YES(O(1),O(n^2)) 159.65/60.08 YES(O(1),O(n^2)) 159.65/60.08 159.65/60.08 We are left with following problem, upon which TcT provides the 159.65/60.08 certificate YES(O(1),O(n^2)). 159.65/60.08 159.65/60.08 Strict Trs: 159.65/60.08 { b(b(b(b(x1)))) -> a(x1) 159.65/60.08 , b(a(x1)) -> a(b(x1)) 159.65/60.08 , a(a(a(x1))) -> b(a(a(b(x1)))) } 159.65/60.08 Obligation: 159.65/60.08 derivational complexity 159.65/60.08 Answer: 159.65/60.08 YES(O(1),O(n^2)) 159.65/60.08 159.65/60.08 We use the processor 'matrix interpretation of dimension 1' to 159.65/60.08 orient following rules strictly. 159.65/60.08 159.65/60.08 Trs: { b(b(b(b(x1)))) -> a(x1) } 159.65/60.08 159.65/60.08 The induced complexity on above rules (modulo remaining rules) is 159.65/60.08 YES(?,O(n^1)) . These rules are moved into the corresponding weak 159.65/60.08 component(s). 159.65/60.08 159.65/60.08 Sub-proof: 159.65/60.08 ---------- 159.65/60.08 TcT has computed the following triangular matrix interpretation. 159.65/60.08 159.65/60.08 [b](x1) = [1] x1 + [16] 159.65/60.08 159.65/60.08 [a](x1) = [1] x1 + [32] 159.65/60.08 159.65/60.08 The order satisfies the following ordering constraints: 159.65/60.08 159.65/60.08 [b(b(b(b(x1))))] = [1] x1 + [64] 159.65/60.08 > [1] x1 + [32] 159.65/60.08 = [a(x1)] 159.65/60.08 159.65/60.08 [b(a(x1))] = [1] x1 + [48] 159.65/60.08 >= [1] x1 + [48] 159.65/60.08 = [a(b(x1))] 159.65/60.08 159.65/60.08 [a(a(a(x1)))] = [1] x1 + [96] 159.65/60.08 >= [1] x1 + [96] 159.65/60.08 = [b(a(a(b(x1))))] 159.65/60.08 159.65/60.08 159.65/60.08 We return to the main proof. 159.65/60.08 159.65/60.08 We are left with following problem, upon which TcT provides the 159.65/60.08 certificate YES(O(1),O(n^2)). 159.65/60.08 159.65/60.08 Strict Trs: 159.65/60.08 { b(a(x1)) -> a(b(x1)) 159.65/60.08 , a(a(a(x1))) -> b(a(a(b(x1)))) } 159.65/60.08 Weak Trs: { b(b(b(b(x1)))) -> a(x1) } 159.65/60.08 Obligation: 159.65/60.08 derivational complexity 159.65/60.08 Answer: 159.65/60.08 YES(O(1),O(n^2)) 159.65/60.08 159.65/60.08 We use the processor 'matrix interpretation of dimension 1' to 159.65/60.08 orient following rules strictly. 159.65/60.08 159.65/60.08 Trs: { a(a(a(x1))) -> b(a(a(b(x1)))) } 159.65/60.08 159.65/60.08 The induced complexity on above rules (modulo remaining rules) is 159.65/60.08 YES(?,O(n^1)) . These rules are moved into the corresponding weak 159.65/60.08 component(s). 159.65/60.08 159.65/60.08 Sub-proof: 159.65/60.08 ---------- 159.65/60.08 TcT has computed the following triangular matrix interpretation. 