YES(O(1),O(n^2)) 164.64/60.04 YES(O(1),O(n^2)) 164.64/60.04 164.64/60.04 We are left with following problem, upon which TcT provides the 164.64/60.04 certificate YES(O(1),O(n^2)). 164.64/60.04 164.64/60.04 Strict Trs: 164.64/60.04 { g(x, h(y, z)) -> h(g(x, y), z) 164.64/60.04 , g(f(x, y), z) -> f(x, g(y, z)) 164.64/60.04 , g(h(x, y), z) -> g(x, f(y, z)) } 164.64/60.04 Obligation: 164.64/60.04 derivational complexity 164.64/60.04 Answer: 164.64/60.04 YES(O(1),O(n^2)) 164.64/60.04 164.64/60.04 We use the processor 'matrix interpretation of dimension 1' to 164.64/60.04 orient following rules strictly. 164.64/60.04 164.64/60.04 Trs: { g(h(x, y), z) -> g(x, f(y, z)) } 164.64/60.04 164.64/60.04 The induced complexity on above rules (modulo remaining rules) is 164.64/60.04 YES(?,O(n^1)) . These rules are moved into the corresponding weak 164.64/60.04 component(s). 164.64/60.04 164.64/60.04 Sub-proof: 164.64/60.04 ---------- 164.64/60.04 TcT has computed the following triangular matrix interpretation. 164.64/60.04 164.64/60.04 [g](x1, x2) = [1] x1 + [1] x2 + [0] 164.64/60.04 164.64/60.04 [f](x1, x2) = [1] x1 + [1] x2 + [0] 164.64/60.04 164.64/60.04 [h](x1, x2) = [1] x1 + [1] x2 + [1] 164.64/60.04 164.64/60.04 The order satisfies the following ordering constraints: 164.64/60.04 164.64/60.04 [g(x, h(y, z))] = [1] x + [1] y + [1] z + [1] 164.64/60.04 >= [1] x + [1] y + [1] z + [1] 164.64/60.04 = [h(g(x, y), z)] 164.64/60.04 164.64/60.04 [g(f(x, y), z)] = [1] x + [1] y + [1] z + [0] 164.64/60.04 >= [1] x + [1] y + [1] z + [0] 164.64/60.04 = [f(x, g(y, z))] 164.64/60.04 164.64/60.04 [g(h(x, y), z)] = [1] x + [1] y + [1] z + [1] 164.64/60.04 > [1] x + [1] y + [1] z + [0] 164.64/60.04 = [g(x, f(y, z))] 164.64/60.04 164.64/60.04 164.64/60.04 We return to the main proof. 164.64/60.04 164.64/60.04 We are left with following problem, upon which TcT provides the 164.64/60.04 certificate YES(O(1),O(n^2)). 164.64/60.04 164.64/60.04 Strict Trs: 164.64/60.04 { g(x, h(y, z)) -> h(g(x, y), z) 164.64/60.04 , g(f(x, y), z) -> f(x, g(y, z)) } 164.64/60.04 Weak Trs: { g(h(x, y), z) -> g(x, f(y, z)) } 164.64/60.04 Obligation: 164.64/60.04 derivational complexity 164.64/60.04 Answer: 164.64/60.04 YES(O(1),O(n^2)) 164.64/60.04 164.64/60.04 We use the processor 'matrix interpretation of dimension 2' to 164.64/60.04 orient following rules strictly. 164.64/60.04 164.64/60.04 Trs: { g(f(x, y), z) -> f(x, g(y, z)) } 164.64/60.04 164.64/60.04 The induced complexity on above rules (modulo remaining rules) is 164.64/60.04 YES(?,O(n^2)) . These rules are moved into the corresponding weak 164.64/60.04 component(s). 164.64/60.04 164.64/60.04 Sub-proof: 164.64/60.04 ---------- 164.64/60.04 TcT has computed the following triangular matrix interpretation. 164.64/60.04 164.64/60.04 [g](x1, x2) = [1 1] x1 + [1 0] x2 + [0] 164.64/60.04 [0 1] [0 0] [0] 164.64/60.04 164.64/60.04 [f](x1, x2) = [1 1] x1 + [1 0] x2 + [0] 164.64/60.04 [0 1] [0 1] [1] 164.64/60.04 164.64/60.04 [h](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 164.64/60.04 [0 1] [0 0] [0] 164.64/60.04 164.64/60.04 The order satisfies the following ordering constraints: 164.64/60.04 164.64/60.04 [g(x, h(y, z))] = [1 1] x + [1 0] y + [1 2] z + [0] 164.64/60.04 [0 1] [0 0] [0 0] [0] 164.64/60.04 >= [1 1] x + [1 0] y + [1 2] z + [0] 164.64/60.04 [0 1] [0 0] [0 0] [0] 164.64/60.04 = [h(g(x, y), z)] 164.64/60.04 164.64/60.04 [g(f(x, y), z)] = [1 2] x + [1 1] y + [1 0] z + [1] 164.64/60.04 [0 1] [0 1] [0 0] [1] 164.64/60.04 > [1 1] x + [1 1] y + [1 0] z + [0] 164.64/60.04 [0 1] [0 1] [0 0] [1] 164.64/60.04 = [f(x, g(y, z))] 164.64/60.04 164.64/60.04 [g(h(x, y), z)] = [1 1] x + [1 2] y + [1 0] z + [0] 164.64/60.04 [0 1] [0 0] [0 0] [0] 164.64/60.04 >= [1 1] x + [1 1] y + [1 0] z + [0] 164.64/60.04 [0 1] [0 0] [0 0] [0] 164.64/60.04 = [g(x, f(y, z))] 164.64/60.04 164.64/60.04 164.64/60.04 We return to the main proof. 164.64/60.04 164.64/60.04 We are left with following problem, upon which TcT provides the 164.64/60.04 certificate YES(O(1),O(n^2)). 164.64/60.04 164.64/60.04 Strict Trs: { g(x, h(y, z)) -> h(g(x, y), z) } 164.64/60.04 Weak Trs: 164.64/60.04 { g(f(x, y), z) -> f(x, g(y, z)) 164.64/60.04 , g(h(x, y), z) -> g(x, f(y, z)) } 164.64/60.04 Obligation: 164.64/60.04 derivational complexity 164.64/60.04 Answer: 164.64/60.04 YES(O(1),O(n^2)) 164.64/60.04 164.64/60.04 We use the processor 'matrix interpretation of dimension 2' to 164.64/60.04 orient following rules strictly. 164.64/60.04 164.64/60.04 Trs: { g(x, h(y, z)) -> h(g(x, y), z) } 164.64/60.04 164.64/60.04 The induced complexity on above rules (modulo remaining rules) is 164.64/60.04 YES(?,O(n^2)) . These rules are moved into the corresponding weak 164.64/60.04 component(s). 164.64/60.04 164.64/60.04 Sub-proof: 164.64/60.04 ---------- 164.64/60.04 TcT has computed the following triangular matrix interpretation. 164.64/60.04 164.64/60.04 [g](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 164.64/60.04 [0 0] [0 1] [0] 164.64/60.04 164.64/60.04 [f](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 164.64/60.04 [0 0] [0 0] [0] 164.64/60.04 164.64/60.04 [h](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 164.64/60.04 [0 1] [0 0] [1] 164.64/60.04 164.64/60.04 The order satisfies the following ordering constraints: 164.64/60.04 164.64/60.04 [g(x, h(y, z))] = [1 0] x + [1 1] y + [1 2] z + [1] 164.64/60.04 [0 0] [0 1] [0 0] [1] 164.64/60.04 > [1 0] x + [1 1] y + [1 2] z + [0] 164.64/60.04 [0 0] [0 1] [0 0] [1] 164.64/60.04 = [h(g(x, y), z)] 164.64/60.04 164.64/60.04 [g(f(x, y), z)] = [1 0] x + [1 0] y + [1 1] z + [0] 164.64/60.04 [0 0] [0 0] [0 1] [0] 164.64/60.04 >= [1 0] x + [1 0] y + [1 1] z + [0] 164.64/60.04 [0 0] [0 0] [0 0] [0] 164.64/60.04 = [f(x, g(y, z))] 164.64/60.04 164.64/60.04 [g(h(x, y), z)] = [1 0] x + [1 2] y + [1 1] z + [0] 164.64/60.04 [0 0] [0 0] [0 1] [0] 164.64/60.04 >= [1 0] x + [1 0] y + [1 0] z + [0] 164.64/60.04 [0 0] [0 0] [0 0] [0] 164.64/60.04 = [g(x, f(y, z))] 164.64/60.04 164.64/60.04 164.64/60.04 We return to the main proof. 164.64/60.04 164.64/60.04 We are left with following problem, upon which TcT provides the 164.64/60.04 certificate YES(O(1),O(1)). 164.64/60.04 164.64/60.04 Weak Trs: 164.64/60.04 { g(x, h(y, z)) -> h(g(x, y), z) 164.64/60.04 , g(f(x, y), z) -> f(x, g(y, z)) 164.64/60.04 , g(h(x, y), z) -> g(x, f(y, z)) } 164.64/60.04 Obligation: 164.64/60.04 derivational complexity 164.64/60.04 Answer: 164.64/60.04 YES(O(1),O(1)) 164.64/60.04 164.64/60.04 Empty rules are trivially bounded 164.64/60.04 164.64/60.04 Hurray, we answered YES(O(1),O(n^2)) 164.64/60.05 EOF