YES(O(1),O(n^2)) 172.96/60.03 YES(O(1),O(n^2)) 172.96/60.03 172.96/60.03 We are left with following problem, upon which TcT provides the 172.96/60.03 certificate YES(O(1),O(n^2)). 172.96/60.03 172.96/60.03 Strict Trs: 172.96/60.03 { a__f(X) -> f(X) 172.96/60.03 , a__f(f(a())) -> c(f(g(f(a())))) 172.96/60.03 , mark(f(X)) -> a__f(mark(X)) 172.96/60.03 , mark(a()) -> a() 172.96/60.03 , mark(c(X)) -> c(X) 172.96/60.03 , mark(g(X)) -> g(mark(X)) } 172.96/60.03 Obligation: 172.96/60.03 derivational complexity 172.96/60.03 Answer: 172.96/60.03 YES(O(1),O(n^2)) 172.96/60.03 172.96/60.03 The weightgap principle applies (using the following nonconstant 172.96/60.03 growth matrix-interpretation) 172.96/60.03 172.96/60.03 TcT has computed the following triangular matrix interpretation. 172.96/60.03 Note that the diagonal of the component-wise maxima of 172.96/60.03 interpretation-entries contains no more than 1 non-zero entries. 172.96/60.03 172.96/60.03 [a__f](x1) = [1] x1 + [1] 172.96/60.03 172.96/60.03 [f](x1) = [1] x1 + [0] 172.96/60.03 172.96/60.03 [a] = [0] 172.96/60.03 172.96/60.03 [c](x1) = [1] x1 + [0] 172.96/60.03 172.96/60.03 [g](x1) = [1] x1 + [0] 172.96/60.03 172.96/60.03 [mark](x1) = [1] x1 + [0] 172.96/60.03 172.96/60.03 The order satisfies the following ordering constraints: 172.96/60.03 172.96/60.03 [a__f(X)] = [1] X + [1] 172.96/60.03 > [1] X + [0] 172.96/60.03 = [f(X)] 172.96/60.03 172.96/60.03 [a__f(f(a()))] = [1] 172.96/60.03 > [0] 172.96/60.03 = [c(f(g(f(a()))))] 172.96/60.03 172.96/60.03 [mark(f(X))] = [1] X + [0] 172.96/60.03 ? [1] X + [1] 172.96/60.03 = [a__f(mark(X))] 172.96/60.03 172.96/60.03 [mark(a())] = [0] 172.96/60.03 >= [0] 172.96/60.03 = [a()] 172.96/60.03 172.96/60.03 [mark(c(X))] = [1] X + [0] 172.96/60.03 >= [1] X + [0] 172.96/60.03 = [c(X)] 172.96/60.03 172.96/60.03 [mark(g(X))] = [1] X + [0] 172.96/60.03 >= [1] X + [0] 172.96/60.03 = [g(mark(X))] 172.96/60.03 172.96/60.03 172.96/60.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 172.96/60.03 172.96/60.03 We are left with following problem, upon which TcT provides the 172.96/60.03 certificate YES(O(1),O(n^2)). 172.96/60.03 172.96/60.03 Strict Trs: 172.96/60.03 { mark(f(X)) -> a__f(mark(X)) 172.96/60.03 , mark(a()) -> a() 172.96/60.03 , mark(c(X)) -> c(X) 172.96/60.03 , mark(g(X)) -> g(mark(X)) } 172.96/60.03 Weak Trs: 172.96/60.03 { a__f(X) -> f(X) 172.96/60.03 , a__f(f(a())) -> c(f(g(f(a())))) } 172.96/60.03 Obligation: 172.96/60.03 derivational complexity 172.96/60.03 Answer: 172.96/60.03 YES(O(1),O(n^2)) 172.96/60.03 172.96/60.03 We use the processor 'matrix interpretation of dimension 1' to 172.96/60.03 orient following rules strictly. 172.96/60.03 172.96/60.03 Trs: 172.96/60.03 { mark(a()) -> a() 172.96/60.03 , mark(c(X)) -> c(X) } 172.96/60.03 172.96/60.03 The induced complexity on above rules (modulo remaining rules) is 172.96/60.03 YES(?,O(n^1)) . These rules are moved into the corresponding weak 172.96/60.03 component(s). 172.96/60.03 172.96/60.03 Sub-proof: 172.96/60.03 ---------- 172.96/60.03 TcT has computed the following triangular matrix interpretation. 