YES(O(1),O(n^2)) 161.67/60.09 YES(O(1),O(n^2)) 161.67/60.09 161.67/60.09 We are left with following problem, upon which TcT provides the 161.67/60.09 certificate YES(O(1),O(n^2)). 161.67/60.09 161.67/60.09 Strict Trs: 161.67/60.09 { a__f(X) -> f(X) 161.67/60.09 , a__f(f(X)) -> a__c(f(g(f(X)))) 161.67/60.09 , a__c(X) -> d(X) 161.67/60.09 , a__c(X) -> c(X) 161.67/60.09 , a__h(X) -> a__c(d(X)) 161.67/60.09 , a__h(X) -> h(X) 161.67/60.09 , mark(f(X)) -> a__f(mark(X)) 161.67/60.09 , mark(g(X)) -> g(X) 161.67/60.09 , mark(d(X)) -> d(X) 161.67/60.09 , mark(c(X)) -> a__c(X) 161.67/60.09 , mark(h(X)) -> a__h(mark(X)) } 161.67/60.09 Obligation: 161.67/60.09 derivational complexity 161.67/60.09 Answer: 161.67/60.09 YES(O(1),O(n^2)) 161.67/60.09 161.67/60.09 The weightgap principle applies (using the following nonconstant 161.67/60.09 growth matrix-interpretation) 161.67/60.09 161.67/60.09 TcT has computed the following triangular matrix interpretation. 161.67/60.09 Note that the diagonal of the component-wise maxima of 161.67/60.09 interpretation-entries contains no more than 1 non-zero entries. 161.67/60.09 161.67/60.09 [a__f](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 [f](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [a__c](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [g](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [d](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [a__h](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [mark](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [c](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 [h](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 The order satisfies the following ordering constraints: 161.67/60.09 161.67/60.09 [a__f(X)] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [f(X)] 161.67/60.09 161.67/60.09 [a__f(f(X))] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [a__c(f(g(f(X))))] 161.67/60.09 161.67/60.09 [a__c(X)] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [a__c(X)] = [1] X + [0] 161.67/60.09 ? [1] X + [1] 161.67/60.09 = [c(X)] 161.67/60.09 161.67/60.09 [a__h(X)] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [a__c(d(X))] 161.67/60.09 161.67/60.09 [a__h(X)] = [1] X + [0] 161.67/60.09 ? [1] X + [1] 161.67/60.09 = [h(X)] 161.67/60.09 161.67/60.09 [mark(f(X))] = [1] X + [0] 161.67/60.09 ? [1] X + [1] 161.67/60.09 = [a__f(mark(X))] 161.67/60.09 161.67/60.09 [mark(g(X))] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [g(X)] 161.67/60.09 161.67/60.09 [mark(d(X))] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [mark(c(X))] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [a__c(X)] 161.67/60.09 161.67/60.09 [mark(h(X))] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [a__h(mark(X))] 161.67/60.09 161.67/60.09 161.67/60.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 161.67/60.09 161.67/60.09 We are left with following problem, upon which TcT provides the 161.67/60.09 certificate YES(O(1),O(n^2)). 161.67/60.09 161.67/60.09 Strict Trs: 161.67/60.09 { a__c(X) -> d(X) 161.67/60.09 , a__c(X) -> c(X) 161.67/60.09 , a__h(X) -> a__c(d(X)) 161.67/60.09 , a__h(X) -> h(X) 161.67/60.09 , mark(f(X)) -> a__f(mark(X)) 161.67/60.09 , mark(g(X)) -> g(X) 161.67/60.09 , mark(d(X)) -> d(X) } 161.67/60.09 Weak Trs: 161.67/60.09 { a__f(X) -> f(X) 161.67/60.09 , a__f(f(X)) -> a__c(f(g(f(X)))) 161.67/60.09 , mark(c(X)) -> a__c(X) 161.67/60.09 , mark(h(X)) -> a__h(mark(X)) } 161.67/60.09 Obligation: 161.67/60.09 derivational complexity 161.67/60.09 Answer: 161.67/60.09 YES(O(1),O(n^2)) 161.67/60.09 161.67/60.09 We use the processor 'matrix interpretation of dimension 1' to 161.67/60.09 orient following rules strictly. 