YES(O(1),O(n^2)) 167.03/60.06 YES(O(1),O(n^2)) 167.03/60.06 167.03/60.06 We are left with following problem, upon which TcT provides the 167.03/60.06 certificate YES(O(1),O(n^2)). 167.03/60.06 167.03/60.06 Strict Trs: 167.03/60.06 { div(X, e()) -> i(X) 167.03/60.06 , div(div(X, Y), Z) -> div(Y, div(i(X), Z)) 167.03/60.06 , i(div(X, Y)) -> div(Y, X) } 167.03/60.06 Obligation: 167.03/60.06 derivational complexity 167.03/60.06 Answer: 167.03/60.06 YES(O(1),O(n^2)) 167.03/60.06 167.03/60.06 We use the processor 'matrix interpretation of dimension 1' to 167.03/60.06 orient following rules strictly. 167.03/60.06 167.03/60.06 Trs: { div(X, e()) -> i(X) } 167.03/60.06 167.03/60.06 The induced complexity on above rules (modulo remaining rules) is 167.03/60.06 YES(?,O(n^1)) . These rules are moved into the corresponding weak 167.03/60.06 component(s). 167.03/60.06 167.03/60.06 Sub-proof: 167.03/60.06 ---------- 167.03/60.06 TcT has computed the following triangular matrix interpretation. 167.03/60.06 167.03/60.06 [div](x1, x2) = [1] x1 + [1] x2 + [2] 167.03/60.06 167.03/60.06 [e] = [1] 167.03/60.06 167.03/60.06 [i](x1) = [1] x1 + [0] 167.03/60.06 167.03/60.06 The order satisfies the following ordering constraints: 167.03/60.06 167.03/60.06 [div(X, e())] = [1] X + [3] 167.03/60.06 > [1] X + [0] 167.03/60.06 = [i(X)] 167.03/60.06 167.03/60.06 [div(div(X, Y), Z)] = [1] X + [1] Y + [1] Z + [4] 167.03/60.06 >= [1] X + [1] Y + [1] Z + [4] 167.03/60.06 = [div(Y, div(i(X), Z))] 167.03/60.06 167.03/60.06 [i(div(X, Y))] = [1] X + [1] Y + [2] 167.03/60.06 >= [1] X + [1] Y + [2] 167.03/60.06 = [div(Y, X)] 167.03/60.06 167.03/60.06 167.03/60.06 We return to the main proof. 167.03/60.06 167.03/60.06 We are left with following problem, upon which TcT provides the 167.03/60.06 certificate YES(O(1),O(n^2)). 167.03/60.06 167.03/60.06 Strict Trs: 167.03/60.06 { div(div(X, Y), Z) -> div(Y, div(i(X), Z)) 167.03/60.06 , i(div(X, Y)) -> div(Y, X) } 167.03/60.06 Weak Trs: { div(X, e()) -> i(X) } 167.03/60.06 Obligation: 167.03/60.06 derivational complexity 167.03/60.06 Answer: 167.03/60.06 YES(O(1),O(n^2)) 167.03/60.06 167.03/60.06 The weightgap principle applies (using the following nonconstant 167.03/60.06 growth matrix-interpretation) 167.03/60.06 167.03/60.06 TcT has computed the following triangular matrix interpretation. 167.03/60.06 Note that the diagonal of the component-wise maxima of 167.03/60.06 interpretation-entries contains no more than 1 non-zero entries. 