YES(O(1),O(n^2)) 176.53/60.02 YES(O(1),O(n^2)) 176.53/60.02 176.53/60.02 We are left with following problem, upon which TcT provides the 176.53/60.02 certificate YES(O(1),O(n^2)). 176.53/60.02 176.53/60.02 Strict Trs: 176.53/60.02 { a__f(X) -> f(X) 176.53/60.02 , a__f(f(a())) -> a__f(g(f(a()))) 176.53/60.02 , mark(f(X)) -> a__f(X) 176.53/60.02 , mark(a()) -> a() 176.53/60.02 , mark(g(X)) -> g(mark(X)) } 176.53/60.02 Obligation: 176.53/60.02 derivational complexity 176.53/60.02 Answer: 176.53/60.02 YES(O(1),O(n^2)) 176.53/60.02 176.53/60.02 We use the processor 'matrix interpretation of dimension 1' to 176.53/60.02 orient following rules strictly. 176.53/60.02 176.53/60.02 Trs: 176.53/60.02 { mark(f(X)) -> a__f(X) 176.53/60.02 , mark(a()) -> a() } 176.53/60.02 176.53/60.02 The induced complexity on above rules (modulo remaining rules) is 176.53/60.02 YES(?,O(n^1)) . These rules are moved into the corresponding weak 176.53/60.02 component(s). 176.53/60.02 176.53/60.02 Sub-proof: 176.53/60.02 ---------- 176.53/60.02 TcT has computed the following triangular matrix interpretation. 176.53/60.02 176.53/60.02 [a__f](x1) = [1] x1 + [0] 176.53/60.02 176.53/60.02 [f](x1) = [1] x1 + [0] 176.53/60.02 176.53/60.02 [a] = [0] 176.53/60.02 176.53/60.02 [g](x1) = [1] x1 + [0] 176.53/60.02 176.53/60.02 [mark](x1) = [1] x1 + [1] 176.53/60.02 176.53/60.02 The order satisfies the following ordering constraints: 176.53/60.02 176.53/60.02 [a__f(X)] = [1] X + [0] 176.53/60.02 >= [1] X + [0] 176.53/60.02 = [f(X)] 176.53/60.02 176.53/60.02 [a__f(f(a()))] = [0] 176.53/60.02 >= [0] 176.53/60.02 = [a__f(g(f(a())))] 176.53/60.02 176.53/60.02 [mark(f(X))] = [1] X + [1] 176.53/60.02 > [1] X + [0] 176.53/60.02 = [a__f(X)] 176.53/60.02 176.53/60.02 [mark(a())] = [1] 176.53/60.02 > [0] 176.53/60.02 = [a()] 176.53/60.02 176.53/60.02 [mark(g(X))] = [1] X + [1] 176.53/60.02 >= [1] X + [1] 176.53/60.02 = [g(mark(X))] 176.53/60.02 176.53/60.02 176.53/60.02 We return to the main proof. 176.53/60.02 176.53/60.02 We are left with following problem, upon which TcT provides the 176.53/60.02 certificate YES(O(1),O(n^2)). 176.53/60.02 176.53/60.02 Strict Trs: 176.53/60.02 { a__f(X) -> f(X) 176.53/60.02 , a__f(f(a())) -> a__f(g(f(a()))) 176.53/60.02 , mark(g(X)) -> g(mark(X)) } 176.53/60.03 Weak Trs: 176.53/60.03 { mark(f(X)) -> a__f(X) 176.53/60.03 , mark(a()) -> a() } 176.53/60.03 Obligation: 176.53/60.03 derivational complexity 176.53/60.03 Answer: 176.53/60.03 YES(O(1),O(n^2)) 176.53/60.03 176.53/60.03 We use the processor 'matrix interpretation of dimension 1' to 176.53/60.03 orient following rules strictly. 176.53/60.03 176.53/60.03 Trs: { a__f(X) -> f(X) } 176.53/60.03 176.53/60.03 The induced complexity on above rules (modulo remaining rules) is 176.53/60.03 YES(?,O(n^1)) . These rules are moved into the corresponding weak 176.53/60.03 component(s). 176.53/60.03 176.53/60.03 Sub-proof: 176.53/60.03 ---------- 176.53/60.03 TcT has computed the following triangular matrix interpretation. 