YES(O(1),O(n^1)) 165.07/60.08 YES(O(1),O(n^1)) 165.07/60.08 165.07/60.08 We are left with following problem, upon which TcT provides the 165.07/60.08 certificate YES(O(1),O(n^1)). 165.07/60.08 165.07/60.08 Strict Trs: 165.07/60.08 { a(a(a(x1))) -> b(b(x1)) 165.07/60.08 , b(b(b(x1))) -> c(d(x1)) 165.07/60.08 , c(x1) -> a(a(x1)) 165.07/60.08 , d(x1) -> c(x1) } 165.07/60.08 Obligation: 165.07/60.08 derivational complexity 165.07/60.08 Answer: 165.07/60.08 YES(O(1),O(n^1)) 165.07/60.08 165.07/60.08 The weightgap principle applies (using the following nonconstant 165.07/60.08 growth matrix-interpretation) 165.07/60.08 165.07/60.08 TcT has computed the following triangular matrix interpretation. 165.07/60.08 Note that the diagonal of the component-wise maxima of 165.07/60.08 interpretation-entries contains no more than 1 non-zero entries. 165.07/60.08 165.07/60.08 [a](x1) = [1] x1 + [0] 165.07/60.08 165.07/60.08 [b](x1) = [1] x1 + [0] 165.07/60.08 165.07/60.08 [c](x1) = [1] x1 + [2] 165.07/60.08 165.07/60.08 [d](x1) = [1] x1 + [2] 165.07/60.08 165.07/60.08 The order satisfies the following ordering constraints: 165.07/60.08 165.07/60.08 [a(a(a(x1)))] = [1] x1 + [0] 165.07/60.08 >= [1] x1 + [0] 165.07/60.08 = [b(b(x1))] 165.07/60.08 165.07/60.08 [b(b(b(x1)))] = [1] x1 + [0] 165.07/60.08 ? [1] x1 + [4] 165.07/60.08 = [c(d(x1))] 165.07/60.08 165.07/60.08 [c(x1)] = [1] x1 + [2] 165.07/60.08 > [1] x1 + [0] 165.07/60.08 = [a(a(x1))] 165.07/60.08 165.07/60.08 [d(x1)] = [1] x1 + [2] 165.07/60.08 >= [1] x1 + [2] 165.07/60.08 = [c(x1)] 165.07/60.08 165.07/60.08 165.07/60.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 165.07/60.08 165.07/60.08 We are left with following problem, upon which TcT provides the 165.07/60.08 certificate YES(O(1),O(n^1)). 165.07/60.08 165.07/60.08 Strict Trs: 165.07/60.08 { a(a(a(x1))) -> b(b(x1)) 165.07/60.08 , b(b(b(x1))) -> c(d(x1)) 165.07/60.08 , d(x1) -> c(x1) } 165.07/60.08 Weak Trs: { c(x1) -> a(a(x1)) } 165.07/60.08 Obligation: 165.07/60.08 derivational complexity 165.07/60.08 Answer: 165.07/60.08 YES(O(1),O(n^1)) 165.07/60.08 165.07/60.08 We use the processor 'matrix interpretation of dimension 1' to 165.07/60.08 orient following rules strictly. 165.07/60.08 165.07/60.08 Trs: 165.07/60.08 { a(a(a(x1))) -> b(b(x1)) 165.07/60.08 , b(b(b(x1))) -> c(d(x1)) 165.07/60.08 , d(x1) -> c(x1) } 165.07/60.08 165.07/60.08 The induced complexity on above rules (modulo remaining rules) is 165.07/60.08 YES(?,O(n^1)) . These rules are moved into the corresponding weak 165.07/60.08 component(s). 165.07/60.08 165.07/60.08 Sub-proof: 165.07/60.08 ---------- 165.07/60.08 TcT has computed the following triangular matrix interpretation. 165.07/60.08 165.07/60.08 [a](x1) = [1] x1 + [15] 165.07/60.08 165.07/60.08 [b](x1) = [1] x1 + [22] 165.07/60.08 165.07/60.08 [c](x1) = [1] x1 + [31] 165.07/60.08 165.07/60.08 [d](x1) = [1] x1 + [32] 165.07/60.08 165.07/60.08 The order satisfies the following ordering constraints: 165.07/60.08 165.07/60.08 [a(a(a(x1)))] = [1] x1 + [45] 165.07/60.08 > [1] x1 + [44] 165.07/60.08 = [b(b(x1))] 165.07/60.08 165.07/60.08 [b(b(b(x1)))] = [1] x1 + [66] 165.07/60.08 > [1] x1 + [63] 165.07/60.08 = [c(d(x1))] 165.07/60.08 165.07/60.08 [c(x1)] = [1] x1 + [31] 165.07/60.08 > [1] x1 + [30] 165.07/60.08 = [a(a(x1))] 165.07/60.08 165.07/60.08 [d(x1)] = [1] x1 + [32] 165.07/60.08 > [1] x1 + [31] 165.07/60.08 = [c(x1)] 165.07/60.08 165.07/60.08 165.07/60.08 We return to the main proof. 165.07/60.08 165.07/60.08 We are left with following problem, upon which TcT provides the 165.07/60.08 certificate YES(O(1),O(1)). 165.07/60.08 165.07/60.08 Weak Trs: 165.07/60.08 { a(a(a(x1))) -> b(b(x1)) 165.07/60.08 , b(b(b(x1))) -> c(d(x1)) 165.07/60.08 , c(x1) -> a(a(x1)) 165.07/60.08 , d(x1) -> c(x1) } 165.07/60.08 Obligation: 165.07/60.08 derivational complexity 165.07/60.08 Answer: 165.07/60.08 YES(O(1),O(1)) 165.07/60.08 165.07/60.08 Empty rules are trivially bounded 165.07/60.08 165.07/60.08 Hurray, we answered YES(O(1),O(n^1)) 165.28/60.15 EOF