YES(O(1),O(n^2)) 167.01/60.08 YES(O(1),O(n^2)) 167.01/60.08 167.01/60.08 We are left with following problem, upon which TcT provides the 167.01/60.08 certificate YES(O(1),O(n^2)). 167.01/60.08 167.01/60.08 Strict Trs: 167.01/60.08 { a(a(x1)) -> d(c(x1)) 167.01/60.08 , a(b(x1)) -> c(c(c(x1))) 167.01/60.08 , d(d(x1)) -> b(a(c(x1))) 167.01/60.08 , c(d(x1)) -> a(a(x1)) 167.01/60.08 , c(c(x1)) -> b(x1) 167.01/60.08 , b(b(x1)) -> a(c(c(x1))) } 167.01/60.08 Obligation: 167.01/60.08 derivational complexity 167.01/60.08 Answer: 167.01/60.08 YES(O(1),O(n^2)) 167.01/60.08 167.01/60.08 The weightgap principle applies (using the following nonconstant 167.01/60.08 growth matrix-interpretation) 167.01/60.08 167.01/60.08 TcT has computed the following triangular matrix interpretation. 167.01/60.08 Note that the diagonal of the component-wise maxima of 167.01/60.08 interpretation-entries contains no more than 1 non-zero entries. 167.01/60.08 167.01/60.08 [a](x1) = [1] x1 + [0] 167.01/60.08 167.01/60.08 [d](x1) = [1] x1 + [0] 167.01/60.08 167.01/60.08 [c](x1) = [1] x1 + [0] 167.01/60.08 167.01/60.08 [b](x1) = [1] x1 + [1] 167.01/60.08 167.01/60.08 The order satisfies the following ordering constraints: 167.01/60.08 167.01/60.08 [a(a(x1))] = [1] x1 + [0] 167.01/60.08 >= [1] x1 + [0] 167.01/60.08 = [d(c(x1))] 167.01/60.08 167.01/60.08 [a(b(x1))] = [1] x1 + [1] 167.01/60.08 > [1] x1 + [0] 167.01/60.08 = [c(c(c(x1)))] 167.01/60.08 167.01/60.08 [d(d(x1))] = [1] x1 + [0] 167.01/60.08 ? [1] x1 + [1] 167.01/60.08 = [b(a(c(x1)))] 167.01/60.08 167.01/60.08 [c(d(x1))] = [1] x1 + [0] 167.01/60.08 >= [1] x1 + [0] 167.01/60.08 = [a(a(x1))] 167.01/60.08 167.01/60.08 [c(c(x1))] = [1] x1 + [0] 167.01/60.08 ? [1] x1 + [1] 167.01/60.08 = [b(x1)] 167.01/60.08 167.01/60.08 [b(b(x1))] = [1] x1 + [2] 167.01/60.08 > [1] x1 + [0] 167.01/60.08 = [a(c(c(x1)))] 167.01/60.08 167.01/60.08 167.01/60.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 167.01/60.08 167.01/60.08 We are left with following problem, upon which TcT provides the 167.01/60.08 certificate YES(O(1),O(n^2)). 167.01/60.08 167.01/60.08 Strict Trs: 167.01/60.08 { a(a(x1)) -> d(c(x1)) 167.01/60.08 , d(d(x1)) -> b(a(c(x1))) 167.01/60.08 , c(d(x1)) -> a(a(x1)) 167.01/60.08 , c(c(x1)) -> b(x1) } 167.01/60.08 Weak Trs: 167.01/60.08 { a(b(x1)) -> c(c(c(x1))) 167.01/60.08 , b(b(x1)) -> a(c(c(x1))) } 167.01/60.08 Obligation: 167.01/60.08 derivational complexity 167.01/60.08 Answer: 167.01/60.08 YES(O(1),O(n^2)) 167.01/60.08 167.01/60.08 The weightgap principle applies (using the following nonconstant 167.01/60.08 growth matrix-interpretation) 167.01/60.08 167.01/60.08 TcT has computed the following triangular matrix interpretation. 