176.53/60.03 176.53/60.03 [a__f](x1) = [1] x1 + [1] 176.53/60.03 176.53/60.03 [f](x1) = [1] x1 + [0] 176.53/60.03 176.53/60.03 [a] = [0] 176.53/60.03 176.53/60.03 [g](x1) = [1] x1 + [0] 176.53/60.03 176.53/60.03 [mark](x1) = [1] x1 + [1] 176.53/60.03 176.53/60.03 The order satisfies the following ordering constraints: 176.53/60.03 176.53/60.03 [a__f(X)] = [1] X + [1] 176.53/60.03 > [1] X + [0] 176.53/60.03 = [f(X)] 176.53/60.03 176.53/60.03 [a__f(f(a()))] = [1] 176.53/60.03 >= [1] 176.53/60.03 = [a__f(g(f(a())))] 176.53/60.03 176.53/60.03 [mark(f(X))] = [1] X + [1] 176.53/60.03 >= [1] X + [1] 176.53/60.03 = [a__f(X)] 176.53/60.03 176.53/60.03 [mark(a())] = [1] 176.53/60.03 > [0] 176.53/60.03 = [a()] 176.53/60.03 176.53/60.03 [mark(g(X))] = [1] X + [1] 176.53/60.03 >= [1] X + [1] 176.53/60.03 = [g(mark(X))] 176.53/60.03 176.53/60.03 176.53/60.03 We return to the main proof. 176.53/60.03 176.53/60.03 We are left with following problem, upon which TcT provides the 176.53/60.03 certificate YES(O(1),O(n^2)). 176.53/60.03 176.53/60.03 Strict Trs: 176.53/60.03 { a__f(f(a())) -> a__f(g(f(a()))) 176.53/60.03 , mark(g(X)) -> g(mark(X)) } 176.53/60.03 Weak Trs: 176.53/60.03 { a__f(X) -> f(X) 176.53/60.03 , mark(f(X)) -> a__f(X) 176.53/60.03 , mark(a()) -> a() } 176.53/60.03 Obligation: 176.53/60.03 derivational complexity 176.53/60.03 Answer: 176.53/60.03 YES(O(1),O(n^2)) 176.53/60.03 176.53/60.03 We use the processor 'matrix interpretation of dimension 2' to 176.53/60.03 orient following rules strictly. 176.53/60.03 176.53/60.03 Trs: { a__f(f(a())) -> a__f(g(f(a()))) } 176.53/60.03 176.53/60.03 The induced complexity on above rules (modulo remaining rules) is 176.53/60.03 YES(?,O(n^1)) . These rules are moved into the corresponding weak 176.53/60.03 component(s). 176.53/60.03 176.53/60.03 Sub-proof: 176.53/60.03 ---------- 176.53/60.03 TcT has computed the following triangular matrix interpretation. 176.53/60.03 Note that the diagonal of the component-wise maxima of 176.53/60.03 interpretation-entries contains no more than 1 non-zero entries. 176.53/60.03 176.53/60.03 [a__f](x1) = [1 1] x1 + [0] 176.53/60.03 [0 0] [2] 176.53/60.03 176.53/60.03 [f](x1) = [1 1] x1 + [0] 176.53/60.03 [0 0] [2] 176.53/60.03 176.53/60.03 [a] = [0] 176.53/60.03 [2] 176.53/60.03 176.53/60.03 [g](x1) = [1 0] x1 + [0] 176.53/60.03 [0 0] [1] 176.53/60.03 176.53/60.03 [mark](x1) = [1 0] x1 + [0] 176.53/60.03 [0 0] [2] 176.53/60.03 176.53/60.03 The order satisfies the following ordering constraints: 176.53/60.03 176.53/60.03 [a__f(X)] = [1 1] X + [0] 176.53/60.03 [0 0] [2] 176.53/60.03 >= [1 1] X + [0] 176.53/60.03 [0 0] [2] 176.53/60.03 = [f(X)] 176.53/60.03 176.53/60.03 [a__f(f(a()))] = [4] 176.53/60.03 [2] 176.53/60.03 > [3] 176.53/60.03 [2] 176.53/60.03 = [a__f(g(f(a())))] 176.53/60.03 176.53/60.03 [mark(f(X))] = [1 1] X + [0] 176.53/60.03 [0 0] [2] 176.53/60.03 >= [1 1] X + [0] 176.53/60.03 [0 0] [2] 176.53/60.03 = [a__f(X)] 176.53/60.03 176.53/60.03 [mark(a())] = [0] 176.53/60.03 [2] 176.53/60.03 >= [0] 176.53/60.03 [2] 176.53/60.03 = [a()] 176.53/60.03 176.53/60.03 [mark(g(X))] = [1 0] X + [0] 176.53/60.03 [0 0] [2] 176.53/60.03 >= [1 0] X + [0] 176.53/60.03 [0 0] [1] 176.53/60.03 = [g(mark(X))] 176.53/60.03 176.53/60.03 176.53/60.03 We return to the main proof. 