YES(O(1),O(n^2)) 159.63/60.08 YES(O(1),O(n^2)) 159.63/60.08 159.63/60.08 We are left with following problem, upon which TcT provides the 159.63/60.08 certificate YES(O(1),O(n^2)). 159.63/60.08 159.63/60.08 Strict Trs: 159.63/60.08 { b(b(x1)) -> a(c(x1)) 159.63/60.08 , b(c(x1)) -> a(x1) 159.63/60.08 , c(c(c(x1))) -> b(x1) 159.63/60.08 , a(x1) -> c(b(x1)) } 159.63/60.08 Obligation: 159.63/60.08 derivational complexity 159.63/60.08 Answer: 159.63/60.08 YES(O(1),O(n^2)) 159.63/60.08 159.63/60.08 We use the processor 'matrix interpretation of dimension 1' to 159.63/60.08 orient following rules strictly. 159.63/60.08 159.63/60.08 Trs: { b(b(x1)) -> a(c(x1)) } 159.63/60.08 159.63/60.08 The induced complexity on above rules (modulo remaining rules) is 159.63/60.08 YES(?,O(n^1)) . These rules are moved into the corresponding weak 159.63/60.08 component(s). 159.63/60.08 159.63/60.08 Sub-proof: 159.63/60.08 ---------- 159.63/60.08 TcT has computed the following triangular matrix interpretation. 159.63/60.08 159.63/60.08 [b](x1) = [1] x1 + [6] 159.63/60.08 159.63/60.08 [c](x1) = [1] x1 + [2] 159.63/60.08 159.63/60.08 [a](x1) = [1] x1 + [8] 159.63/60.08 159.63/60.08 The order satisfies the following ordering constraints: 159.63/60.08 159.63/60.08 [b(b(x1))] = [1] x1 + [12] 159.63/60.08 > [1] x1 + [10] 159.63/60.08 = [a(c(x1))] 159.63/60.08 159.63/60.08 [b(c(x1))] = [1] x1 + [8] 159.63/60.08 >= [1] x1 + [8] 159.63/60.08 = [a(x1)] 159.63/60.08 159.63/60.08 [c(c(c(x1)))] = [1] x1 + [6] 159.63/60.08 >= [1] x1 + [6] 159.63/60.08 = [b(x1)] 159.63/60.08 159.63/60.08 [a(x1)] = [1] x1 + [8] 159.63/60.08 >= [1] x1 + [8] 159.63/60.08 = [c(b(x1))] 159.63/60.08 159.63/60.08 159.63/60.08 We return to the main proof. 159.63/60.08 159.63/60.08 We are left with following problem, upon which TcT provides the 159.63/60.08 certificate YES(O(1),O(n^2)). 159.63/60.08 159.63/60.08 Strict Trs: 159.63/60.08 { b(c(x1)) -> a(x1) 159.63/60.08 , c(c(c(x1))) -> b(x1) 159.63/60.08 , a(x1) -> c(b(x1)) } 159.63/60.08 Weak Trs: { b(b(x1)) -> a(c(x1)) } 159.63/60.08 Obligation: 159.63/60.08 derivational complexity 159.63/60.08 Answer: 159.63/60.08 YES(O(1),O(n^2)) 159.63/60.08 159.63/60.08 We use the processor 'matrix interpretation of dimension 1' to 159.63/60.08 orient following rules strictly. 159.63/60.08 159.63/60.08 Trs: { c(c(c(x1))) -> b(x1) } 159.63/60.08 159.63/60.08 The induced complexity on above rules (modulo remaining rules) is 159.63/60.08 YES(?,O(n^1)) . These rules are moved into the corresponding weak 159.63/60.08 component(s). 159.63/60.08 159.63/60.08 Sub-proof: 159.63/60.08 ---------- 159.63/60.08 TcT has computed the following triangular matrix interpretation. 159.63/60.08 159.63/60.08 [b](x1) = [1] x1 + [2] 159.63/60.08 159.63/60.08 [c](x1) = [1] x1 + [1] 159.63/60.08 159.63/60.08 [a](x1) = [1] x1 + [3] 159.63/60.08 159.63/60.08 The order satisfies the following ordering constraints: 159.63/60.08 159.63/60.08 [b(b(x1))] = [1] x1 + [4] 159.63/60.08 >= [1] x1 + [4] 159.63/60.08 = [a(c(x1))] 159.63/60.08 159.63/60.08 [b(c(x1))] = [1] x1 + [3] 159.63/60.08 >= [1] x1 + [3] 159.63/60.08 = [a(x1)] 159.63/60.08 159.63/60.08 [c(c(c(x1)))] = [1] x1 + [3] 159.63/60.08 > [1] x1 + [2] 159.63/60.08 = [b(x1)] 159.63/60.08 159.63/60.08 [a(x1)] = [1] x1 + [3] 159.63/60.08 >= [1] x1 + [3] 159.63/60.08 = [c(b(x1))] 159.63/60.08 159.63/60.08 159.63/60.08 We return to the main proof. 159.63/60.08 159.63/60.08 We are left with following problem, upon which TcT provides the 159.63/60.08 certificate YES(O(1),O(n^2)). 