YES(O(1),O(n^1)) 182.77/60.03 YES(O(1),O(n^1)) 182.77/60.03 182.77/60.03 We are left with following problem, upon which TcT provides the 182.77/60.03 certificate YES(O(1),O(n^1)). 182.77/60.03 182.77/60.03 Strict Trs: 182.77/60.03 { a(a(x1)) -> b(c(x1)) 182.77/60.03 , b(x1) -> a(x1) 182.77/60.03 , b(b(x1)) -> c(d(x1)) 182.77/60.03 , c(c(x1)) -> d(f(x1)) 182.77/60.03 , d(x1) -> b(x1) 182.77/60.03 , d(d(x1)) -> f(f(f(x1))) 182.77/60.03 , f(f(x1)) -> g(a(x1)) 182.77/60.03 , g(g(x1)) -> a(x1) } 182.77/60.03 Obligation: 182.77/60.03 derivational complexity 182.77/60.03 Answer: 182.77/60.03 YES(O(1),O(n^1)) 182.77/60.03 182.77/60.03 The weightgap principle applies (using the following nonconstant 182.77/60.03 growth matrix-interpretation) 182.77/60.03 182.77/60.03 TcT has computed the following triangular matrix interpretation. 182.77/60.03 Note that the diagonal of the component-wise maxima of 182.77/60.03 interpretation-entries contains no more than 1 non-zero entries. 182.77/60.03 182.77/60.03 [a](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 [b](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 [c](x1) = [1] x1 + [1] 182.77/60.03 182.77/60.03 [d](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 [f](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 [g](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 The order satisfies the following ordering constraints: 182.77/60.03 182.77/60.03 [a(a(x1))] = [1] x1 + [4] 182.77/60.03 > [1] x1 + [3] 182.77/60.03 = [b(c(x1))] 182.77/60.03 182.77/60.03 [b(x1)] = [1] x1 + [2] 182.77/60.03 >= [1] x1 + [2] 182.77/60.03 = [a(x1)] 182.77/60.03 182.77/60.03 [b(b(x1))] = [1] x1 + [4] 182.77/60.03 > [1] x1 + [3] 182.77/60.03 = [c(d(x1))] 182.77/60.03 182.77/60.03 [c(c(x1))] = [1] x1 + [2] 182.77/60.03 ? [1] x1 + [4] 182.77/60.03 = [d(f(x1))] 182.77/60.03 182.77/60.03 [d(x1)] = [1] x1 + [2] 182.77/60.03 >= [1] x1 + [2] 182.77/60.03 = [b(x1)] 182.77/60.03 182.77/60.03 [d(d(x1))] = [1] x1 + [4] 182.77/60.03 ? [1] x1 + [6] 182.77/60.03 = [f(f(f(x1)))] 182.77/60.03 182.77/60.03 [f(f(x1))] = [1] x1 + [4] 182.77/60.03 >= [1] x1 + [4] 182.77/60.03 = [g(a(x1))] 182.77/60.03 182.77/60.03 [g(g(x1))] = [1] x1 + [4] 182.77/60.03 > [1] x1 + [2] 182.77/60.03 = [a(x1)] 182.77/60.03 182.77/60.03 182.77/60.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 182.77/60.03 182.77/60.03 We are left with following problem, upon which TcT provides the 182.77/60.03 certificate YES(O(1),O(n^1)). 182.77/60.03 182.77/60.03 Strict Trs: 182.77/60.03 { b(x1) -> a(x1) 182.77/60.03 , c(c(x1)) -> d(f(x1)) 182.77/60.03 , d(x1) -> b(x1) 182.77/60.03 , d(d(x1)) -> f(f(f(x1))) 182.77/60.03 , f(f(x1)) -> g(a(x1)) } 182.77/60.03 Weak Trs: 182.77/60.03 { a(a(x1)) -> b(c(x1)) 182.77/60.03 , b(b(x1)) -> c(d(x1)) 182.77/60.03 , g(g(x1)) -> a(x1) } 182.77/60.03 Obligation: 182.77/60.03 derivational complexity 182.77/60.03 Answer: 182.77/60.03 YES(O(1),O(n^1)) 182.77/60.03 182.77/60.03 The weightgap principle applies (using the following nonconstant 182.77/60.03 growth matrix-interpretation) 182.77/60.03 182.77/60.03 TcT has computed the following triangular matrix interpretation. 