YES(O(1),O(n^2)) 166.81/60.07 YES(O(1),O(n^2)) 166.81/60.07 166.81/60.07 We are left with following problem, upon which TcT provides the 166.81/60.07 certificate YES(O(1),O(n^2)). 166.81/60.07 166.81/60.07 Strict Trs: 166.81/60.07 { a__f(X) -> g(h(f(X))) 166.81/60.07 , a__f(X) -> f(X) 166.81/60.07 , mark(g(X)) -> g(X) 166.81/60.07 , mark(h(X)) -> h(mark(X)) 166.81/60.07 , mark(f(X)) -> a__f(mark(X)) } 166.81/60.07 Obligation: 166.81/60.07 derivational complexity 166.81/60.07 Answer: 166.81/60.07 YES(O(1),O(n^2)) 166.81/60.07 166.81/60.07 We use the processor 'matrix interpretation of dimension 1' to 166.81/60.07 orient following rules strictly. 166.81/60.07 166.81/60.07 Trs: { mark(g(X)) -> g(X) } 166.81/60.07 166.81/60.07 The induced complexity on above rules (modulo remaining rules) is 166.81/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 166.81/60.07 component(s). 166.81/60.07 166.81/60.07 Sub-proof: 166.81/60.07 ---------- 166.81/60.07 TcT has computed the following triangular matrix interpretation. 166.81/60.07 166.81/60.07 [a__f](x1) = [1] x1 + [0] 166.81/60.07 166.81/60.07 [g](x1) = [1] x1 + [0] 166.81/60.07 166.81/60.07 [h](x1) = [1] x1 + [0] 166.81/60.07 166.81/60.07 [f](x1) = [1] x1 + [0] 166.81/60.07 166.81/60.07 [mark](x1) = [1] x1 + [1] 166.81/60.07 166.81/60.07 The order satisfies the following ordering constraints: 166.81/60.07 166.81/60.07 [a__f(X)] = [1] X + [0] 166.81/60.07 >= [1] X + [0] 166.81/60.07 = [g(h(f(X)))] 166.81/60.07 166.81/60.07 [a__f(X)] = [1] X + [0] 166.81/60.07 >= [1] X + [0] 166.81/60.07 = [f(X)] 166.81/60.07 166.81/60.07 [mark(g(X))] = [1] X + [1] 166.81/60.07 > [1] X + [0] 166.81/60.07 = [g(X)] 166.81/60.07 166.81/60.07 [mark(h(X))] = [1] X + [1] 166.81/60.07 >= [1] X + [1] 166.81/60.07 = [h(mark(X))] 166.81/60.07 166.81/60.07 [mark(f(X))] = [1] X + [1] 166.81/60.07 >= [1] X + [1] 166.81/60.07 = [a__f(mark(X))] 166.81/60.07 166.81/60.07 166.81/60.07 We return to the main proof. 166.81/60.07 166.81/60.07 We are left with following problem, upon which TcT provides the 166.81/60.07 certificate YES(O(1),O(n^2)). 166.81/60.07 166.81/60.07 Strict Trs: 166.81/60.07 { a__f(X) -> g(h(f(X))) 166.81/60.07 , a__f(X) -> f(X) 166.81/60.07 , mark(h(X)) -> h(mark(X)) 166.81/60.07 , mark(f(X)) -> a__f(mark(X)) } 166.81/60.07 Weak Trs: { mark(g(X)) -> g(X) } 166.81/60.07 Obligation: 166.81/60.07 derivational complexity 166.81/60.07 Answer: 166.81/60.07 YES(O(1),O(n^2)) 166.81/60.07 166.81/60.07 The weightgap principle applies (using the following nonconstant 166.81/60.07 growth matrix-interpretation) 166.81/60.07 166.81/60.07 TcT has computed the following triangular matrix interpretation. 166.81/60.07 Note that the diagonal of the component-wise maxima of 166.81/60.07 interpretation-entries contains no more than 1 non-zero entries. 