161.67/60.09 161.67/60.09 Trs: { a__h(X) -> a__c(d(X)) } 161.67/60.09 161.67/60.09 The induced complexity on above rules (modulo remaining rules) is 161.67/60.09 YES(?,O(n^1)) . These rules are moved into the corresponding weak 161.67/60.09 component(s). 161.67/60.09 161.67/60.09 Sub-proof: 161.67/60.09 ---------- 161.67/60.09 TcT has computed the following triangular matrix interpretation. 161.67/60.09 161.67/60.09 [a__f](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [f](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [a__c](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [g](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [d](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [a__h](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 [mark](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [c](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [h](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 The order satisfies the following ordering constraints: 161.67/60.09 161.67/60.09 [a__f(X)] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [f(X)] 161.67/60.09 161.67/60.09 [a__f(f(X))] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [a__c(f(g(f(X))))] 161.67/60.09 161.67/60.09 [a__c(X)] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [a__c(X)] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [c(X)] 161.67/60.09 161.67/60.09 [a__h(X)] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [a__c(d(X))] 161.67/60.09 161.67/60.09 [a__h(X)] = [1] X + [1] 161.67/60.09 >= [1] X + [1] 161.67/60.09 = [h(X)] 161.67/60.09 161.67/60.09 [mark(f(X))] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [a__f(mark(X))] 161.67/60.09 161.67/60.09 [mark(g(X))] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [g(X)] 161.67/60.09 161.67/60.09 [mark(d(X))] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [mark(c(X))] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [a__c(X)] 161.67/60.09 161.67/60.09 [mark(h(X))] = [1] X + [1] 161.67/60.09 >= [1] X + [1] 161.67/60.09 = [a__h(mark(X))] 161.67/60.09 161.67/60.09 161.67/60.09 We return to the main proof. 161.67/60.09 161.67/60.09 We are left with following problem, upon which TcT provides the 161.67/60.09 certificate YES(O(1),O(n^2)). 161.67/60.09 161.67/60.09 Strict Trs: 161.67/60.09 { a__c(X) -> d(X) 161.67/60.09 , a__c(X) -> c(X) 161.67/60.09 , a__h(X) -> h(X) 161.67/60.09 , mark(f(X)) -> a__f(mark(X)) 161.67/60.09 , mark(g(X)) -> g(X) 161.67/60.09 , mark(d(X)) -> d(X) } 161.67/60.09 Weak Trs: 161.67/60.09 { a__f(X) -> f(X) 161.67/60.09 , a__f(f(X)) -> a__c(f(g(f(X)))) 161.67/60.09 , a__h(X) -> a__c(d(X)) 161.67/60.09 , mark(c(X)) -> a__c(X) 161.67/60.09 , mark(h(X)) -> a__h(mark(X)) } 161.67/60.09 Obligation: 161.67/60.09 derivational complexity 161.67/60.09 Answer: 161.67/60.09 YES(O(1),O(n^2)) 161.67/60.09 161.67/60.09 We use the processor 'matrix interpretation of dimension 1' to 161.67/60.09 orient following rules strictly. 161.67/60.09 161.67/60.09 Trs: 161.67/60.09 { mark(g(X)) -> g(X) 161.67/60.09 , mark(d(X)) -> d(X) } 161.67/60.09 161.67/60.09 The induced complexity on above rules (modulo remaining rules) is 161.67/60.09 YES(?,O(n^1)) . These rules are moved into the corresponding weak 161.67/60.09 component(s). 161.67/60.09 161.67/60.09 Sub-proof: 161.67/60.09 ---------- 161.67/60.09 TcT has computed the following triangular matrix interpretation. 