167.03/60.06 167.03/60.06 [div](x1, x2) = [1] x1 + [1] x2 + [0] 167.03/60.06 167.03/60.06 [e] = [2] 167.03/60.06 167.03/60.06 [i](x1) = [1] x1 + [1] 167.03/60.06 167.03/60.06 The order satisfies the following ordering constraints: 167.03/60.06 167.03/60.06 [div(X, e())] = [1] X + [2] 167.03/60.06 > [1] X + [1] 167.03/60.06 = [i(X)] 167.03/60.06 167.03/60.06 [div(div(X, Y), Z)] = [1] X + [1] Y + [1] Z + [0] 167.03/60.06 ? [1] X + [1] Y + [1] Z + [1] 167.03/60.06 = [div(Y, div(i(X), Z))] 167.03/60.06 167.03/60.06 [i(div(X, Y))] = [1] X + [1] Y + [1] 167.03/60.06 > [1] X + [1] Y + [0] 167.03/60.06 = [div(Y, X)] 167.03/60.06 167.03/60.06 167.03/60.06 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 167.03/60.06 167.03/60.06 We are left with following problem, upon which TcT provides the 167.03/60.06 certificate YES(O(1),O(n^2)). 167.03/60.06 167.03/60.06 Strict Trs: { div(div(X, Y), Z) -> div(Y, div(i(X), Z)) } 167.03/60.06 Weak Trs: 167.03/60.06 { div(X, e()) -> i(X) 167.03/60.06 , i(div(X, Y)) -> div(Y, X) } 167.03/60.06 Obligation: 167.03/60.06 derivational complexity 167.03/60.06 Answer: 167.03/60.06 YES(O(1),O(n^2)) 167.03/60.06 167.03/60.06 We use the processor 'matrix interpretation of dimension 2' to 167.03/60.06 orient following rules strictly. 167.03/60.06 167.03/60.06 Trs: { div(div(X, Y), Z) -> div(Y, div(i(X), Z)) } 167.03/60.06 167.03/60.06 The induced complexity on above rules (modulo remaining rules) is 167.03/60.06 YES(?,O(n^2)) . These rules are moved into the corresponding weak 167.03/60.06 component(s). 167.03/60.06 167.03/60.06 Sub-proof: 167.03/60.06 ---------- 167.03/60.06 TcT has computed the following triangular matrix interpretation. 167.03/60.06 167.03/60.06 [div](x1, x2) = [1 1] x1 + [1 0] x2 + [0] 167.03/60.06 [0 1] [0 1] [1] 167.03/60.06 167.03/60.06 [e] = [0] 167.03/60.06 [0] 167.03/60.06 167.03/60.06 [i](x1) = [1 1] x1 + [0] 167.03/60.06 [0 1] [0] 167.03/60.06 167.03/60.06 The order satisfies the following ordering constraints: 167.03/60.06 167.03/60.06 [div(X, e())] = [1 1] X + [0] 167.03/60.06 [0 1] [1] 167.03/60.06 >= [1 1] X + [0] 167.03/60.06 [0 1] [0] 167.03/60.06 = [i(X)] 167.03/60.06 167.03/60.06 [div(div(X, Y), Z)] = [1 2] X + [1 1] Y + [1 0] Z + [1] 167.03/60.06 [0 1] [0 1] [0 1] [2] 167.03/60.06 > [1 2] X + [1 1] Y + [1 0] Z + [0] 167.03/60.06 [0 1] [0 1] [0 1] [2] 167.03/60.06 = [div(Y, div(i(X), Z))] 167.03/60.06 167.03/60.06 [i(div(X, Y))] = [1 2] X + [1 1] Y + [1] 167.03/60.06 [0 1] [0 1] [1] 167.03/60.06 > [1 0] X + [1 1] Y + [0] 167.03/60.06 [0 1] [0 1] [1] 167.03/60.06 = [div(Y, X)] 167.03/60.06 167.03/60.06 167.03/60.06 We return to the main proof. 167.03/60.06 167.03/60.06 We are left with following problem, upon which TcT provides the 167.03/60.06 certificate YES(O(1),O(1)). 167.03/60.06 167.03/60.06 Weak Trs: 167.03/60.06 { div(X, e()) -> i(X) 167.03/60.06 , div(div(X, Y), Z) -> div(Y, div(i(X), Z)) 167.03/60.06 , i(div(X, Y)) -> div(Y, X) } 167.03/60.06 Obligation: 167.03/60.06 derivational complexity 167.03/60.06 Answer: 167.03/60.06 YES(O(1),O(1)) 167.03/60.06 167.03/60.06 Empty rules are trivially bounded 167.03/60.06 167.03/60.06 Hurray, we answered YES(O(1),O(n^2)) 167.24/60.10 EOF