166.81/60.07 166.81/60.07 [a__f](x1) = [1] x1 + [1] 166.81/60.07 166.81/60.07 [g](x1) = [1] x1 + [0] 166.81/60.07 166.81/60.07 [h](x1) = [1] x1 + [0] 166.81/60.07 166.81/60.07 [f](x1) = [1] x1 + [0] 166.81/60.07 166.81/60.07 [mark](x1) = [1] x1 + [0] 166.81/60.07 166.81/60.07 The order satisfies the following ordering constraints: 166.81/60.07 166.81/60.07 [a__f(X)] = [1] X + [1] 166.81/60.07 > [1] X + [0] 166.81/60.07 = [g(h(f(X)))] 166.81/60.07 166.81/60.07 [a__f(X)] = [1] X + [1] 166.81/60.07 > [1] X + [0] 166.81/60.07 = [f(X)] 166.81/60.07 166.81/60.07 [mark(g(X))] = [1] X + [0] 166.81/60.07 >= [1] X + [0] 166.81/60.07 = [g(X)] 166.81/60.07 166.81/60.07 [mark(h(X))] = [1] X + [0] 166.81/60.07 >= [1] X + [0] 166.81/60.07 = [h(mark(X))] 166.81/60.07 166.81/60.07 [mark(f(X))] = [1] X + [0] 166.81/60.07 ? [1] X + [1] 166.81/60.07 = [a__f(mark(X))] 166.81/60.07 166.81/60.07 166.81/60.07 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 166.81/60.07 166.81/60.07 We are left with following problem, upon which TcT provides the 166.81/60.07 certificate YES(O(1),O(n^2)). 166.81/60.07 166.81/60.07 Strict Trs: 166.81/60.07 { mark(h(X)) -> h(mark(X)) 166.81/60.07 , mark(f(X)) -> a__f(mark(X)) } 166.81/60.07 Weak Trs: 166.81/60.07 { a__f(X) -> g(h(f(X))) 166.81/60.07 , a__f(X) -> f(X) 166.81/60.07 , mark(g(X)) -> g(X) } 166.81/60.07 Obligation: 166.81/60.07 derivational complexity 166.81/60.07 Answer: 166.81/60.07 YES(O(1),O(n^2)) 166.81/60.07 166.81/60.07 We use the processor 'matrix interpretation of dimension 2' to 166.81/60.07 orient following rules strictly. 166.81/60.07 166.81/60.07 Trs: { mark(f(X)) -> a__f(mark(X)) } 166.81/60.07 166.81/60.07 The induced complexity on above rules (modulo remaining rules) is 166.81/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 166.81/60.07 component(s). 166.81/60.07 166.81/60.07 Sub-proof: 166.81/60.07 ---------- 166.81/60.07 TcT has computed the following triangular matrix interpretation. 166.81/60.07 166.81/60.07 [a__f](x1) = [1 0] x1 + [0] 166.81/60.07 [0 1] [2] 166.81/60.07 166.81/60.07 [g](x1) = [1 0] x1 + [0] 166.81/60.07 [0 0] [0] 166.81/60.07 166.81/60.07 [h](x1) = [1 0] x1 + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 166.81/60.07 [f](x1) = [1 0] x1 + [0] 166.81/60.07 [0 1] [2] 166.81/60.07 166.81/60.07 [mark](x1) = [1 1] x1 + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 166.81/60.07 The order satisfies the following ordering constraints: 166.81/60.07 166.81/60.07 [a__f(X)] = [1 0] X + [0] 166.81/60.07 [0 1] [2] 166.81/60.07 >= [1 0] X + [0] 166.81/60.07 [0 0] [0] 166.81/60.07 = [g(h(f(X)))] 166.81/60.07 166.81/60.07 [a__f(X)] = [1 0] X + [0] 166.81/60.07 [0 1] [2] 166.81/60.07 >= [1 0] X + [0] 166.81/60.07 [0 1] [2] 166.81/60.07 = [f(X)] 166.81/60.07 166.81/60.07 [mark(g(X))] = [1 0] X + [0] 166.81/60.07 [0 0] [0] 166.81/60.07 >= [1 0] X + [0] 166.81/60.07 [0 0] [0] 166.81/60.07 = [g(X)] 166.81/60.07 166.81/60.07 [mark(h(X))] = [1 1] X + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 >= [1 1] X + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 = [h(mark(X))] 166.81/60.07 166.81/60.07 [mark(f(X))] = [1 1] X + [2] 166.81/60.07 [0 1] [2] 166.81/60.07 > [1 1] X + [0] 166.81/60.07 [0 1] [2] 166.81/60.07 = [a__f(mark(X))] 166.81/60.07 166.81/60.07 166.81/60.07 We return to the main proof. 