167.01/60.08 Note that the diagonal of the component-wise maxima of 167.01/60.08 interpretation-entries contains no more than 1 non-zero entries. 167.01/60.08 167.01/60.08 [a](x1) = [1] x1 + [2] 167.01/60.08 167.01/60.08 [d](x1) = [1] x1 + [0] 167.01/60.08 167.01/60.08 [c](x1) = [1] x1 + [0] 167.01/60.08 167.01/60.08 [b](x1) = [1] x1 + [2] 167.01/60.08 167.01/60.08 The order satisfies the following ordering constraints: 167.01/60.08 167.01/60.08 [a(a(x1))] = [1] x1 + [4] 167.01/60.08 > [1] x1 + [0] 167.01/60.08 = [d(c(x1))] 167.01/60.08 167.01/60.08 [a(b(x1))] = [1] x1 + [4] 167.01/60.08 > [1] x1 + [0] 167.01/60.08 = [c(c(c(x1)))] 167.01/60.08 167.01/60.08 [d(d(x1))] = [1] x1 + [0] 167.01/60.08 ? [1] x1 + [4] 167.01/60.08 = [b(a(c(x1)))] 167.01/60.08 167.01/60.08 [c(d(x1))] = [1] x1 + [0] 167.01/60.08 ? [1] x1 + [4] 167.01/60.08 = [a(a(x1))] 167.01/60.08 167.01/60.08 [c(c(x1))] = [1] x1 + [0] 167.01/60.08 ? [1] x1 + [2] 167.01/60.08 = [b(x1)] 167.01/60.08 167.01/60.08 [b(b(x1))] = [1] x1 + [4] 167.01/60.08 > [1] x1 + [2] 167.01/60.08 = [a(c(c(x1)))] 167.01/60.08 167.01/60.08 167.01/60.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 167.01/60.08 167.01/60.08 We are left with following problem, upon which TcT provides the 167.01/60.08 certificate YES(O(1),O(n^2)). 167.01/60.08 167.01/60.08 Strict Trs: 167.01/60.08 { d(d(x1)) -> b(a(c(x1))) 167.01/60.08 , c(d(x1)) -> a(a(x1)) 167.01/60.08 , c(c(x1)) -> b(x1) } 167.01/60.08 Weak Trs: 167.01/60.08 { a(a(x1)) -> d(c(x1)) 167.01/60.08 , a(b(x1)) -> c(c(c(x1))) 167.01/60.08 , b(b(x1)) -> a(c(c(x1))) } 167.01/60.08 Obligation: 167.01/60.08 derivational complexity 167.01/60.08 Answer: 167.01/60.08 YES(O(1),O(n^2)) 167.01/60.08 167.01/60.08 The weightgap principle applies (using the following nonconstant 167.01/60.08 growth matrix-interpretation) 167.01/60.08 167.01/60.08 TcT has computed the following triangular matrix interpretation. 167.01/60.08 Note that the diagonal of the component-wise maxima of 167.01/60.08 interpretation-entries contains no more than 1 non-zero entries. 167.01/60.08 167.01/60.08 [a](x1) = [1] x1 + [2] 167.01/60.08 167.01/60.08 [d](x1) = [1] x1 + [2] 167.01/60.08 167.01/60.08 [c](x1) = [1] x1 + [0] 167.01/60.08 167.01/60.08 [b](x1) = [1] x1 + [1] 167.01/60.08 167.01/60.08 The order satisfies the following ordering constraints: 167.01/60.08 167.01/60.08 [a(a(x1))] = [1] x1 + [4] 167.01/60.08 > [1] x1 + [2] 167.01/60.08 = [d(c(x1))] 167.01/60.08 167.01/60.08 [a(b(x1))] = [1] x1 + [3] 167.01/60.08 > [1] x1 + [0] 167.01/60.08 = [c(c(c(x1)))] 167.01/60.08 167.01/60.08 [d(d(x1))] = [1] x1 + [4] 167.01/60.08 > [1] x1 + [3] 167.01/60.08 = [b(a(c(x1)))] 167.01/60.08 167.01/60.08 [c(d(x1))] = [1] x1 + [2] 167.01/60.08 ? [1] x1 + [4] 167.01/60.08 = [a(a(x1))] 167.01/60.08 167.01/60.08 [c(c(x1))] = [1] x1 + [0] 167.01/60.08 ? [1] x1 + [1] 167.01/60.08 = [b(x1)] 167.01/60.08 167.01/60.08 [b(b(x1))] = [1] x1 + [2] 167.01/60.08 >= [1] x1 + [2] 167.01/60.08 = [a(c(c(x1)))] 167.01/60.08 167.01/60.08 167.01/60.