159.65/60.08 159.65/60.08 [b](x1) = [1] x1 + [16] 159.65/60.08 159.65/60.08 [a](x1) = [1] x1 + [33] 159.65/60.08 159.65/60.08 The order satisfies the following ordering constraints: 159.65/60.08 159.65/60.08 [b(b(b(b(x1))))] = [1] x1 + [64] 159.65/60.08 > [1] x1 + [33] 159.65/60.08 = [a(x1)] 159.65/60.08 159.65/60.08 [b(a(x1))] = [1] x1 + [49] 159.65/60.08 >= [1] x1 + [49] 159.65/60.08 = [a(b(x1))] 159.65/60.08 159.65/60.08 [a(a(a(x1)))] = [1] x1 + [99] 159.65/60.08 > [1] x1 + [98] 159.65/60.08 = [b(a(a(b(x1))))] 159.65/60.08 159.65/60.08 159.65/60.08 We return to the main proof. 159.65/60.08 159.65/60.08 We are left with following problem, upon which TcT provides the 159.65/60.08 certificate YES(O(1),O(n^2)). 159.65/60.08 159.65/60.08 Strict Trs: { b(a(x1)) -> a(b(x1)) } 159.65/60.08 Weak Trs: 159.65/60.08 { b(b(b(b(x1)))) -> a(x1) 159.65/60.08 , a(a(a(x1))) -> b(a(a(b(x1)))) } 159.65/60.08 Obligation: 159.65/60.08 derivational complexity 159.65/60.08 Answer: 159.65/60.08 YES(O(1),O(n^2)) 159.65/60.08 159.65/60.08 We use the processor 'matrix interpretation of dimension 2' to 159.65/60.08 orient following rules strictly. 159.65/60.08 159.65/60.08 Trs: { b(a(x1)) -> a(b(x1)) } 159.65/60.08 159.65/60.08 The induced complexity on above rules (modulo remaining rules) is 159.65/60.08 YES(?,O(n^2)) . These rules are moved into the corresponding weak 159.65/60.08 component(s). 159.65/60.08 159.65/60.08 Sub-proof: 159.65/60.08 ---------- 159.65/60.08 TcT has computed the following triangular matrix interpretation. 159.65/60.08 159.65/60.08 [b](x1) = [1 3] x1 + [0] 159.65/60.08 [0 1] [1] 159.65/60.08 159.65/60.08 [a](x1) = [1 6] x1 + [0] 159.65/60.08 [0 1] [3] 159.65/60.08 159.65/60.08 The order satisfies the following ordering constraints: 159.65/60.08 159.65/60.08 [b(b(b(b(x1))))] = [1 12] x1 + [18] 159.65/60.08 [0 1] [4] 159.65/60.08 > [1 6] x1 + [0] 159.65/60.08 [0 1] [3] 159.65/60.08 = [a(x1)] 159.65/60.08 159.65/60.08 [b(a(x1))] = [1 9] x1 + [9] 159.65/60.08 [0 1] [4] 159.65/60.08 > [1 9] x1 + [6] 159.65/60.08 [0 1] [4] 159.65/60.08 = [a(b(x1))] 159.65/60.08 159.65/60.08 [a(a(a(x1)))] = [1 18] x1 + [54] 159.65/60.08 [0 1] [9] 159.65/60.08 > [1 18] x1 + [51] 159.65/60.08 [0 1] [8] 159.65/60.08 = [b(a(a(b(x1))))] 159.65/60.08 159.65/60.08 159.65/60.08 We return to the main proof. 159.65/60.08 159.65/60.08 We are left with following problem, upon which TcT provides the 159.65/60.08 certificate YES(O(1),O(1)). 159.65/60.08 159.65/60.08 Weak Trs: 159.65/60.08 { b(b(b(b(x1)))) -> a(x1) 159.65/60.08 , b(a(x1)) -> a(b(x1)) 159.65/60.08 , a(a(a(x1))) -> b(a(a(b(x1)))) } 159.65/60.08 Obligation: 159.65/60.08 derivational complexity 159.65/60.08 Answer: 159.65/60.08 YES(O(1),O(1)) 159.65/60.08 159.65/60.08 Empty rules are trivially bounded 159.65/60.08 159.65/60.08 Hurray, we answered YES(O(1),O(n^2)) 159.86/60.11 EOF