YES(O(1),O(n^2)) 176.24/60.07 YES(O(1),O(n^2)) 176.24/60.07 176.24/60.07 We are left with following problem, upon which TcT provides the 176.24/60.07 certificate YES(O(1),O(n^2)). 176.24/60.07 176.24/60.07 Strict Trs: 176.24/60.07 { g(x, s(y)) -> g(f(x, y), 0()) 176.24/60.07 , g(0(), f(x, x)) -> x 176.24/60.07 , g(f(x, y), 0()) -> f(g(x, 0()), g(y, 0())) 176.24/60.07 , g(s(x), y) -> g(f(x, y), 0()) } 176.24/60.07 Obligation: 176.24/60.07 derivational complexity 176.24/60.07 Answer: 176.24/60.07 YES(O(1),O(n^2)) 176.24/60.07 176.24/60.07 We use the processor 'matrix interpretation of dimension 1' to 176.24/60.07 orient following rules strictly. 176.24/60.07 176.24/60.07 Trs: 176.24/60.07 { g(x, s(y)) -> g(f(x, y), 0()) 176.24/60.07 , g(s(x), y) -> g(f(x, y), 0()) } 176.24/60.07 176.24/60.07 The induced complexity on above rules (modulo remaining rules) is 176.24/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 176.24/60.07 component(s). 176.24/60.07 176.24/60.07 Sub-proof: 176.24/60.07 ---------- 176.24/60.07 TcT has computed the following triangular matrix interpretation. 176.24/60.07 176.24/60.07 [g](x1, x2) = [1] x1 + [1] x2 + [0] 176.24/60.07 176.24/60.07 [0] = [0] 176.24/60.07 176.24/60.07 [f](x1, x2) = [1] x1 + [1] x2 + [0] 176.24/60.07 176.24/60.07 [s](x1) = [1] x1 + [2] 176.24/60.07 176.24/60.07 The order satisfies the following ordering constraints: 176.24/60.07 176.24/60.07 [g(x, s(y))] = [1] x + [1] y + [2] 176.24/60.07 > [1] x + [1] y + [0] 176.24/60.07 = [g(f(x, y), 0())] 176.24/60.07 176.24/60.07 [g(0(), f(x, x))] = [2] x + [0] 176.24/60.07 >= [1] x + [0] 176.24/60.07 = [x] 176.24/60.07 176.24/60.07 [g(f(x, y), 0())] = [1] x + [1] y + [0] 176.24/60.07 >= [1] x + [1] y + [0] 176.24/60.07 = [f(g(x, 0()), g(y, 0()))] 176.24/60.07 176.24/60.07 [g(s(x), y)] = [1] x + [1] y + [2] 176.24/60.07 > [1] x + [1] y + [0] 176.24/60.07 = [g(f(x, y), 0())] 176.24/60.07 176.24/60.07 176.24/60.07 We return to the main proof. 176.24/60.07 176.24/60.07 We are left with following problem, upon which TcT provides the 176.24/60.07 certificate YES(O(1),O(n^2)). 176.24/60.07 176.24/60.07 Strict Trs: 176.24/60.07 { g(0(), f(x, x)) -> x 176.24/60.07 , g(f(x, y), 0()) -> f(g(x, 0()), g(y, 0())) } 176.24/60.07 Weak Trs: 176.24/60.07 { g(x, s(y)) -> g(f(x, y), 0()) 176.24/60.07 , g(s(x), y) -> g(f(x, y), 0()) } 176.24/60.07 Obligation: 176.24/60.07 derivational complexity 176.24/60.07 Answer: 176.24/60.07 YES(O(1),O(n^2)) 176.24/60.07 176.24/60.07 We use the processor 'matrix interpretation of dimension 1' to 176.24/60.07 orient following rules strictly. 176.24/60.07 176.24/60.07 Trs: { g(0(), f(x, x)) -> x } 176.24/60.07 176.24/60.07 The induced complexity on above rules (modulo remaining rules) is 176.24/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 176.24/60.07 component(s). 176.24/60.07 176.24/60.07 Sub-proof: 176.24/60.07 ---------- 176.24/60.07 TcT has computed the following triangular matrix interpretation. 