(1),O(1)) 0.00/0.36 0.00/0.36 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.36 orient following rules strictly. 0.00/0.36 0.00/0.36 DPs: 0.00/0.36 { 1: f^#(X) -> c_1(X) 0.00/0.36 , 3: activate^#(X) -> c_3(X) 0.00/0.36 , 4: activate^#(n__f(X)) -> c_4(f^#(X)) } 0.00/0.36 0.00/0.36 Sub-proof: 0.00/0.36 ---------- 0.00/0.36 The following argument positions are usable: 0.00/0.36 Uargs(c_2) = {1}, Uargs(c_4) = {1} 0.00/0.36 0.00/0.36 TcT has computed the following constructor-restricted matrix 0.00/0.36 interpretation. Note that the diagonal of the component-wise maxima 0.00/0.36 of interpretation-entries (of constructors) contains no more than 0 0.00/0.36 non-zero entries. 0.00/0.36 0.00/0.36 [f](x1) = [7] x1 + [7] 0.00/0.36 0.00/0.36 [a] = [4] 0.00/0.36 0.00/0.36 [g](x1) = [4] 0.00/0.36 0.00/0.36 [n__f](x1) = [2] 0.00/0.36 0.00/0.36 [f^#](x1) = [1] 0.00/0.36 0.00/0.36 [c_1](x1) = [0] 0.00/0.36 0.00/0.36 [c_2](x1) = [1] x1 + [0] 0.00/0.36 0.00/0.36 [activate^#](x1) = [4] x1 + [7] 0.00/0.36 0.00/0.36 [c_3](x1) = [4] x1 + [6] 0.00/0.36 0.00/0.36 [c_4](x1) = [1] x1 + [5] 0.00/0.36 0.00/0.36 The order satisfies the following ordering constraints: 0.00/0.36 0.00/0.36 [f^#(X)] = [1] 0.00/0.36 > [0] 0.00/0.36 = [c_1(X)] 0.00/0.36 0.00/0.36 [f^#(f(a()))] = [1] 0.00/0.36 >= [1] 0.00/0.36 = [c_2(f^#(g(n__f(a()))))] 0.00/0.36 0.00/0.36 [activate^#(X)] = [4] X + [7] 0.00/0.36 > [4] X + [6] 0.00/0.36 = [c_3(X)] 0.00/0.36 0.00/0.36 [activate^#(n__f(X))] = [15] 0.00/0.36 > [6] 0.00/0.36 = [c_4(f^#(X))] 0.00/0.36 0.00/0.36 0.00/0.36 We return to the main proof. Consider the set of all dependency 0.00/0.36 pairs 0.00/0.36 0.00/0.36 : 0.00/0.36 { 1: f^#(X) -> c_1(X) 0.00/0.36 , 2: f^#(f(a())) -> c_2(f^#(g(n__f(a())))) 0.00/0.36 , 3: activate^#(X) -> c_3(X) 0.00/0.36 , 4: activate^#(n__f(X)) -> c_4(f^#(X)) } 0.00/0.36 0.00/0.36 Processor 'matrix interpretation of dimension 1' induces the 0.00/0.36 complexity certificate YES(?,O(1)) on application of dependency 0.00/0.36 pairs {1,3,4}. These cover all (indirect) predecessors of 0.00/0.36 dependency pairs {1,2,3,4}, their number of application is equally 0.00/0.36 bounded. The dependency pairs are shifted into the weak component. 0.00/0.36 0.00/0.36 We are left with following problem, upon which TcT provides the 0.00/0.36 certificate YES(O(1),O(1)). 0.00/0.36 0.00/0.36 Weak DPs: 0.00/0.36 { f^#(X) -> c_1(X) 0.00/0.36 , f^#(f(a())) -> c_2(f^#(g(n__f(a())))) 0.00/0.36 , activate^#(X) -> c_3(X) 0.00/0.36 , activate^#(n__f(X)) -> c_4(f^#(X)) } 0.00/0.36 Obligation: 0.00/0.36 runtime complexity 0.00/0.36 Answer: 0.00/0.36 YES(O(1),O(1)) 0.00/0.36 0.00/0.36 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.36 closed under successors. The DPs are removed. 0.00/0.36 0.00/0.36 { f^#(X) -> c_1(X) 0.00/0.36 , f^#(f(a())) -> c_2(f^#(g(n__f(a())))) 0.00/0.36 , activate^#(X) -> c_3(X) 0.00/0.36 , activate^#(n__f(X)) -> c_4(f^#(X)) } 0.00/0.36 0.00/0.36 We are left with following problem, upon which TcT provides the 0.00/0.36 certificate YES(O(1),O(1)). 0.00/0.36 0.00/0.36 Rules: Empty 0.00/0.36 Obligation: 0.00/0.36 runtime complexity 0.00/0.36 Answer: 0.00/0.36 YES(O(1),O(1)) 0.00/0.36 0.00/0.36 Empty rules are trivially bounded 0.00/0.36 0.00/0.36 Hurray, we answered YES(O(1),O(1)) 0.00/0.36 EOF