182.77/60.03 Note that the diagonal of the component-wise maxima of 182.77/60.03 interpretation-entries contains no more than 1 non-zero entries. 182.77/60.03 182.77/60.03 [a](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 [b](x1) = [1] x1 + [1] 182.77/60.03 182.77/60.03 [c](x1) = [1] x1 + [0] 182.77/60.03 182.77/60.03 [d](x1) = [1] x1 + [1] 182.77/60.03 182.77/60.03 [f](x1) = [1] x1 + [0] 182.77/60.03 182.77/60.03 [g](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 The order satisfies the following ordering constraints: 182.77/60.03 182.77/60.03 [a(a(x1))] = [1] x1 + [4] 182.77/60.03 > [1] x1 + [1] 182.77/60.03 = [b(c(x1))] 182.77/60.03 182.77/60.03 [b(x1)] = [1] x1 + [1] 182.77/60.03 ? [1] x1 + [2] 182.77/60.03 = [a(x1)] 182.77/60.03 182.77/60.03 [b(b(x1))] = [1] x1 + [2] 182.77/60.03 > [1] x1 + [1] 182.77/60.03 = [c(d(x1))] 182.77/60.03 182.77/60.03 [c(c(x1))] = [1] x1 + [0] 182.77/60.03 ? [1] x1 + [1] 182.77/60.03 = [d(f(x1))] 182.77/60.03 182.77/60.03 [d(x1)] = [1] x1 + [1] 182.77/60.03 >= [1] x1 + [1] 182.77/60.03 = [b(x1)] 182.77/60.03 182.77/60.03 [d(d(x1))] = [1] x1 + [2] 182.77/60.03 > [1] x1 + [0] 182.77/60.03 = [f(f(f(x1)))] 182.77/60.03 182.77/60.03 [f(f(x1))] = [1] x1 + [0] 182.77/60.03 ? [1] x1 + [4] 182.77/60.03 = [g(a(x1))] 182.77/60.03 182.77/60.03 [g(g(x1))] = [1] x1 + [4] 182.77/60.03 > [1] x1 + [2] 182.77/60.03 = [a(x1)] 182.77/60.03 182.77/60.03 182.77/60.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 182.77/60.03 182.77/60.03 We are left with following problem, upon which TcT provides the 182.77/60.03 certificate YES(O(1),O(n^1)). 182.77/60.03 182.77/60.03 Strict Trs: 182.77/60.03 { b(x1) -> a(x1) 182.77/60.03 , c(c(x1)) -> d(f(x1)) 182.77/60.03 , d(x1) -> b(x1) 182.77/60.03 , f(f(x1)) -> g(a(x1)) } 182.77/60.03 Weak Trs: 182.77/60.03 { a(a(x1)) -> b(c(x1)) 182.77/60.03 , b(b(x1)) -> c(d(x1)) 182.77/60.03 , d(d(x1)) -> f(f(f(x1))) 182.77/60.03 , g(g(x1)) -> a(x1) } 182.77/60.03 Obligation: 182.77/60.03 derivational complexity 182.77/60.03 Answer: 182.77/60.03 YES(O(1),O(n^1)) 182.77/60.03 182.77/60.03 The weightgap principle applies (using the following nonconstant 182.77/60.03 growth matrix-interpretation) 182.77/60.03 182.77/60.03 TcT has computed the following triangular matrix interpretation. 182.77/60.03 Note that the diagonal of the component-wise maxima of 182.77/60.03 interpretation-entries contains no more than 1 non-zero entries. 182.77/60.03 182.77/60.03 [a](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 [b](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 [c](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 [d](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 [f](x1) = [1] x1 + [0] 182.77/60.03 182.77/60.03 [g](x1) = [1] x1 + [2] 182.77/60.03 182.77/60.03 The order satisfies the following ordering constraints: 182.77/60.03 182.77/60.03 [a(a(x1))] = [1] x1 + [4] 182.77/60.03 >= [1] x1 + [4] 182.77/60.03 = [b(c(x1))] 182.77/60.03 182.77/60.03 [b(x1)] = [1] x1 + [2] 182.77/60.03 >= [1] x1 + [2] 182.77/60.03 = [a(x1)] 182.77/60.03 182.77/60.03 [b(b(x1))] = [1] x1 + [4] 182.77/60.03 >= [1] x1 + [4] 182.77/60.03 = [c(d(x1))] 182.77/60.03 182.77/60.03 [c(c(x1))] = [1] x1 + [4] 182.77/60.03 > [1] x1 + [2] 182.77/60.03 = [d(f(x1))] 182.77/60.03 182.77/60.03 [d(x1)] = [1] x1 + [2] 182.77/60.03 >= [1] x1 + [2] 182.77/60.03 = [b(x1)] 182.77/60.03 182.77/60.03 [d(d(x1))] = [1] x1 + [4] 182.77/60.03 > [1] x1 + [0] 182.77/60.03 = [f(f(f(x1)))] 182.77/60.03 182.77/60.03 [f(f(x1))] = [1] x1 + [0] 182.77/60.03 ? [1] x1 + [4] 182.77/60.03 = [g(a(x1))] 182.77/60.03 182.77/60.03 [g(g(x1))] = [1] x1 + [4] 182.77/60.03 > [1] x1 + [2] 182.77/60.03 = [a(x1)] 182.77/60.03 182.77/60.03 182.77/60.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 182.77/60.03 182.77/60.03 We are left with following problem, upon which TcT provides the 182.77/60.03 certificate YES(O(1),O(n^1)). 182.77/60.03 182.77/60.03 Strict Trs: 182.77/60.03 { b(x1) -> a(x1) 182.77/60.03 , d(x1) -> b(x1) 182.77/60.03 , f(f(x1)) -> g(a(x1)) } 182.77/60.03 Weak Trs: 182.77/60.03 { a(a(x1)) -> b(c(x1)) 182.77/60.03 , b(b(x1)) -> c(d(x1)) 182.77/60.03 , c(c(x1)) -> d(f(x1)) 182.77/60.03 , d(d(x1)) -> f(f(f(x1))) 182.77/60.03 , g(g(x1)) -> a(x1) } 182.77/60.03 Obligation: 182.77/60.03 derivational complexity 182.77/60.03 Answer: 182.77/60.03 YES(O(1),O(n^1)) 182.77/60.03 182.77/60.03 We use the processor 'matrix interpretation of dimension 1' to 182.77/60.03 orient following rules strictly. 182.77/60.03 182.77/60.03 Trs: 182.77/60.03 { b(x1) -> a(x1) 182.77/60.03 , d(x1) -> b(x1) } 182.77/60.03 182.77/60.03 The induced complexity on above rules (modulo remaining rules) is 182.77/60.03 YES(?,O(n^1)) . These rules are moved into the corresponding weak 182.77/60.03 component(s). 182.77/60.03 182.77/60.03 Sub-proof: 182.77/60.03 ---------- 182.77/60.03 TcT has computed the following triangular matrix interpretation. 182.77/60.03 182.77/60.03 [a](x1) = [1] x1 + [32] 182.77/60.03 182.77/60.03 [b](x1) = [1] x1 + [33] 182.77/60.03 182.77/60.03 [c](x1) = [1] x1 + [30] 182.77/60.03 182.77/60.03 [d](x1) = [1] x1 + [36] 182.77/60.03 182.77/60.03 [f](x1) = [1] x1 + [24] 182.77/60.03 182.77/60.03 [g](x1) = [1] x1 + [16] 182.77/60.03 182.77/60.03 The order satisfies the following ordering constraints: 182.77/60.03 182.77/60.03 [a(a(x1))] = [1] x1 + [64] 182.77/60.03 > [1] x1 + [63] 182.77/60.03 = [b(c(x1))] 182.77/60.03 182.77/60.03 [b(x1)] = [1] x1 + [33] 182.77/60.03 > [1] x1 + [32] 182.77/60.03 = [a(x1)] 182.77/60.03 182.77/60.03 [b(b(x1))] = [1] x1 + [66] 182.77/60.03 >= [1] x1 + [66] 182.77/60.03 = [c(d(x1))] 182.77/60.03 182.77/60.03 [c(c(x1))] = [1] x1 + [60] 182.77/60.03 >= [1] x1 + [60] 182.77/60.03 = [d(f(x1))] 182.77/60.03 182.77/60.03 [d(x1)] = [1] x1 + [36] 182.77/60.03 > [1] x1 + [33] 182.77/60.03 = [b(x1)] 182.77/60.03 182.77/60.03 [d(d(x1))] = [1] x1 + [72] 182.77/60.03 >= [1] x1 + [72] 182.77/60.03 = [f(f(f(x1)))] 182.77/60.03 182.77/60.03 [f(f(x1))] = [1] x1 + [48] 182.77/60.03 >= [1] x1 + [48] 182.77/60.03 = [g(a(x1))] 182.77/60.03 182.77/60.03 [g(g(x1))] = [1] x1 + [32] 182.77/60.03 >= [1] x1 + [32] 182.77/60.03 = [a(x1)] 182.77/60.03 182.77/60.03 182.77/60.03 We return to the main proof. 