161.67/60.09 161.67/60.09 [a__f](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [f](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [a__c](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [g](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [d](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [a__h](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [mark](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 [c](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [h](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 The order satisfies the following ordering constraints: 161.67/60.09 161.67/60.09 [a__f(X)] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [f(X)] 161.67/60.09 161.67/60.09 [a__f(f(X))] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [a__c(f(g(f(X))))] 161.67/60.09 161.67/60.09 [a__c(X)] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [a__c(X)] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [c(X)] 161.67/60.09 161.67/60.09 [a__h(X)] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [a__c(d(X))] 161.67/60.09 161.67/60.09 [a__h(X)] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [h(X)] 161.67/60.09 161.67/60.09 [mark(f(X))] = [1] X + [1] 161.67/60.09 >= [1] X + [1] 161.67/60.09 = [a__f(mark(X))] 161.67/60.09 161.67/60.09 [mark(g(X))] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [g(X)] 161.67/60.09 161.67/60.09 [mark(d(X))] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [mark(c(X))] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [a__c(X)] 161.67/60.09 161.67/60.09 [mark(h(X))] = [1] X + [1] 161.67/60.09 >= [1] X + [1] 161.67/60.09 = [a__h(mark(X))] 161.67/60.09 161.67/60.09 161.67/60.09 We return to the main proof. 161.67/60.09 161.67/60.09 We are left with following problem, upon which TcT provides the 161.67/60.09 certificate YES(O(1),O(n^2)). 161.67/60.09 161.67/60.09 Strict Trs: 161.67/60.09 { a__c(X) -> d(X) 161.67/60.09 , a__c(X) -> c(X) 161.67/60.09 , a__h(X) -> h(X) 161.67/60.09 , mark(f(X)) -> a__f(mark(X)) } 161.67/60.09 Weak Trs: 161.67/60.09 { a__f(X) -> f(X) 161.67/60.09 , a__f(f(X)) -> a__c(f(g(f(X)))) 161.67/60.09 , a__h(X) -> a__c(d(X)) 161.67/60.09 , mark(g(X)) -> g(X) 161.67/60.09 , mark(d(X)) -> d(X) 161.67/60.09 , mark(c(X)) -> a__c(X) 161.67/60.09 , mark(h(X)) -> a__h(mark(X)) } 161.67/60.09 Obligation: 161.67/60.09 derivational complexity 161.67/60.09 Answer: 161.67/60.09 YES(O(1),O(n^2)) 161.67/60.09 161.67/60.09 The weightgap principle applies (using the following nonconstant 161.67/60.09 growth matrix-interpretation) 161.67/60.09 161.67/60.09 TcT has computed the following triangular matrix interpretation. 161.67/60.09 Note that the diagonal of the component-wise maxima of 161.67/60.09 interpretation-entries contains no more than 1 non-zero entries. 161.67/60.09 161.67/60.09 [a__f](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 [f](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [a__c](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 [g](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [d](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [a__h](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 [mark](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [c](x1) = [1] x1 + [2] 161.67/60.09 161.67/60.09 [h](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 The order satisfies the following ordering constraints: 161.67/60.09 161.67/60.09 [a__f(X)] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [f(X)] 161.67/60.09 161.67/60.09 [a__f(f(X))] = [1] X + [1] 161.67/60.09 >= [1] X + [1] 161.67/60.09 = [a__c(f(g(f(X))))] 161.67/60.09 161.67/60.09 [a__c(X)] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [a__c(X)] = [1] X + [1] 161.67/60.09 ? [1] X + [2] 161.67/60.09 = [c(X)] 161.67/60.09 161.67/60.09 [a__h(X)] = [1] X + [1] 161.67/60.09 >= [1] X + [1] 161.67/60.09 = [a__c(d(X))] 161.67/60.09 161.67/60.09 [a__h(X)] = [1] X + [1] 161.67/60.09 >= [1] X + [1] 161.67/60.09 = [h(X)] 161.67/60.09 161.67/60.09 [mark(f(X))] = [1] X + [0] 161.67/60.09 ? [1] X + [1] 161.67/60.09 = [a__f(mark(X))] 161.67/60.09 161.67/60.09 [mark(g(X))] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [g(X)] 161.67/60.09 161.67/60.09 [mark(d(X))] = [1] X + [0] 161.67/60.09 >= [1] X + [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [mark(c(X))] = [1] X + [2] 161.67/60.09 > [1] X + [1] 161.67/60.09 = [a__c(X)] 161.67/60.09 161.67/60.09 [mark(h(X))] = [1] X + [1] 161.67/60.09 >= [1] X + [1] 161.67/60.09 = [a__h(mark(X))] 161.67/60.09 161.67/60.09 161.67/60.