YES(O(1),O(n^2)) 170.38/60.07 YES(O(1),O(n^2)) 170.38/60.07 170.38/60.07 We are left with following problem, upon which TcT provides the 170.38/60.07 certificate YES(O(1),O(n^2)). 170.38/60.07 170.38/60.07 Strict Trs: 170.38/60.07 { g(s(x)) -> f(x) 170.38/60.07 , g(0()) -> 0() 170.38/60.07 , f(s(x)) -> s(s(g(x))) 170.38/60.07 , f(0()) -> s(0()) } 170.38/60.07 Obligation: 170.38/60.07 derivational complexity 170.38/60.07 Answer: 170.38/60.07 YES(O(1),O(n^2)) 170.38/60.07 170.38/60.07 We use the processor 'matrix interpretation of dimension 1' to 170.38/60.07 orient following rules strictly. 170.38/60.07 170.38/60.07 Trs: 170.38/60.07 { g(0()) -> 0() 170.38/60.07 , f(0()) -> s(0()) } 170.38/60.07 170.38/60.07 The induced complexity on above rules (modulo remaining rules) is 170.38/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 170.38/60.07 component(s). 170.38/60.07 170.38/60.07 Sub-proof: 170.38/60.07 ---------- 170.38/60.07 TcT has computed the following triangular matrix interpretation. 170.38/60.07 170.38/60.07 [g](x1) = [1] x1 + [1] 170.38/60.07 170.38/60.07 [s](x1) = [1] x1 + [0] 170.38/60.07 170.38/60.07 [f](x1) = [1] x1 + [1] 170.38/60.07 170.38/60.07 [0] = [0] 170.38/60.07 170.38/60.07 The order satisfies the following ordering constraints: 170.38/60.07 170.38/60.07 [g(s(x))] = [1] x + [1] 170.38/60.07 >= [1] x + [1] 170.38/60.07 = [f(x)] 170.38/60.07 170.38/60.07 [g(0())] = [1] 170.38/60.07 > [0] 170.38/60.07 = [0()] 170.38/60.07 170.38/60.07 [f(s(x))] = [1] x + [1] 170.38/60.07 >= [1] x + [1] 170.38/60.07 = [s(s(g(x)))] 170.38/60.07 170.38/60.07 [f(0())] = [1] 170.38/60.07 > [0] 170.38/60.07 = [s(0())] 170.38/60.07 170.38/60.07 170.38/60.07 We return to the main proof. 170.38/60.07 170.38/60.07 We are left with following problem, upon which TcT provides the 170.38/60.07 certificate YES(O(1),O(n^2)). 170.38/60.07 170.38/60.07 Strict Trs: 170.38/60.07 { g(s(x)) -> f(x) 170.38/60.07 , f(s(x)) -> s(s(g(x))) } 170.38/60.07 Weak Trs: 170.38/60.07 { g(0()) -> 0() 170.38/60.07 , f(0()) -> s(0()) } 170.38/60.07 Obligation: 170.38/60.07 derivational complexity 170.38/60.07 Answer: 170.38/60.07 YES(O(1),O(n^2)) 170.38/60.07 170.38/60.07 The weightgap principle applies (using the following nonconstant 170.38/60.07 growth matrix-interpretation) 170.38/60.07 170.38/60.07 TcT has computed the following triangular matrix interpretation. 170.38/60.07 Note that the diagonal of the component-wise maxima of 170.38/60.07 interpretation-entries contains no more than 1 non-zero entries. 