176.53/60.03 176.53/60.03 We are left with following problem, upon which TcT provides the 176.53/60.03 certificate YES(O(1),O(n^2)). 176.53/60.03 176.53/60.03 Strict Trs: { mark(g(X)) -> g(mark(X)) } 176.53/60.03 Weak Trs: 176.53/60.03 { a__f(X) -> f(X) 176.53/60.03 , a__f(f(a())) -> a__f(g(f(a()))) 176.53/60.03 , mark(f(X)) -> a__f(X) 176.53/60.03 , mark(a()) -> a() } 176.53/60.03 Obligation: 176.53/60.03 derivational complexity 176.53/60.03 Answer: 176.53/60.03 YES(O(1),O(n^2)) 176.53/60.03 176.53/60.03 We use the processor 'matrix interpretation of dimension 2' to 176.53/60.03 orient following rules strictly. 176.53/60.03 176.53/60.03 Trs: { mark(g(X)) -> g(mark(X)) } 176.53/60.03 176.53/60.03 The induced complexity on above rules (modulo remaining rules) is 176.53/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak 176.53/60.03 component(s). 176.53/60.03 176.53/60.03 Sub-proof: 176.53/60.03 ---------- 176.53/60.03 TcT has computed the following triangular matrix interpretation. 176.53/60.03 176.53/60.03 [a__f](x1) = [1 0] x1 + [0] 176.53/60.03 [0 0] [0] 176.53/60.03 176.53/60.03 [f](x1) = [1 0] x1 + [0] 176.53/60.03 [0 0] [0] 176.53/60.03 176.53/60.03 [a] = [0] 176.53/60.03 [0] 176.53/60.03 176.53/60.03 [g](x1) = [1 0] x1 + [0] 176.53/60.03 [0 1] [2] 176.53/60.03 176.53/60.03 [mark](x1) = [1 1] x1 + [0] 176.53/60.03 [0 1] [0] 176.53/60.03 176.53/60.03 The order satisfies the following ordering constraints: 176.53/60.03 176.53/60.03 [a__f(X)] = [1 0] X + [0] 176.53/60.03 [0 0] [0] 176.53/60.03 >= [1 0] X + [0] 176.53/60.03 [0 0] [0] 176.53/60.03 = [f(X)] 176.53/60.03 176.53/60.03 [a__f(f(a()))] = [0] 176.53/60.03 [0] 176.53/60.03 >= [0] 176.53/60.03 [0] 176.53/60.03 = [a__f(g(f(a())))] 176.53/60.03 176.53/60.03 [mark(f(X))] = [1 0] X + [0] 176.53/60.03 [0 0] [0] 176.53/60.03 >= [1 0] X + [0] 176.53/60.03 [0 0] [0] 176.53/60.03 = [a__f(X)] 176.53/60.03 176.53/60.03 [mark(a())] = [0] 176.53/60.03 [0] 176.53/60.03 >= [0] 176.53/60.03 [0] 176.53/60.03 = [a()] 176.53/60.03 176.53/60.03 [mark(g(X))] = [1 1] X + [2] 176.53/60.03 [0 1] [2] 176.53/60.03 > [1 1] X + [0] 176.53/60.03 [0 1] [2] 176.53/60.03 = [g(mark(X))] 176.53/60.03 176.53/60.03 176.53/60.03 We return to the main proof. 176.53/60.03 176.53/60.03 We are left with following problem, upon which TcT provides the 176.53/60.03 certificate YES(O(1),O(1)). 176.53/60.03 176.53/60.03 Weak Trs: 176.53/60.03 { a__f(X) -> f(X) 176.53/60.03 , a__f(f(a())) -> a__f(g(f(a()))) 176.53/60.03 , mark(f(X)) -> a__f(X) 176.53/60.03 , mark(a()) -> a() 176.53/60.03 , mark(g(X)) -> g(mark(X)) } 176.53/60.03 Obligation: 176.53/60.03 derivational complexity 176.53/60.03 Answer: 176.53/60.03 YES(O(1),O(1)) 176.53/60.03 176.53/60.03 Empty rules are trivially bounded 176.53/60.03 176.53/60.03 Hurray, we answered YES(O(1),O(n^2)) 176.53/60.03 EOF