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 161.67/60.09 161.67/60.09 We are left with following problem, upon which TcT provides the 161.67/60.09 certificate YES(O(1),O(n^2)). 161.67/60.09 161.67/60.09 Strict Trs: 161.67/60.09 { a__c(X) -> c(X) 161.67/60.09 , a__h(X) -> h(X) 161.67/60.09 , mark(f(X)) -> a__f(mark(X)) } 161.67/60.09 Weak Trs: 161.67/60.09 { a__f(X) -> f(X) 161.67/60.09 , a__f(f(X)) -> a__c(f(g(f(X)))) 161.67/60.09 , a__c(X) -> d(X) 161.67/60.09 , a__h(X) -> a__c(d(X)) 161.67/60.09 , mark(g(X)) -> g(X) 161.67/60.09 , mark(d(X)) -> d(X) 161.67/60.09 , mark(c(X)) -> a__c(X) 161.67/60.09 , mark(h(X)) -> a__h(mark(X)) } 161.67/60.09 Obligation: 161.67/60.09 derivational complexity 161.67/60.09 Answer: 161.67/60.09 YES(O(1),O(n^2)) 161.67/60.09 161.67/60.09 The weightgap principle applies (using the following nonconstant 161.67/60.09 growth matrix-interpretation) 161.67/60.09 161.67/60.09 TcT has computed the following triangular matrix interpretation. 161.67/60.09 Note that the diagonal of the component-wise maxima of 161.67/60.09 interpretation-entries contains no more than 1 non-zero entries. 161.67/60.09 161.67/60.09 [a__f](x1) = [1] x1 + [2] 161.67/60.09 161.67/60.09 [f](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [a__c](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 [g](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [d](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [a__h](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 [mark](x1) = [1] x1 + [1] 161.67/60.09 161.67/60.09 [c](x1) = [1] x1 + [0] 161.67/60.09 161.67/60.09 [h](x1) = [1] x1 + [2] 161.67/60.09 161.67/60.09 The order satisfies the following ordering constraints: 161.67/60.09 161.67/60.09 [a__f(X)] = [1] X + [2] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [f(X)] 161.67/60.09 161.67/60.09 [a__f(f(X))] = [1] X + [2] 161.67/60.09 > [1] X + [1] 161.67/60.09 = [a__c(f(g(f(X))))] 161.67/60.09 161.67/60.09 [a__c(X)] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [a__c(X)] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [c(X)] 161.67/60.09 161.67/60.09 [a__h(X)] = [1] X + [1] 161.67/60.09 >= [1] X + [1] 161.67/60.09 = [a__c(d(X))] 161.67/60.09 161.67/60.09 [a__h(X)] = [1] X + [1] 161.67/60.09 ? [1] X + [2] 161.67/60.09 = [h(X)] 161.67/60.09 161.67/60.09 [mark(f(X))] = [1] X + [1] 161.67/60.09 ? [1] X + [3] 161.67/60.09 = [a__f(mark(X))] 161.67/60.09 161.67/60.09 [mark(g(X))] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [g(X)] 161.67/60.09 161.67/60.09 [mark(d(X))] = [1] X + [1] 161.67/60.09 > [1] X + [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [mark(c(X))] = [1] X + [1] 161.67/60.09 >= [1] X + [1] 161.67/60.09 = [a__c(X)] 161.67/60.09 161.67/60.09 [mark(h(X))] = [1] X + [3] 161.67/60.09 > [1] X + [2] 161.67/60.09 = [a__h(mark(X))] 161.67/60.09 161.67/60.09 161.67/60.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 161.67/60.09 161.67/60.09 We are left with following problem, upon which TcT provides the 161.67/60.09 certificate YES(O(1),O(n^2)). 