172.96/60.03 172.96/60.03 [a__f](x1) = [1] x1 + [0] 172.96/60.03 172.96/60.03 [f](x1) = [1] x1 + [0] 172.96/60.03 172.96/60.03 [a] = [0] 172.96/60.03 172.96/60.03 [c](x1) = [1] x1 + [0] 172.96/60.03 172.96/60.03 [g](x1) = [1] x1 + [0] 172.96/60.03 172.96/60.03 [mark](x1) = [1] x1 + [1] 172.96/60.03 172.96/60.03 The order satisfies the following ordering constraints: 172.96/60.03 172.96/60.03 [a__f(X)] = [1] X + [0] 172.96/60.03 >= [1] X + [0] 172.96/60.03 = [f(X)] 172.96/60.03 172.96/60.03 [a__f(f(a()))] = [0] 172.96/60.03 >= [0] 172.96/60.03 = [c(f(g(f(a()))))] 172.96/60.03 172.96/60.03 [mark(f(X))] = [1] X + [1] 172.96/60.03 >= [1] X + [1] 172.96/60.03 = [a__f(mark(X))] 172.96/60.03 172.96/60.03 [mark(a())] = [1] 172.96/60.03 > [0] 172.96/60.03 = [a()] 172.96/60.03 172.96/60.03 [mark(c(X))] = [1] X + [1] 172.96/60.03 > [1] X + [0] 172.96/60.03 = [c(X)] 172.96/60.03 172.96/60.03 [mark(g(X))] = [1] X + [1] 172.96/60.03 >= [1] X + [1] 172.96/60.03 = [g(mark(X))] 172.96/60.03 172.96/60.03 172.96/60.03 We return to the main proof. 172.96/60.03 172.96/60.03 We are left with following problem, upon which TcT provides the 172.96/60.03 certificate YES(O(1),O(n^2)). 172.96/60.03 172.96/60.03 Strict Trs: 172.96/60.03 { mark(f(X)) -> a__f(mark(X)) 172.96/60.03 , mark(g(X)) -> g(mark(X)) } 172.96/60.03 Weak Trs: 172.96/60.03 { a__f(X) -> f(X) 172.96/60.03 , a__f(f(a())) -> c(f(g(f(a())))) 172.96/60.03 , mark(a()) -> a() 172.96/60.03 , mark(c(X)) -> c(X) } 172.96/60.03 Obligation: 172.96/60.03 derivational complexity 172.96/60.03 Answer: 172.96/60.03 YES(O(1),O(n^2)) 172.96/60.03 172.96/60.03 We use the processor 'matrix interpretation of dimension 2' to 172.96/60.03 orient following rules strictly. 172.96/60.03 172.96/60.03 Trs: { mark(g(X)) -> g(mark(X)) } 172.96/60.03 172.96/60.03 The induced complexity on above rules (modulo remaining rules) is 172.96/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak 172.96/60.03 component(s). 172.96/60.03 172.96/60.03 Sub-proof: 172.96/60.03 ---------- 172.96/60.03 TcT has computed the following triangular matrix interpretation. 172.96/60.03 172.96/60.03 [a__f](x1) = [1 0] x1 + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 172.96/60.03 [f](x1) = [1 0] x1 + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 172.96/60.03 [a] = [0] 172.96/60.03 [0] 172.96/60.03 172.96/60.03 [c](x1) = [1 0] x1 + [0] 172.96/60.03 [0 0] [0] 172.96/60.03 172.96/60.03 [g](x1) = [1 0] x1 + [0] 172.96/60.03 [0 1] [2] 172.96/60.03 172.96/60.03 [mark](x1) = [1 1] x1 + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 172.96/60.03 The order satisfies the following ordering constraints: 172.96/60.03 172.96/60.03 [a__f(X)] = [1 0] X + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 >= [1 0] X + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 = [f(X)] 172.96/60.03 172.96/60.03 [a__f(f(a()))] = [0] 172.96/60.03 [0] 172.96/60.03 >= [0] 172.96/60.03 [0] 172.96/60.03 = [c(f(g(f(a()))))] 172.96/60.03 172.96/60.03 [mark(f(X))] = [1 1] X + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 >= [1 1] X + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 = [a__f(mark(X))] 172.96/60.03 172.96/60.03 [mark(a())] = [0] 172.96/60.03 [0] 172.96/60.03 >= [0] 172.96/60.03 [0] 172.96/60.03 = [a()] 172.96/60.03 172.96/60.03 [mark(c(X))] = [1 0] X + [0] 172.96/60.03 [0 0] [0] 172.96/60.03 >= [1 0] X + [0] 172.96/60.03 [0 0] [0] 172.96/60.03 = [c(X)] 172.96/60.03 172.96/60.03 [mark(g(X))] = [1 1] X + [2] 172.96/60.03 [0 1] [2] 172.96/60.03 > [1 1] X + [0] 172.96/60.03 [0 1] [2] 172.96/60.03 = [g(mark(X))] 172.96/60.03 172.96/60.03 172.96/60.03 We return to the main proof. 