YES(O(1),O(n^2)) 163.64/60.07 YES(O(1),O(n^2)) 163.64/60.07 163.64/60.07 We are left with following problem, upon which TcT provides the 163.64/60.07 certificate YES(O(1),O(n^2)). 163.64/60.07 163.64/60.07 Strict Trs: 163.64/60.07 { s(s(x)) -> x 163.64/60.07 , s(a()) -> a() 163.64/60.07 , s(f(x, y)) -> f(s(y), s(x)) 163.64/60.07 , s(g(x, y)) -> g(s(x), s(y)) 163.64/60.07 , f(x, a()) -> x 163.64/60.07 , f(a(), y) -> y 163.64/60.07 , f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v)) 163.64/60.07 , g(a(), a()) -> a() } 163.64/60.07 Obligation: 163.64/60.07 derivational complexity 163.64/60.07 Answer: 163.64/60.07 YES(O(1),O(n^2)) 163.64/60.07 163.64/60.07 We use the processor 'matrix interpretation of dimension 1' to 163.64/60.07 orient following rules strictly. 163.64/60.07 163.64/60.07 Trs: 163.64/60.07 { f(x, a()) -> x 163.64/60.07 , f(a(), y) -> y 163.64/60.07 , g(a(), a()) -> a() } 163.64/60.07 163.64/60.07 The induced complexity on above rules (modulo remaining rules) is 163.64/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 163.64/60.07 component(s). 163.64/60.07 163.64/60.07 Sub-proof: 163.64/60.07 ---------- 163.64/60.07 TcT has computed the following triangular matrix interpretation. 163.64/60.07 163.64/60.07 [s](x1) = [1] x1 + [0] 163.64/60.07 163.64/60.07 [a] = [1] 163.64/60.07 163.64/60.07 [f](x1, x2) = [1] x1 + [1] x2 + [0] 163.64/60.07 163.64/60.07 [g](x1, x2) = [1] x1 + [1] x2 + [0] 163.64/60.07 163.64/60.07 The order satisfies the following ordering constraints: 163.64/60.07 163.64/60.07 [s(s(x))] = [1] x + [0] 163.64/60.07 >= [1] x + [0] 163.64/60.07 = [x] 163.64/60.07 163.64/60.07 [s(a())] = [1] 163.64/60.07 >= [1] 163.64/60.07 = [a()] 163.64/60.07 163.64/60.07 [s(f(x, y))] = [1] x + [1] y + [0] 163.64/60.07 >= [1] x + [1] y + [0] 163.64/60.07 = [f(s(y), s(x))] 163.64/60.07 163.64/60.07 [s(g(x, y))] = [1] x + [1] y + [0] 163.64/60.07 >= [1] x + [1] y + [0] 163.64/60.07 = [g(s(x), s(y))] 163.64/60.07 163.64/60.07 [f(x, a())] = [1] x + [1] 163.64/60.07 > [1] x + [0] 163.64/60.07 = [x] 163.64/60.07 163.64/60.07 [f(a(), y)] = [1] y + [1] 163.64/60.07 > [1] y + [0] 163.64/60.07 = [y] 163.64/60.07 163.64/60.07 [f(g(x, y), g(u, v))] = [1] x + [1] y + [1] u + [1] v + [0] 163.64/60.07 >= [1] x + [1] y + [1] u + [1] v + [0] 163.64/60.07 = [g(f(x, u), f(y, v))] 163.64/60.07 163.64/60.07 [g(a(), a())] = [2] 163.64/60.07 > [1] 163.64/60.07 = [a()] 163.64/60.07 163.64/60.07 163.64/60.07 We return to the main proof. 163.64/60.07 163.64/60.07 We are left with following problem, upon which TcT provides the 163.64/60.07 certificate YES(O(1),O(n^2)). 163.64/60.07 163.64/60.07 Strict Trs: 163.64/60.07 { s(s(x)) -> x 163.64/60.07 , s(a()) -> a() 163.64/60.07 , s(f(x, y)) -> f(s(y), s(x)) 163.64/60.07 , s(g(x, y)) -> g(s(x), s(y)) 163.64/60.07 , f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v)) } 163.64/60.07 Weak Trs: 163.64/60.07 { f(x, a()) -> x 163.64/60.07 , f(a(), y) -> y 163.64/60.07 , g(a(), a()) -> a() } 163.64/60.07 Obligation: 163.64/60.07 derivational complexity 163.64/60.07 Answer: 163.64/60.07 YES(O(1),O(n^2)) 163.64/60.07 163.64/60.07 We use the processor 'matrix interpretation of dimension 1' to 163.64/60.07 orient following rules strictly. 163.64/60.07 163.64/60.07 Trs: { f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v)) } 163.64/60.07 163.64/60.07 The induced complexity on above rules (modulo remaining rules) is 163.64/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 163.64/60.07 component(s). 163.64/60.07 163.64/60.07 Sub-proof: 163.64/60.07 ---------- 163.64/60.07 TcT has computed the following triangular matrix interpretation. 