161.67/60.09 161.67/60.09 Strict Trs: 161.67/60.09 { a__h(X) -> h(X) 161.67/60.09 , mark(f(X)) -> a__f(mark(X)) } 161.67/60.09 Weak Trs: 161.67/60.09 { a__f(X) -> f(X) 161.67/60.09 , a__f(f(X)) -> a__c(f(g(f(X)))) 161.67/60.09 , a__c(X) -> d(X) 161.67/60.09 , a__c(X) -> c(X) 161.67/60.09 , a__h(X) -> a__c(d(X)) 161.67/60.09 , mark(g(X)) -> g(X) 161.67/60.09 , mark(d(X)) -> d(X) 161.67/60.09 , mark(c(X)) -> a__c(X) 161.67/60.09 , mark(h(X)) -> a__h(mark(X)) } 161.67/60.09 Obligation: 161.67/60.09 derivational complexity 161.67/60.09 Answer: 161.67/60.09 YES(O(1),O(n^2)) 161.67/60.09 161.67/60.09 We use the processor 'matrix interpretation of dimension 2' to 161.67/60.09 orient following rules strictly. 161.67/60.09 161.67/60.09 Trs: { a__h(X) -> h(X) } 161.67/60.09 161.67/60.09 The induced complexity on above rules (modulo remaining rules) is 161.67/60.09 YES(?,O(n^2)) . These rules are moved into the corresponding weak 161.67/60.09 component(s). 161.67/60.09 161.67/60.09 Sub-proof: 161.67/60.09 ---------- 161.67/60.09 TcT has computed the following triangular matrix interpretation. 161.67/60.09 161.67/60.09 [a__f](x1) = [1 0] x1 + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 161.67/60.09 [f](x1) = [1 0] x1 + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 161.67/60.09 [a__c](x1) = [1 0] x1 + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 161.67/60.09 [g](x1) = [1 0] x1 + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 161.67/60.09 [d](x1) = [1 0] x1 + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 161.67/60.09 [a__h](x1) = [1 0] x1 + [1] 161.67/60.09 [0 1] [2] 161.67/60.09 161.67/60.09 [mark](x1) = [1 1] x1 + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 161.67/60.09 [c](x1) = [1 0] x1 + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 161.67/60.09 [h](x1) = [1 0] x1 + [0] 161.67/60.09 [0 1] [2] 161.67/60.09 161.67/60.09 The order satisfies the following ordering constraints: 161.67/60.09 161.67/60.09 [a__f(X)] = [1 0] X + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 = [f(X)] 161.67/60.09 161.67/60.09 [a__f(f(X))] = [1 0] X + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [a__c(f(g(f(X))))] 161.67/60.09 161.67/60.09 [a__c(X)] = [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [a__c(X)] = [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [c(X)] 161.67/60.09 161.67/60.09 [a__h(X)] = [1 0] X + [1] 161.67/60.09 [0 1] [2] 161.67/60.09 > [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [a__c(d(X))] 161.67/60.09 161.67/60.09 [a__h(X)] = [1 0] X + [1] 161.67/60.09 [0 1] [2] 161.67/60.09 > [1 0] X + [0] 161.67/60.09 [0 1] [2] 161.67/60.09 = [h(X)] 161.67/60.09 161.67/60.09 [mark(f(X))] = [1 1] X + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 >= [1 1] X + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 = [a__f(mark(X))] 161.67/60.09 161.67/60.09 [mark(g(X))] = [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [g(X)] 161.67/60.09 161.67/60.09 [mark(d(X))] = [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [mark(c(X))] = [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [a__c(X)] 161.67/60.09 161.67/60.09 [mark(h(X))] = [1 1] X + [2] 161.67/60.09 [0 1] [2] 161.67/60.09 > [1 1] X + [1] 161.67/60.09 [0 1] [2] 161.67/60.09 = [a__h(mark(X))] 161.67/60.09 161.67/60.09 161.67/60.09 We return to the main proof. 161.67/60.09 161.67/60.09 We are left with following problem, upon which TcT provides the 161.67/60.09 certificate YES(O(1),O(n^2)). 161.67/60.09 161.67/60.09 Strict Trs: { mark(f(X)) -> a__f(mark(X)) } 161.67/60.09 Weak Trs: 161.67/60.09 { a__f(X) -> f(X) 161.67/60.09 , a__f(f(X)) -> a__c(f(g(f(X)))) 161.67/60.09 , a__c(X) -> d(X) 161.67/60.09 , a__c(X) -> c(X) 161.67/60.09 , a__h(X) -> a__c(d(X)) 161.67/60.09 , a__h(X) -> h(X) 161.67/60.09 , mark(g(X)) -> g(X) 161.67/60.09 , mark(d(X)) -> d(X) 161.67/60.09 , mark(c(X)) -> a__c(X) 161.67/60.09 , mark(h(X)) -> a__h(mark(X)) } 161.67/60.09 Obligation: 161.67/60.09 derivational complexity 161.67/60.09 Answer: 161.67/60.09 YES(O(1),O(n^2)) 161.67/60.09 161.67/60.09 We use the processor 'matrix interpretation of dimension 2' to 161.67/60.09 orient following rules strictly. 