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 167.01/60.08 167.01/60.08 We are left with following problem, upon which TcT provides the 167.01/60.08 certificate YES(O(1),O(n^2)). 167.01/60.08 167.01/60.08 Strict Trs: 167.01/60.08 { c(d(x1)) -> a(a(x1)) 167.01/60.08 , c(c(x1)) -> b(x1) } 167.01/60.08 Weak Trs: 167.01/60.08 { a(a(x1)) -> d(c(x1)) 167.01/60.08 , a(b(x1)) -> c(c(c(x1))) 167.01/60.08 , d(d(x1)) -> b(a(c(x1))) 167.01/60.08 , b(b(x1)) -> a(c(c(x1))) } 167.01/60.08 Obligation: 167.01/60.08 derivational complexity 167.01/60.08 Answer: 167.01/60.08 YES(O(1),O(n^2)) 167.01/60.08 167.01/60.08 We use the processor 'matrix interpretation of dimension 1' to 167.01/60.08 orient following rules strictly. 167.01/60.08 167.01/60.08 Trs: { c(c(x1)) -> b(x1) } 167.01/60.08 167.01/60.08 The induced complexity on above rules (modulo remaining rules) is 167.01/60.08 YES(?,O(n^1)) . These rules are moved into the corresponding weak 167.01/60.08 component(s). 167.01/60.08 167.01/60.08 Sub-proof: 167.01/60.08 ---------- 167.01/60.08 TcT has computed the following triangular matrix interpretation. 167.01/60.08 167.01/60.08 [a](x1) = [1] x1 + [32] 167.01/60.08 167.01/60.08 [d](x1) = [1] x1 + [44] 167.01/60.08 167.01/60.08 [c](x1) = [1] x1 + [20] 167.01/60.08 167.01/60.08 [b](x1) = [1] x1 + [36] 167.01/60.08 167.01/60.08 The order satisfies the following ordering constraints: 167.01/60.08 167.01/60.08 [a(a(x1))] = [1] x1 + [64] 167.01/60.08 >= [1] x1 + [64] 167.01/60.08 = [d(c(x1))] 167.01/60.08 167.01/60.08 [a(b(x1))] = [1] x1 + [68] 167.01/60.08 > [1] x1 + [60] 167.01/60.08 = [c(c(c(x1)))] 167.01/60.08 167.01/60.08 [d(d(x1))] = [1] x1 + [88] 167.01/60.08 >= [1] x1 + [88] 167.01/60.08 = [b(a(c(x1)))] 167.01/60.08 167.01/60.08 [c(d(x1))] = [1] x1 + [64] 167.01/60.08 >= [1] x1 + [64] 167.01/60.08 = [a(a(x1))] 167.01/60.08 167.01/60.08 [c(c(x1))] = [1] x1 + [40] 167.01/60.08 > [1] x1 + [36] 167.01/60.08 = [b(x1)] 167.01/60.08 167.01/60.08 [b(b(x1))] = [1] x1 + [72] 167.01/60.08 >= [1] x1 + [72] 167.01/60.08 = [a(c(c(x1)))] 167.01/60.08 167.01/60.08 167.01/60.08 We return to the main proof. 167.01/60.08 167.01/60.08 We are left with following problem, upon which TcT provides the 167.01/60.08 certificate YES(O(1),O(n^2)). 