YES(O(1),O(n^2)) 296.15/100.46 YES(O(1),O(n^2)) 296.15/100.46 296.15/100.46 We are left with following problem, upon which TcT provides the 296.15/100.46 certificate YES(O(1),O(n^2)). 296.15/100.46 296.15/100.46 Strict Trs: 296.15/100.46 { a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y)))))) 296.15/100.46 , f(a(x), a(y)) -> a(f(x, y)) 296.15/100.46 , f(b(x), b(y)) -> b(f(x, y)) } 296.15/100.46 Obligation: 296.15/100.46 derivational complexity 296.15/100.46 Answer: 296.15/100.46 YES(O(1),O(n^2)) 296.15/100.46 296.15/100.46 The weightgap principle applies (using the following nonconstant 296.15/100.46 growth matrix-interpretation) 296.15/100.46 296.15/100.46 TcT has computed the following triangular matrix interpretation. 296.15/100.46 Note that the diagonal of the component-wise maxima of 296.15/100.46 interpretation-entries contains no more than 1 non-zero entries. 296.15/100.46 296.15/100.46 [a](x1) = [1] x1 + [0] 296.15/100.46 296.15/100.46 [f](x1, x2) = [1] x1 + [1] x2 + [0] 296.15/100.46 296.15/100.46 [b](x1) = [1] x1 + [1] 296.15/100.46 296.15/100.46 The order satisfies the following ordering constraints: 296.15/100.46 296.15/100.46 [a(a(f(x, y)))] = [1] x + [1] y + [0] 296.15/100.46 ? [1] x + [1] y + [4] 296.15/100.46 = [f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))] 296.15/100.46 296.15/100.46 [f(a(x), a(y))] = [1] x + [1] y + [0] 296.15/100.46 >= [1] x + [1] y + [0] 296.15/100.46 = [a(f(x, y))] 296.15/100.46 296.15/100.46 [f(b(x), b(y))] = [1] x + [1] y + [2] 296.15/100.46 > [1] x + [1] y + [1] 296.15/100.46 = [b(f(x, y))] 296.15/100.46 296.15/100.46 296.15/100.46 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 296.15/100.46 296.15/100.46 We are left with following problem, upon which TcT provides the 296.15/100.46 certificate YES(O(1),O(n^2)). 296.15/100.46 296.15/100.46 Strict Trs: 296.15/100.46 { a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y)))))) 296.15/100.46 , f(a(x), a(y)) -> a(f(x, y)) } 296.15/100.46 Weak Trs: { f(b(x), b(y)) -> b(f(x, y)) } 296.15/100.46 Obligation: 296.15/100.46 derivational complexity 296.15/100.46 Answer: 296.15/100.46 YES(O(1),O(n^2)) 296.15/100.46 296.15/100.46 The weightgap principle applies (using the following nonconstant 296.15/100.46 growth matrix-interpretation) 296.15/100.46 296.15/100.46 TcT has computed the following triangular matrix interpretation. 296.15/100.46 Note that the diagonal of the component-wise maxima of 296.15/100.46 interpretation-entries contains no more than 1 non-zero entries. 296.15/100.46 296.15/100.46 [a](x1) = [1] x1 + [1] 296.15/100.46 296.15/100.46 [f](x1, x2) = [1] x1 + [1] x2 + [0] 296.15/100.46 296.15/100.46 [b](x1) = [1] x1 + [0] 296.15/100.46 296.15/100.46 The order satisfies the following ordering constraints: 296.15/100.46 296.15/100.46 [a(a(f(x, y)))] = [1] x + [1] y + [2] 296.15/100.46 ? [1] x + [1] y + [6] 296.15/100.46 = [f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))] 296.15/100.46 296.15/100.46 [f(a(x), a(y))] = [1] x + [1] y + [2] 296.15/100.46 > [1] x + [1] y + [1] 296.15/100.46 = [a(f(x, y))] 296.15/100.46 296.15/100.46 [f(b(x), b(y))] = [1] x + [1] y + [0] 296.15/100.46 >= [1] x + [1] y + [0] 296.15/100.46 = [b(f(x, y))] 296.15/100.46 296.15/100.46 296.15/100.46 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 296.15/100.46 296.15/100.46 We are left with following problem, upon which TcT provides the 296.15/100.46 certificate YES(O(1),O(n^2)). 