176.24/60.07 176.24/60.07 [g](x1, x2) = [1] x1 + [1] x2 + [0] 176.24/60.07 176.24/60.07 [0] = [0] 176.24/60.07 176.24/60.07 [f](x1, x2) = [1] x1 + [1] x2 + [2] 176.24/60.07 176.24/60.07 [s](x1) = [1] x1 + [2] 176.24/60.07 176.24/60.07 The order satisfies the following ordering constraints: 176.24/60.07 176.24/60.07 [g(x, s(y))] = [1] x + [1] y + [2] 176.24/60.07 >= [1] x + [1] y + [2] 176.24/60.07 = [g(f(x, y), 0())] 176.24/60.07 176.24/60.07 [g(0(), f(x, x))] = [2] x + [2] 176.24/60.07 > [1] x + [0] 176.24/60.07 = [x] 176.24/60.07 176.24/60.07 [g(f(x, y), 0())] = [1] x + [1] y + [2] 176.24/60.07 >= [1] x + [1] y + [2] 176.24/60.07 = [f(g(x, 0()), g(y, 0()))] 176.24/60.07 176.24/60.07 [g(s(x), y)] = [1] x + [1] y + [2] 176.24/60.07 >= [1] x + [1] y + [2] 176.24/60.07 = [g(f(x, y), 0())] 176.24/60.07 176.24/60.07 176.24/60.07 We return to the main proof. 176.24/60.07 176.24/60.07 We are left with following problem, upon which TcT provides the 176.24/60.07 certificate YES(O(1),O(n^2)). 176.24/60.07 176.24/60.07 Strict Trs: { g(f(x, y), 0()) -> f(g(x, 0()), g(y, 0())) } 176.24/60.07 Weak Trs: 176.24/60.07 { g(x, s(y)) -> g(f(x, y), 0()) 176.24/60.07 , g(0(), f(x, x)) -> x 176.24/60.07 , g(s(x), y) -> g(f(x, y), 0()) } 176.24/60.07 Obligation: 176.24/60.07 derivational complexity 176.24/60.07 Answer: 176.24/60.07 YES(O(1),O(n^2)) 176.24/60.07 176.24/60.07 We use the processor 'matrix interpretation of dimension 2' to 176.24/60.07 orient following rules strictly. 176.24/60.07 176.24/60.07 Trs: { g(f(x, y), 0()) -> f(g(x, 0()), g(y, 0())) } 176.24/60.07 176.24/60.07 The induced complexity on above rules (modulo remaining rules) is 176.24/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 176.24/60.07 component(s). 176.24/60.07 176.24/60.07 Sub-proof: 176.24/60.07 ---------- 176.24/60.07 TcT has computed the following triangular matrix interpretation. 176.24/60.07 176.24/60.07 [g](x1, x2) = [1 1] x1 + [1 1] x2 + [0] 176.24/60.07 [0 1] [0 1] [0] 176.24/60.07 176.24/60.07 [0] = [0] 176.24/60.07 [0] 176.24/60.07 176.24/60.07 [f](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 176.24/60.07 [0 1] [0 1] [1] 176.24/60.07 176.24/60.07 [s](x1) = [1 0] x1 + [0] 176.24/60.07 [0 1] [1] 176.24/60.07 176.24/60.07 The order satisfies the following ordering constraints: 176.24/60.07 176.24/60.07 [g(x, s(y))] = [1 1] x + [1 1] y + [1] 176.24/60.07 [0 1] [0 1] [1] 176.24/60.07 >= [1 1] x + [1 1] y + [1] 176.24/60.07 [0 1] [0 1] [1] 176.24/60.07 = [g(f(x, y), 0())] 176.24/60.07 176.24/60.07 [g(0(), f(x, x))] = [2 2] x + [1] 176.24/60.07 [0 2] [1] 176.24/60.07 > [1 0] x + [0] 176.24/60.07 [0 1] [0] 176.24/60.07 = [x] 176.24/60.07 176.24/60.07 [g(f(x, y), 0())] = [1 1] x + [1 1] y + [1] 176.24/60.07 [0 1] [0 1] [1] 176.24/60.07 > [1 1] x + [1 1] y + [0] 176.24/60.07 [0 1] [0 1] [1] 176.24/60.07 = [f(g(x, 0()), g(y, 0()))] 176.24/60.07 176.24/60.07 [g(s(x), y)] = [1 1] x + [1 1] y + [1] 176.24/60.07 [0 1] [0 1] [1] 176.24/60.07 >= [1 1] x + [1 1] y + [1] 176.24/60.07 [0 1] [0 1] [1] 176.24/60.07 = [g(f(x, y), 0())] 176.24/60.07 176.24/60.07 176.24/60.07 We return to the main proof. 176.24/60.07 176.24/60.07 We are left with following problem, upon which TcT provides the 176.24/60.07 certificate YES(O(1),O(1)). 176.24/60.07 176.24/60.07 Weak Trs: 176.24/60.07 { g(x, s(y)) -> g(f(x, y), 0()) 176.24/60.07 , g(0(), f(x, x)) -> x 176.24/60.07 , g(f(x, y), 0()) -> f(g(x, 0()), g(y, 0())) 176.24/60.07 , g(s(x), y) -> g(f(x, y), 0()) } 176.24/60.07 Obligation: 176.24/60.07 derivational complexity 176.24/60.07 Answer: 176.24/60.07 YES(O(1),O(1)) 176.24/60.07 176.24/60.07 Empty rules are trivially bounded 176.24/60.07 176.24/60.07 Hurray, we answered YES(O(1),O(n^2)) 176.24/60.08 EOF