167.01/60.08 167.01/60.08 Strict Trs: { c(d(x1)) -> a(a(x1)) } 167.01/60.08 Weak Trs: 167.01/60.08 { a(a(x1)) -> d(c(x1)) 167.01/60.08 , a(b(x1)) -> c(c(c(x1))) 167.01/60.08 , d(d(x1)) -> b(a(c(x1))) 167.01/60.08 , c(c(x1)) -> b(x1) 167.01/60.08 , b(b(x1)) -> a(c(c(x1))) } 167.01/60.08 Obligation: 167.01/60.08 derivational complexity 167.01/60.08 Answer: 167.01/60.08 YES(O(1),O(n^2)) 167.01/60.08 167.01/60.08 We use the processor 'matrix interpretation of dimension 2' to 167.01/60.08 orient following rules strictly. 167.01/60.08 167.01/60.08 Trs: { c(d(x1)) -> a(a(x1)) } 167.01/60.08 167.01/60.08 The induced complexity on above rules (modulo remaining rules) is 167.01/60.08 YES(?,O(n^2)) . These rules are moved into the corresponding weak 167.01/60.08 component(s). 167.01/60.08 167.01/60.08 Sub-proof: 167.01/60.08 ---------- 167.01/60.08 TcT has computed the following triangular matrix interpretation. 167.01/60.08 167.01/60.08 [a](x1) = [1 8] x1 + [0] 167.01/60.08 [0 1] [2] 167.01/60.08 167.01/60.08 [d](x1) = [1 11] x1 + [4] 167.01/60.08 [0 1] [3] 167.01/60.08 167.01/60.08 [c](x1) = [1 5] x1 + [0] 167.01/60.08 [0 1] [1] 167.01/60.08 167.01/60.08 [b](x1) = [1 9] x1 + [4] 167.01/60.08 [0 1] [2] 167.01/60.08 167.01/60.08 The order satisfies the following ordering constraints: 167.01/60.08 167.01/60.08 [a(a(x1))] = [1 16] x1 + [16] 167.01/60.08 [0 1] [4] 167.01/60.08 > [1 16] x1 + [15] 167.01/60.08 [0 1] [4] 167.01/60.08 = [d(c(x1))] 167.01/60.08 167.01/60.08 [a(b(x1))] = [1 17] x1 + [20] 167.01/60.08 [0 1] [4] 167.01/60.08 > [1 15] x1 + [15] 167.01/60.08 [0 1] [3] 167.01/60.08 = [c(c(c(x1)))] 167.01/60.08 167.01/60.08 [d(d(x1))] = [1 22] x1 + [41] 167.01/60.08 [0 1] [6] 167.01/60.08 > [1 22] x1 + [39] 167.01/60.08 [0 1] [5] 167.01/60.08 = [b(a(c(x1)))] 167.01/60.08 167.01/60.08 [c(d(x1))] = [1 16] x1 + [19] 167.01/60.08 [0 1] [4] 167.01/60.08 > [1 16] x1 + [16] 167.01/60.08 [0 1] [4] 167.01/60.08 = [a(a(x1))] 167.01/60.08 167.01/60.08 [c(c(x1))] = [1 10] x1 + [5] 167.01/60.08 [0 1] [2] 167.01/60.08 > [1 9] x1 + [4] 167.01/60.08 [0 1] [2] 167.01/60.08 = [b(x1)] 167.01/60.08 167.01/60.08 [b(b(x1))] = [1 18] x1 + [26] 167.01/60.08 [0 1] [4] 167.01/60.08 > [1 18] x1 + [21] 167.01/60.08 [0 1] [4] 167.01/60.08 = [a(c(c(x1)))] 167.01/60.08 167.01/60.08 167.01/60.08 We return to the main proof. 167.01/60.08 167.01/60.08 We are left with following problem, upon which TcT provides the 167.01/60.08 certificate YES(O(1),O(1)). 167.01/60.08 167.01/60.08 Weak Trs: 167.01/60.08 { a(a(x1)) -> d(c(x1)) 167.01/60.08 , a(b(x1)) -> c(c(c(x1))) 167.01/60.08 , d(d(x1)) -> b(a(c(x1))) 167.01/60.08 , c(d(x1)) -> a(a(x1)) 167.01/60.08 , c(c(x1)) -> b(x1) 167.01/60.08 , b(b(x1)) -> a(c(c(x1))) } 167.01/60.08 Obligation: 167.01/60.08 derivational complexity 167.01/60.08 Answer: 167.01/60.08 YES(O(1),O(1)) 167.01/60.08 167.01/60.08 Empty rules are trivially bounded 167.01/60.08 167.01/60.08 Hurray, we answered YES(O(1),O(n^2)) 167.24/60.13 EOF