YES(O(1),O(n^1)) 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(n^1)). 0.00/0.86 0.00/0.86 Strict Trs: 0.00/0.86 { a__f(X) -> f(X) 0.00/0.86 , a__f(f(a())) -> a__f(g(f(a()))) 0.00/0.86 , mark(f(X)) -> a__f(X) 0.00/0.86 , mark(a()) -> a() 0.00/0.86 , mark(g(X)) -> g(mark(X)) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 We add the following weak dependency pairs: 0.00/0.86 0.00/0.86 Strict DPs: 0.00/0.86 { a__f^#(X) -> c_1() 0.00/0.86 , a__f^#(f(a())) -> c_2(a__f^#(g(f(a())))) 0.00/0.86 , mark^#(f(X)) -> c_3(a__f^#(X)) 0.00/0.86 , mark^#(a()) -> c_4() 0.00/0.86 , mark^#(g(X)) -> c_5(mark^#(X)) } 0.00/0.86 0.00/0.86 and mark the set of starting terms. 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(n^1)). 0.00/0.86 0.00/0.86 Strict DPs: 0.00/0.86 { a__f^#(X) -> c_1() 0.00/0.86 , a__f^#(f(a())) -> c_2(a__f^#(g(f(a())))) 0.00/0.86 , mark^#(f(X)) -> c_3(a__f^#(X)) 0.00/0.86 , mark^#(a()) -> c_4() 0.00/0.86 , mark^#(g(X)) -> c_5(mark^#(X)) } 0.00/0.86 Strict Trs: 0.00/0.86 { a__f(X) -> f(X) 0.00/0.86 , a__f(f(a())) -> a__f(g(f(a()))) 0.00/0.86 , mark(f(X)) -> a__f(X) 0.00/0.86 , mark(a()) -> a() 0.00/0.86 , mark(g(X)) -> g(mark(X)) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 No rule is usable, rules are removed from the input problem. 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(n^1)). 0.00/0.86 0.00/0.86 Strict DPs: 0.00/0.86 { a__f^#(X) -> c_1() 0.00/0.86 , a__f^#(f(a())) -> c_2(a__f^#(g(f(a())))) 0.00/0.86 , mark^#(f(X)) -> c_3(a__f^#(X)) 0.00/0.86 , mark^#(a()) -> c_4() 0.00/0.86 , mark^#(g(X)) -> c_5(mark^#(X)) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 The weightgap principle applies (using the following constant 0.00/0.86 growth matrix-interpretation) 0.00/0.86 0.00/0.86 The following argument positions are usable: 0.00/0.86 Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1} 0.00/0.86 0.00/0.86 TcT has computed the following constructor-restricted matrix 0.00/0.86 interpretation. 0.00/0.86 0.00/0.86 [f](x1) = [0] 0.00/0.86 [0] 0.00/0.86 0.00/0.86 [a] = [0] 0.00/0.86 [0] 0.00/0.86 0.00/0.86 [g](x1) = [0] 0.00/0.86 [0] 0.00/0.86 0.00/0.86 [a__f^#](x1) = [0] 0.00/0.86 [0] 0.00/0.86 0.00/0.86 [c_1] = [1] 0.00/0.86 [0] 0.00/0.86 0.00/0.86 [c_2](x1) = [1 0] x1 + [1] 0.00/0.86 [0 1] [0] 0.00/0.86 0.00/0.86 [mark^#](x1) = [1] 0.00/0.86 [0] 0.00/0.86 0.00/0.86 [c_3](x1) = [1 0] x1 + [0] 0.00/0.86 [0 1] [0] 0.00/0.86 0.00/0.86 [c_4] = [0] 0.00/0.86 [0] 0.00/0.86 0.00/0.86 [c_5](x1) = [1 0] x1 + [1] 0.00/0.86 [0 1] [0] 0.00/0.86 0.00/0.86 The order satisfies the following ordering constraints: 0.00/0.86 0.00/0.86 [a__f^#(X)] = [0] 0.00/0.86 [0] 0.00/0.86 ? [1] 0.00/0.86 [0] 0.00/0.86 = [c_1()] 0.00/0.86 0.00/0.86 [a__f^#(f(a()))] = [0] 0.00/0.86 [0] 0.00/0.86 ? [1] 0.00/0.86 [0] 0.00/0.86 = [c_2(a__f^#(g(f(a()))))] 0.00/0.86 0.00/0.86 [mark^#(f(X))] = [1] 0.00/0.86 [0] 0.00/0.86 > [0] 0.00/0.86 [0] 0.00/0.86 = [c_3(a__f^#(X))] 0.00/0.86 0.00/0.86 [mark^#(a())] = [1] 0.00/0.86 [0] 0.00/0.86 > [0] 0.00/0.86 [0] 0.00/0.86 = [c_4()] 0.00/0.86 0.00/0.86 [mark^#(g(X))] = [1] 0.00/0.86 [0] 0.00/0.86 ? [2] 0.00/0.86 [0] 0.00/0.86 = [c_5(mark^#(X))] 0.00/0.86 0.00/0.86 0.00/0.86 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(n^1)). 