161.67/60.09 161.67/60.09 Trs: { mark(f(X)) -> a__f(mark(X)) } 161.67/60.09 161.67/60.09 The induced complexity on above rules (modulo remaining rules) is 161.67/60.09 YES(?,O(n^2)) . These rules are moved into the corresponding weak 161.67/60.09 component(s). 161.67/60.09 161.67/60.09 Sub-proof: 161.67/60.09 ---------- 161.67/60.09 TcT has computed the following triangular matrix interpretation. 161.67/60.09 161.67/60.09 [a__f](x1) = [1 2] x1 + [0] 161.67/60.09 [0 1] [2] 161.67/60.09 161.67/60.09 [f](x1) = [1 2] x1 + [0] 161.67/60.09 [0 1] [2] 161.67/60.09 161.67/60.09 [a__c](x1) = [1 0] x1 + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 161.67/60.09 [g](x1) = [1 2] x1 + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 161.67/60.09 [d](x1) = [1 0] x1 + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 161.67/60.09 [a__h](x1) = [1 0] x1 + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 161.67/60.09 [mark](x1) = [1 2] x1 + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 161.67/60.09 [c](x1) = [1 0] x1 + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 161.67/60.09 [h](x1) = [1 0] x1 + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 161.67/60.09 The order satisfies the following ordering constraints: 161.67/60.09 161.67/60.09 [a__f(X)] = [1 2] X + [0] 161.67/60.09 [0 1] [2] 161.67/60.09 >= [1 2] X + [0] 161.67/60.09 [0 1] [2] 161.67/60.09 = [f(X)] 161.67/60.09 161.67/60.09 [a__f(f(X))] = [1 4] X + [4] 161.67/60.09 [0 1] [4] 161.67/60.09 >= [1 4] X + [4] 161.67/60.09 [0 0] [0] 161.67/60.09 = [a__c(f(g(f(X))))] 161.67/60.09 161.67/60.09 [a__c(X)] = [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [a__c(X)] = [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [c(X)] 161.67/60.09 161.67/60.09 [a__h(X)] = [1 0] X + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [a__c(d(X))] 161.67/60.09 161.67/60.09 [a__h(X)] = [1 0] X + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 = [h(X)] 161.67/60.09 161.67/60.09 [mark(f(X))] = [1 4] X + [4] 161.67/60.09 [0 1] [2] 161.67/60.09 > [1 4] X + [0] 161.67/60.09 [0 1] [2] 161.67/60.09 = [a__f(mark(X))] 161.67/60.09 161.67/60.09 [mark(g(X))] = [1 2] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 >= [1 2] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [g(X)] 161.67/60.09 161.67/60.09 [mark(d(X))] = [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [d(X)] 161.67/60.09 161.67/60.09 [mark(c(X))] = [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 >= [1 0] X + [0] 161.67/60.09 [0 0] [0] 161.67/60.09 = [a__c(X)] 161.67/60.09 161.67/60.09 [mark(h(X))] = [1 2] X + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 >= [1 2] X + [0] 161.67/60.09 [0 1] [0] 161.67/60.09 = [a__h(mark(X))] 161.67/60.09 161.67/60.09 161.67/60.09 We return to the main proof. 161.67/60.09 161.67/60.09 We are left with following problem, upon which TcT provides the 161.67/60.09 certificate YES(O(1),O(1)). 161.67/60.09 161.67/60.09 Weak Trs: 161.67/60.09 { a__f(X) -> f(X) 161.67/60.09 , a__f(f(X)) -> a__c(f(g(f(X)))) 161.67/60.09 , a__c(X) -> d(X) 161.67/60.09 , a__c(X) -> c(X) 161.67/60.09 , a__h(X) -> a__c(d(X)) 161.67/60.09 , a__h(X) -> h(X) 161.67/60.09 , mark(f(X)) -> a__f(mark(X)) 161.67/60.09 , mark(g(X)) -> g(X) 161.67/60.09 , mark(d(X)) -> d(X) 161.67/60.09 , mark(c(X)) -> a__c(X) 161.67/60.09 , mark(h(X)) -> a__h(mark(X)) } 161.67/60.09 Obligation: 161.67/60.09 derivational complexity 161.67/60.09 Answer: 161.67/60.09 YES(O(1),O(1)) 161.67/60.09 161.67/60.09 Empty rules are trivially bounded 161.67/60.09 161.67/60.09 Hurray, we answered YES(O(1),O(n^2)) 161.94/60.11 EOF