182.77/60.03 182.77/60.03 We are left with following problem, upon which TcT provides the 182.77/60.03 certificate YES(O(1),O(n^1)). 182.77/60.03 182.77/60.03 Strict Trs: { f(f(x1)) -> g(a(x1)) } 182.77/60.03 Weak Trs: 182.77/60.03 { a(a(x1)) -> b(c(x1)) 182.77/60.03 , b(x1) -> a(x1) 182.77/60.03 , b(b(x1)) -> c(d(x1)) 182.77/60.03 , c(c(x1)) -> d(f(x1)) 182.77/60.03 , d(x1) -> b(x1) 182.77/60.03 , d(d(x1)) -> f(f(f(x1))) 182.77/60.03 , g(g(x1)) -> a(x1) } 182.77/60.03 Obligation: 182.77/60.03 derivational complexity 182.77/60.03 Answer: 182.77/60.03 YES(O(1),O(n^1)) 182.77/60.03 182.77/60.03 We use the processor 'matrix interpretation of dimension 1' to 182.77/60.03 orient following rules strictly. 182.77/60.03 182.77/60.03 Trs: { f(f(x1)) -> g(a(x1)) } 182.77/60.03 182.77/60.03 The induced complexity on above rules (modulo remaining rules) is 182.77/60.03 YES(?,O(n^1)) . These rules are moved into the corresponding weak 182.77/60.03 component(s). 182.77/60.03 182.77/60.03 Sub-proof: 182.77/60.03 ---------- 182.77/60.03 TcT has computed the following triangular matrix interpretation. 182.77/60.03 182.77/60.03 [a](x1) = [1] x1 + [42] 182.77/60.03 182.77/60.03 [b](x1) = [1] x1 + [44] 182.77/60.03 182.77/60.03 [c](x1) = [1] x1 + [40] 182.77/60.03 182.77/60.03 [d](x1) = [1] x1 + [48] 182.77/60.03 182.77/60.03 [f](x1) = [1] x1 + [32] 182.77/60.03 182.77/60.03 [g](x1) = [1] x1 + [21] 182.77/60.03 182.77/60.03 The order satisfies the following ordering constraints: 182.77/60.03 182.77/60.03 [a(a(x1))] = [1] x1 + [84] 182.77/60.03 >= [1] x1 + [84] 182.77/60.03 = [b(c(x1))] 182.77/60.03 182.77/60.03 [b(x1)] = [1] x1 + [44] 182.77/60.03 > [1] x1 + [42] 182.77/60.03 = [a(x1)] 182.77/60.03 182.77/60.03 [b(b(x1))] = [1] x1 + [88] 182.77/60.03 >= [1] x1 + [88] 182.77/60.03 = [c(d(x1))] 182.77/60.03 182.77/60.03 [c(c(x1))] = [1] x1 + [80] 182.77/60.03 >= [1] x1 + [80] 182.77/60.03 = [d(f(x1))] 182.77/60.03 182.77/60.03 [d(x1)] = [1] x1 + [48] 182.77/60.03 > [1] x1 + [44] 182.77/60.03 = [b(x1)] 182.77/60.03 182.77/60.03 [d(d(x1))] = [1] x1 + [96] 182.77/60.03 >= [1] x1 + [96] 182.77/60.03 = [f(f(f(x1)))] 182.77/60.03 182.77/60.03 [f(f(x1))] = [1] x1 + [64] 182.77/60.03 > [1] x1 + [63] 182.77/60.03 = [g(a(x1))] 182.77/60.03 182.77/60.03 [g(g(x1))] = [1] x1 + [42] 182.77/60.03 >= [1] x1 + [42] 182.77/60.03 = [a(x1)] 182.77/60.03 182.77/60.03 182.77/60.03 We return to the main proof. 182.77/60.03 182.77/60.03 We are left with following problem, upon which TcT provides the 182.77/60.03 certificate YES(O(1),O(1)). 182.77/60.03 182.77/60.03 Weak Trs: 182.77/60.03 { a(a(x1)) -> b(c(x1)) 182.77/60.03 , b(x1) -> a(x1) 182.77/60.03 , b(b(x1)) -> c(d(x1)) 182.77/60.03 , c(c(x1)) -> d(f(x1)) 182.77/60.03 , d(x1) -> b(x1) 182.77/60.03 , d(d(x1)) -> f(f(f(x1))) 182.77/60.03 , f(f(x1)) -> g(a(x1)) 182.77/60.03 , g(g(x1)) -> a(x1) } 182.77/60.03 Obligation: 182.77/60.03 derivational complexity 182.77/60.03 Answer: 182.77/60.03 YES(O(1),O(1)) 182.77/60.03 182.77/60.03 Empty rules are trivially bounded 182.77/60.03 182.77/60.03 Hurray, we answered YES(O(1),O(n^1)) 182.77/60.05 EOF