170.38/60.07 170.38/60.07 [g](x1) = [1] x1 + [0] 170.38/60.07 170.38/60.07 [s](x1) = [1] x1 + [0] 170.38/60.07 170.38/60.07 [f](x1) = [1] x1 + [1] 170.38/60.07 170.38/60.07 [0] = [0] 170.38/60.07 170.38/60.07 The order satisfies the following ordering constraints: 170.38/60.07 170.38/60.07 [g(s(x))] = [1] x + [0] 170.38/60.07 ? [1] x + [1] 170.38/60.07 = [f(x)] 170.38/60.07 170.38/60.07 [g(0())] = [0] 170.38/60.07 >= [0] 170.38/60.07 = [0()] 170.38/60.07 170.38/60.07 [f(s(x))] = [1] x + [1] 170.38/60.07 > [1] x + [0] 170.38/60.07 = [s(s(g(x)))] 170.38/60.07 170.38/60.07 [f(0())] = [1] 170.38/60.07 > [0] 170.38/60.07 = [s(0())] 170.38/60.07 170.38/60.07 170.38/60.07 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 170.38/60.07 170.38/60.07 We are left with following problem, upon which TcT provides the 170.38/60.07 certificate YES(O(1),O(n^2)). 170.38/60.07 170.38/60.07 Strict Trs: { g(s(x)) -> f(x) } 170.38/60.07 Weak Trs: 170.38/60.07 { g(0()) -> 0() 170.38/60.07 , f(s(x)) -> s(s(g(x))) 170.38/60.07 , f(0()) -> s(0()) } 170.38/60.07 Obligation: 170.38/60.07 derivational complexity 170.38/60.07 Answer: 170.38/60.07 YES(O(1),O(n^2)) 170.38/60.07 170.38/60.07 We use the processor 'matrix interpretation of dimension 2' to 170.38/60.07 orient following rules strictly. 170.38/60.07 170.38/60.07 Trs: { g(s(x)) -> f(x) } 170.38/60.07 170.38/60.07 The induced complexity on above rules (modulo remaining rules) is 170.38/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 170.38/60.07 component(s). 170.38/60.07 170.38/60.07 Sub-proof: 170.38/60.07 ---------- 170.38/60.07 TcT has computed the following triangular matrix interpretation. 170.38/60.07 170.38/60.07 [g](x1) = [1 1] x1 + [0] 170.38/60.07 [0 1] [0] 170.38/60.07 170.38/60.07 [s](x1) = [1 0] x1 + [0] 170.38/60.07 [0 1] [1] 170.38/60.07 170.38/60.07 [f](x1) = [1 1] x1 + [0] 170.38/60.07 [0 1] [1] 170.38/60.07 170.38/60.07 [0] = [0] 170.38/60.07 [0] 170.38/60.07 170.38/60.07 The order satisfies the following ordering constraints: 170.38/60.07 170.38/60.07 [g(s(x))] = [1 1] x + [1] 170.38/60.07 [0 1] [1] 170.38/60.07 > [1 1] x + [0] 170.38/60.07 [0 1] [1] 170.38/60.07 = [f(x)] 170.38/60.07 170.38/60.07 [g(0())] = [0] 170.38/60.07 [0] 170.38/60.07 >= [0] 170.38/60.07 [0] 170.38/60.07 = [0()] 170.38/60.07 170.38/60.07 [f(s(x))] = [1 1] x + [1] 170.38/60.07 [0 1] [2] 170.38/60.07 > [1 1] x + [0] 170.38/60.07 [0 1] [2] 170.38/60.07 = [s(s(g(x)))] 170.38/60.07 170.38/60.07 [f(0())] = [0] 170.38/60.07 [1] 170.38/60.07 >= [0] 170.38/60.07 [1] 170.38/60.07 = [s(0())] 170.38/60.07 170.38/60.07 170.38/60.07 We return to the main proof. 170.38/60.07 170.38/60.07 We are left with following problem, upon which TcT provides the 170.38/60.07 certificate YES(O(1),O(1)). 170.38/60.07 170.38/60.07 Weak Trs: 170.38/60.07 { g(s(x)) -> f(x) 170.38/60.07 , g(0()) -> 0() 170.38/60.07 , f(s(x)) -> s(s(g(x))) 170.38/60.07 , f(0()) -> s(0()) } 170.38/60.07 Obligation: 170.38/60.07 derivational complexity 170.38/60.07 Answer: 170.38/60.07 YES(O(1),O(1)) 170.38/60.07 170.38/60.07 Empty rules are trivially bounded 170.38/60.07 170.38/60.07 Hurray, we answered YES(O(1),O(n^2)) 170.38/60.08 EOF