296.15/100.46 296.15/100.46 Strict Trs: 296.15/100.46 { a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y)))))) } 296.15/100.46 Weak Trs: 296.15/100.46 { f(a(x), a(y)) -> a(f(x, y)) 296.15/100.46 , f(b(x), b(y)) -> b(f(x, y)) } 296.15/100.46 Obligation: 296.15/100.46 derivational complexity 296.15/100.46 Answer: 296.15/100.46 YES(O(1),O(n^2)) 296.15/100.46 296.15/100.46 We use the processor 'matrix interpretation of dimension 2' to 296.15/100.46 orient following rules strictly. 296.15/100.46 296.15/100.46 Trs: { a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y)))))) } 296.15/100.46 296.15/100.46 The induced complexity on above rules (modulo remaining rules) is 296.15/100.46 YES(?,O(n^2)) . These rules are moved into the corresponding weak 296.15/100.46 component(s). 296.15/100.46 296.15/100.46 Sub-proof: 296.15/100.46 ---------- 296.15/100.46 TcT has computed the following triangular matrix interpretation. 296.15/100.46 296.15/100.46 [a](x1) = [1 1] x1 + [0] 296.15/100.46 [0 1] [1] 296.15/100.46 296.15/100.46 [f](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 296.15/100.46 [0 1] [0 1] [0] 296.15/100.46 296.15/100.46 [b](x1) = [1 0] x1 + [0] 296.15/100.46 [0 0] [0] 296.15/100.46 296.15/100.46 The order satisfies the following ordering constraints: 296.15/100.46 296.15/100.46 [a(a(f(x, y)))] = [1 2] x + [1 2] y + [1] 296.15/100.46 [0 1] [0 1] [2] 296.15/100.46 > [1 1] x + [1 1] y + [0] 296.15/100.46 [0 0] [0 0] [2] 296.15/100.46 = [f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))] 296.15/100.46 296.15/100.46 [f(a(x), a(y))] = [1 1] x + [1 1] y + [0] 296.15/100.46 [0 1] [0 1] [2] 296.15/100.46 >= [1 1] x + [1 1] y + [0] 296.15/100.46 [0 1] [0 1] [1] 296.15/100.46 = [a(f(x, y))] 296.15/100.46 296.15/100.46 [f(b(x), b(y))] = [1 0] x + [1 0] y + [0] 296.15/100.46 [0 0] [0 0] [0] 296.15/100.46 >= [1 0] x + [1 0] y + [0] 296.15/100.46 [0 0] [0 0] [0] 296.15/100.46 = [b(f(x, y))] 296.15/100.46 296.15/100.46 296.15/100.46 We return to the main proof. 296.15/100.46 296.15/100.46 We are left with following problem, upon which TcT provides the 296.15/100.46 certificate YES(O(1),O(1)). 296.15/100.46 296.15/100.46 Weak Trs: 296.15/100.46 { a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y)))))) 296.15/100.46 , f(a(x), a(y)) -> a(f(x, y)) 296.15/100.46 , f(b(x), b(y)) -> b(f(x, y)) } 296.15/100.46 Obligation: 296.15/100.46 derivational complexity 296.15/100.46 Answer: 296.15/100.46 YES(O(1),O(1)) 296.15/100.46 296.15/100.46 Empty rules are trivially bounded 296.15/100.46 296.15/100.46 Hurray, we answered YES(O(1),O(n^2)) 296.43/100.51 EOF