166.81/60.07 166.81/60.07 We are left with following problem, upon which TcT provides the 166.81/60.07 certificate YES(O(1),O(n^2)). 166.81/60.07 166.81/60.07 Strict Trs: { mark(h(X)) -> h(mark(X)) } 166.81/60.07 Weak Trs: 166.81/60.07 { a__f(X) -> g(h(f(X))) 166.81/60.07 , a__f(X) -> f(X) 166.81/60.07 , mark(g(X)) -> g(X) 166.81/60.07 , mark(f(X)) -> a__f(mark(X)) } 166.81/60.07 Obligation: 166.81/60.07 derivational complexity 166.81/60.07 Answer: 166.81/60.07 YES(O(1),O(n^2)) 166.81/60.07 166.81/60.07 We use the processor 'matrix interpretation of dimension 2' to 166.81/60.07 orient following rules strictly. 166.81/60.07 166.81/60.07 Trs: { mark(h(X)) -> h(mark(X)) } 166.81/60.07 166.81/60.07 The induced complexity on above rules (modulo remaining rules) is 166.81/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 166.81/60.07 component(s). 166.81/60.07 166.81/60.07 Sub-proof: 166.81/60.07 ---------- 166.81/60.07 TcT has computed the following triangular matrix interpretation. 166.81/60.07 166.81/60.07 [a__f](x1) = [1 0] x1 + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 166.81/60.07 [g](x1) = [1 0] x1 + [0] 166.81/60.07 [0 0] [0] 166.81/60.07 166.81/60.07 [h](x1) = [1 0] x1 + [0] 166.81/60.07 [0 1] [1] 166.81/60.07 166.81/60.07 [f](x1) = [1 0] x1 + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 166.81/60.07 [mark](x1) = [1 1] x1 + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 166.81/60.07 The order satisfies the following ordering constraints: 166.81/60.07 166.81/60.07 [a__f(X)] = [1 0] X + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 >= [1 0] X + [0] 166.81/60.07 [0 0] [0] 166.81/60.07 = [g(h(f(X)))] 166.81/60.07 166.81/60.07 [a__f(X)] = [1 0] X + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 >= [1 0] X + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 = [f(X)] 166.81/60.07 166.81/60.07 [mark(g(X))] = [1 0] X + [0] 166.81/60.07 [0 0] [0] 166.81/60.07 >= [1 0] X + [0] 166.81/60.07 [0 0] [0] 166.81/60.07 = [g(X)] 166.81/60.07 166.81/60.07 [mark(h(X))] = [1 1] X + [1] 166.81/60.07 [0 1] [1] 166.81/60.07 > [1 1] X + [0] 166.81/60.07 [0 1] [1] 166.81/60.07 = [h(mark(X))] 166.81/60.07 166.81/60.07 [mark(f(X))] = [1 1] X + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 >= [1 1] X + [0] 166.81/60.07 [0 1] [0] 166.81/60.07 = [a__f(mark(X))] 166.81/60.07 166.81/60.07 166.81/60.07 We return to the main proof. 166.81/60.07 166.81/60.07 We are left with following problem, upon which TcT provides the 166.81/60.07 certificate YES(O(1),O(1)). 166.81/60.07 166.81/60.07 Weak Trs: 166.81/60.07 { a__f(X) -> g(h(f(X))) 166.81/60.07 , a__f(X) -> f(X) 166.81/60.07 , mark(g(X)) -> g(X) 166.81/60.07 , mark(h(X)) -> h(mark(X)) 166.81/60.07 , mark(f(X)) -> a__f(mark(X)) } 166.81/60.07 Obligation: 166.81/60.07 derivational complexity 166.81/60.07 Answer: 166.81/60.07 YES(O(1),O(1)) 166.81/60.07 166.81/60.07 Empty rules are trivially bounded 166.81/60.07 166.81/60.07 Hurray, we answered YES(O(1),O(n^2)) 166.81/60.08 EOF