172.96/60.03 172.96/60.03 We are left with following problem, upon which TcT provides the 172.96/60.03 certificate YES(O(1),O(n^2)). 172.96/60.03 172.96/60.03 Strict Trs: { mark(f(X)) -> a__f(mark(X)) } 172.96/60.03 Weak Trs: 172.96/60.03 { a__f(X) -> f(X) 172.96/60.03 , a__f(f(a())) -> c(f(g(f(a())))) 172.96/60.03 , mark(a()) -> a() 172.96/60.03 , mark(c(X)) -> c(X) 172.96/60.03 , mark(g(X)) -> g(mark(X)) } 172.96/60.03 Obligation: 172.96/60.03 derivational complexity 172.96/60.03 Answer: 172.96/60.03 YES(O(1),O(n^2)) 172.96/60.03 172.96/60.03 We use the processor 'matrix interpretation of dimension 2' to 172.96/60.03 orient following rules strictly. 172.96/60.03 172.96/60.03 Trs: { mark(f(X)) -> a__f(mark(X)) } 172.96/60.03 172.96/60.03 The induced complexity on above rules (modulo remaining rules) is 172.96/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak 172.96/60.03 component(s). 172.96/60.03 172.96/60.03 Sub-proof: 172.96/60.03 ---------- 172.96/60.03 TcT has computed the following triangular matrix interpretation. 172.96/60.03 172.96/60.03 [a__f](x1) = [1 0] x1 + [2] 172.96/60.03 [0 1] [2] 172.96/60.03 172.96/60.03 [f](x1) = [1 0] x1 + [2] 172.96/60.03 [0 1] [2] 172.96/60.03 172.96/60.03 [a] = [0] 172.96/60.03 [0] 172.96/60.03 172.96/60.03 [c](x1) = [1 0] x1 + [0] 172.96/60.03 [0 0] [0] 172.96/60.03 172.96/60.03 [g](x1) = [1 0] x1 + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 172.96/60.03 [mark](x1) = [1 1] x1 + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 172.96/60.03 The order satisfies the following ordering constraints: 172.96/60.03 172.96/60.03 [a__f(X)] = [1 0] X + [2] 172.96/60.03 [0 1] [2] 172.96/60.03 >= [1 0] X + [2] 172.96/60.03 [0 1] [2] 172.96/60.03 = [f(X)] 172.96/60.03 172.96/60.03 [a__f(f(a()))] = [4] 172.96/60.03 [4] 172.96/60.03 >= [4] 172.96/60.03 [0] 172.96/60.03 = [c(f(g(f(a()))))] 172.96/60.03 172.96/60.03 [mark(f(X))] = [1 1] X + [4] 172.96/60.03 [0 1] [2] 172.96/60.03 > [1 1] X + [2] 172.96/60.03 [0 1] [2] 172.96/60.03 = [a__f(mark(X))] 172.96/60.03 172.96/60.03 [mark(a())] = [0] 172.96/60.03 [0] 172.96/60.03 >= [0] 172.96/60.03 [0] 172.96/60.03 = [a()] 172.96/60.03 172.96/60.03 [mark(c(X))] = [1 0] X + [0] 172.96/60.03 [0 0] [0] 172.96/60.03 >= [1 0] X + [0] 172.96/60.03 [0 0] [0] 172.96/60.03 = [c(X)] 172.96/60.03 172.96/60.03 [mark(g(X))] = [1 1] X + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 >= [1 1] X + [0] 172.96/60.03 [0 1] [0] 172.96/60.03 = [g(mark(X))] 172.96/60.03 172.96/60.03 172.96/60.03 We return to the main proof. 172.96/60.03 172.96/60.03 We are left with following problem, upon which TcT provides the 172.96/60.03 certificate YES(O(1),O(1)). 172.96/60.03 172.96/60.03 Weak Trs: 172.96/60.03 { a__f(X) -> f(X) 172.96/60.03 , a__f(f(a())) -> c(f(g(f(a())))) 172.96/60.03 , mark(f(X)) -> a__f(mark(X)) 172.96/60.03 , mark(a()) -> a() 172.96/60.03 , mark(c(X)) -> c(X) 172.96/60.03 , mark(g(X)) -> g(mark(X)) } 172.96/60.03 Obligation: 172.96/60.03 derivational complexity 172.96/60.03 Answer: 172.96/60.03 YES(O(1),O(1)) 172.96/60.03 172.96/60.03 Empty rules are trivially bounded 172.96/60.03 172.96/60.03 Hurray, we answered YES(O(1),O(n^2)) 172.96/60.05 EOF