159.63/60.08 159.63/60.08 Strict Trs: 159.63/60.08 { b(c(x1)) -> a(x1) 159.63/60.08 , a(x1) -> c(b(x1)) } 159.63/60.08 Weak Trs: 159.63/60.08 { b(b(x1)) -> a(c(x1)) 159.63/60.08 , c(c(c(x1))) -> b(x1) } 159.63/60.08 Obligation: 159.63/60.08 derivational complexity 159.63/60.08 Answer: 159.63/60.08 YES(O(1),O(n^2)) 159.63/60.08 159.63/60.08 The weightgap principle applies (using the following nonconstant 159.63/60.08 growth matrix-interpretation) 159.63/60.08 159.63/60.08 TcT has computed the following triangular matrix interpretation. 159.63/60.08 Note that the diagonal of the component-wise maxima of 159.63/60.08 interpretation-entries contains no more than 1 non-zero entries. 159.63/60.08 159.63/60.08 [b](x1) = [1] x1 + [1] 159.63/60.08 159.63/60.08 [c](x1) = [1] x1 + [1] 159.63/60.08 159.63/60.08 [a](x1) = [1] x1 + [0] 159.63/60.08 159.63/60.08 The order satisfies the following ordering constraints: 159.63/60.08 159.63/60.08 [b(b(x1))] = [1] x1 + [2] 159.63/60.08 > [1] x1 + [1] 159.63/60.08 = [a(c(x1))] 159.63/60.08 159.63/60.08 [b(c(x1))] = [1] x1 + [2] 159.63/60.08 > [1] x1 + [0] 159.63/60.08 = [a(x1)] 159.63/60.08 159.63/60.08 [c(c(c(x1)))] = [1] x1 + [3] 159.63/60.08 > [1] x1 + [1] 159.63/60.08 = [b(x1)] 159.63/60.08 159.63/60.08 [a(x1)] = [1] x1 + [0] 159.63/60.08 ? [1] x1 + [2] 159.63/60.08 = [c(b(x1))] 159.63/60.08 159.63/60.08 159.63/60.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 159.63/60.08 159.63/60.08 We are left with following problem, upon which TcT provides the 159.63/60.08 certificate YES(O(1),O(n^2)). 159.63/60.08 159.63/60.08 Strict Trs: { a(x1) -> c(b(x1)) } 159.63/60.08 Weak Trs: 159.63/60.08 { b(b(x1)) -> a(c(x1)) 159.63/60.08 , b(c(x1)) -> a(x1) 159.63/60.08 , c(c(c(x1))) -> b(x1) } 159.63/60.08 Obligation: 159.63/60.08 derivational complexity 159.63/60.08 Answer: 159.63/60.08 YES(O(1),O(n^2)) 159.63/60.08 159.63/60.08 We use the processor 'matrix interpretation of dimension 2' to 159.63/60.08 orient following rules strictly. 159.63/60.08 159.63/60.08 Trs: { a(x1) -> c(b(x1)) } 159.63/60.08 159.63/60.08 The induced complexity on above rules (modulo remaining rules) is 159.63/60.08 YES(?,O(n^2)) . These rules are moved into the corresponding weak 159.63/60.08 component(s). 159.63/60.08 159.63/60.08 Sub-proof: 159.63/60.08 ---------- 159.63/60.08 TcT has computed the following triangular matrix interpretation. 159.63/60.08 159.63/60.08 [b](x1) = [1 22] x1 + [4] 159.63/60.08 [0 1] [2] 159.63/60.08 159.63/60.08 [c](x1) = [1 8] x1 + [0] 159.63/60.08 [0 1] [1] 159.63/60.08 159.63/60.08 [a](x1) = [1 30] x1 + [21] 159.63/60.08 [0 1] [3] 159.63/60.08 159.63/60.08 The order satisfies the following ordering constraints: 159.63/60.08 159.63/60.08 [b(b(x1))] = [1 44] x1 + [52] 159.63/60.08 [0 1] [4] 159.63/60.08 > [1 38] x1 + [51] 159.63/60.08 [0 1] [4] 159.63/60.08 = [a(c(x1))] 159.63/60.08 159.63/60.08 [b(c(x1))] = [1 30] x1 + [26] 159.63/60.08 [0 1] [3] 159.63/60.08 > [1 30] x1 + [21] 159.63/60.08 [0 1] [3] 159.63/60.08 = [a(x1)] 159.63/60.08 159.63/60.08 [c(c(c(x1)))] = [1 24] x1 + [24] 159.63/60.08 [0 1] [3] 159.63/60.08 > [1 22] x1 + [4] 159.63/60.08 [0 1] [2] 159.63/60.08 = [b(x1)] 159.63/60.08 159.63/60.08 [a(x1)] = [1 30] x1 + [21] 159.63/60.08 [0 1] [3] 159.63/60.08 > [1 30] x1 + [20] 159.63/60.08 [0 1] [3] 159.63/60.08 = [c(b(x1))] 159.63/60.08 159.63/60.08 159.63/60.08 We return to the main proof. 159.63/60.08 159.63/60.08 We are left with following problem, upon which TcT provides the 159.63/60.08 certificate YES(O(1),O(1)). 159.63/60.08 159.63/60.08 Weak Trs: 159.63/60.08 { b(b(x1)) -> a(c(x1)) 159.63/60.08 , b(c(x1)) -> a(x1) 159.63/60.08 , c(c(c(x1))) -> b(x1) 159.63/60.08 , a(x1) -> c(b(x1)) } 159.63/60.08 Obligation: 159.63/60.08 derivational complexity 159.63/60.08 Answer: 159.63/60.08 YES(O(1),O(1)) 159.63/60.08 159.63/60.08 Empty rules are trivially bounded 159.63/60.08 159.63/60.08 Hurray, we answered YES(O(1),O(n^2)) 159.63/60.09 EOF