0.00/0.86 0.00/0.86 Strict DPs: 0.00/0.86 { a__f^#(X) -> c_1() 0.00/0.86 , a__f^#(f(a())) -> c_2(a__f^#(g(f(a())))) 0.00/0.86 , mark^#(g(X)) -> c_5(mark^#(X)) } 0.00/0.86 Weak DPs: 0.00/0.86 { mark^#(f(X)) -> c_3(a__f^#(X)) 0.00/0.86 , mark^#(a()) -> c_4() } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.86 closed under successors. The DPs are removed. 0.00/0.86 0.00/0.86 { mark^#(a()) -> c_4() } 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(n^1)). 0.00/0.86 0.00/0.86 Strict DPs: 0.00/0.86 { a__f^#(X) -> c_1() 0.00/0.86 , a__f^#(f(a())) -> c_2(a__f^#(g(f(a())))) 0.00/0.86 , mark^#(g(X)) -> c_5(mark^#(X)) } 0.00/0.86 Weak DPs: { mark^#(f(X)) -> c_3(a__f^#(X)) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.86 orient following rules strictly. 0.00/0.86 0.00/0.86 DPs: 0.00/0.86 { 3: mark^#(g(X)) -> c_5(mark^#(X)) } 0.00/0.86 0.00/0.86 Sub-proof: 0.00/0.86 ---------- 0.00/0.86 The following argument positions are usable: 0.00/0.86 Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1} 0.00/0.86 0.00/0.86 TcT has computed the following constructor-based matrix 0.00/0.86 interpretation satisfying not(EDA). 0.00/0.86 0.00/0.86 [f](x1) = [0] 0.00/0.86 0.00/0.86 [a] = [0] 0.00/0.86 0.00/0.86 [g](x1) = [1] x1 + [1] 0.00/0.86 0.00/0.86 [a__f^#](x1) = [0] 0.00/0.86 0.00/0.86 [c_1] = [0] 0.00/0.86 0.00/0.86 [c_2](x1) = [4] x1 + [0] 0.00/0.86 0.00/0.86 [mark^#](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [c_3](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [c_5](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 The order satisfies the following ordering constraints: 0.00/0.86 0.00/0.86 [a__f^#(X)] = [0] 0.00/0.86 >= [0] 0.00/0.86 = [c_1()] 0.00/0.86 0.00/0.86 [a__f^#(f(a()))] = [0] 0.00/0.86 >= [0] 0.00/0.86 = [c_2(a__f^#(g(f(a()))))] 0.00/0.86 0.00/0.86 [mark^#(f(X))] = [0] 0.00/0.86 >= [0] 0.00/0.86 = [c_3(a__f^#(X))] 0.00/0.86 0.00/0.86 [mark^#(g(X))] = [1] X + [1] 0.00/0.86 > [1] X + [0] 0.00/0.86 = [c_5(mark^#(X))] 0.00/0.86 0.00/0.86 0.00/0.86 We return to the main proof. Consider the set of all dependency 0.00/0.86 pairs 0.00/0.86 0.00/0.86 : 0.00/0.86 { 1: a__f^#(X) -> c_1() 0.00/0.86 , 2: a__f^#(f(a())) -> c_2(a__f^#(g(f(a())))) 0.00/0.86 , 3: mark^#(g(X)) -> c_5(mark^#(X)) 0.00/0.86 , 4: mark^#(f(X)) -> c_3(a__f^#(X)) } 0.00/0.86 0.00/0.86 Processor 'matrix interpretation of dimension 1' induces the 0.00/0.86 complexity certificate YES(?,O(n^1)) on application of dependency 0.00/0.86 pairs {3}. These cover all (indirect) predecessors of dependency 0.00/0.86 pairs {1,2,3,4}, their number of application is equally bounded. 0.00/0.86 The dependency pairs are shifted into the weak component. 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(1)). 0.00/0.86 0.00/0.86 Weak DPs: 0.00/0.86 { a__f^#(X) -> c_1() 0.00/0.86 , a__f^#(f(a())) -> c_2(a__f^#(g(f(a())))) 0.00/0.86 , mark^#(f(X)) -> c_3(a__f^#(X)) 0.00/0.86 , mark^#(g(X)) -> c_5(mark^#(X)) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(1)) 0.00/0.86 0.00/0.86 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.86 closed under successors. The DPs are removed. 0.00/0.86 0.00/0.86 { a__f^#(X) -> c_1() 0.00/0.86 , a__f^#(f(a())) -> c_2(a__f^#(g(f(a())))) 0.00/0.86 , mark^#(f(X)) -> c_3(a__f^#(X)) 0.00/0.86 , mark^#(g(X)) -> c_5(mark^#(X)) } 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(1)). 0.00/0.86 0.00/0.86 Rules: Empty 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(1)) 0.00/0.86 0.00/0.86 Empty rules are trivially bounded 0.00/0.86 0.00/0.86 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.87 EOF