163.64/60.07 163.64/60.07 [s](x1) = [1] x1 + [0] 163.64/60.07 163.64/60.07 [a] = [0] 163.64/60.07 163.64/60.07 [f](x1, x2) = [1] x1 + [1] x2 + [0] 163.64/60.07 163.64/60.07 [g](x1, x2) = [1] x1 + [1] x2 + [1] 163.64/60.07 163.64/60.07 The order satisfies the following ordering constraints: 163.64/60.07 163.64/60.07 [s(s(x))] = [1] x + [0] 163.64/60.07 >= [1] x + [0] 163.64/60.07 = [x] 163.64/60.07 163.64/60.07 [s(a())] = [0] 163.64/60.07 >= [0] 163.64/60.07 = [a()] 163.64/60.07 163.64/60.07 [s(f(x, y))] = [1] x + [1] y + [0] 163.64/60.07 >= [1] x + [1] y + [0] 163.64/60.07 = [f(s(y), s(x))] 163.64/60.07 163.64/60.07 [s(g(x, y))] = [1] x + [1] y + [1] 163.64/60.07 >= [1] x + [1] y + [1] 163.64/60.07 = [g(s(x), s(y))] 163.64/60.07 163.64/60.07 [f(x, a())] = [1] x + [0] 163.64/60.07 >= [1] x + [0] 163.64/60.07 = [x] 163.64/60.07 163.64/60.07 [f(a(), y)] = [1] y + [0] 163.64/60.07 >= [1] y + [0] 163.64/60.07 = [y] 163.64/60.07 163.64/60.07 [f(g(x, y), g(u, v))] = [1] x + [1] y + [1] u + [1] v + [2] 163.64/60.07 > [1] x + [1] y + [1] u + [1] v + [1] 163.64/60.07 = [g(f(x, u), f(y, v))] 163.64/60.07 163.64/60.07 [g(a(), a())] = [1] 163.64/60.07 > [0] 163.64/60.07 = [a()] 163.64/60.07 163.64/60.07 163.64/60.07 We return to the main proof. 163.64/60.07 163.64/60.07 We are left with following problem, upon which TcT provides the 163.64/60.07 certificate YES(O(1),O(n^2)). 163.64/60.07 163.64/60.07 Strict Trs: 163.64/60.07 { s(s(x)) -> x 163.64/60.07 , s(a()) -> a() 163.64/60.07 , s(f(x, y)) -> f(s(y), s(x)) 163.64/60.07 , s(g(x, y)) -> g(s(x), s(y)) } 163.64/60.07 Weak Trs: 163.64/60.07 { f(x, a()) -> x 163.64/60.07 , f(a(), y) -> y 163.64/60.07 , f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v)) 163.64/60.07 , g(a(), a()) -> a() } 163.64/60.07 Obligation: 163.64/60.07 derivational complexity 163.64/60.07 Answer: 163.64/60.07 YES(O(1),O(n^2)) 163.64/60.07 163.64/60.07 The weightgap principle applies (using the following nonconstant 163.64/60.07 growth matrix-interpretation) 163.64/60.07 163.64/60.07 TcT has computed the following triangular matrix interpretation. 163.64/60.07 Note that the diagonal of the component-wise maxima of 163.64/60.07 interpretation-entries contains no more than 1 non-zero entries. 163.64/60.07 163.64/60.07 [s](x1) = [1] x1 + [1] 163.64/60.07 163.64/60.07 [a] = [1] 163.64/60.07 163.64/60.07 [f](x1, x2) = [1] x1 + [1] x2 + [0] 163.64/60.07 163.64/60.07 [g](x1, x2) = [1] x1 + [1] x2 + [0] 163.64/60.07 163.64/60.07 The order satisfies the following ordering constraints: 163.64/60.07 163.64/60.07 [s(s(x))] = [1] x + [2] 163.64/60.07 > [1] x + [0] 163.64/60.07 = [x] 163.64/60.07 163.64/60.07 [s(a())] = [2] 163.64/60.07 > [1] 163.64/60.07 = [a()] 163.64/60.07 163.64/60.07 [s(f(x, y))] = [1] x + [1] y + [1] 163.64/60.07 ? [1] x + [1] y + [2] 163.64/60.07 = [f(s(y), s(x))] 163.64/60.07 163.64/60.07 [s(g(x, y))] = [1] x + [1] y + [1] 163.64/60.07 ? [1] x + [1] y + [2] 163.64/60.07 = [g(s(x), s(y))] 163.64/60.07 163.64/60.07 [f(x, a())] = [1] x + [1] 163.64/60.07 > [1] x + [0] 163.64/60.07 = [x] 163.64/60.07 163.64/60.07 [f(a(), y)] = [1] y + [1] 163.64/60.07 > [1] y + [0] 163.64/60.07 = [y] 163.64/60.07 163.64/60.07 [f(g(x, y), g(u, v))] = [1] x + [1] y + [1] u + [1] v + [0] 163.64/60.07 >= [1] x + [1] y + [1] u + [1] v + [0] 163.64/60.07 = [g(f(x, u), f(y, v))] 163.64/60.07 163.64/60.07 [g(a(), a())] = [2] 163.64/60.07 > [1] 163.64/60.07 = [a()] 163.64/60.07 163.64/60.07 163.64/60.07 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 163.64/60.07 163.64/60.07 We are left with following problem, upon which TcT provides the 163.64/60.07 certificate YES(O(1),O(n^2)). 163.64/60.07 163.64/60.07 Strict Trs: 163.64/60.07 { s(f(x, y)) -> f(s(y), s(x)) 163.64/60.07 , s(g(x, y)) -> g(s(x), s(y)) } 163.64/60.07 Weak Trs: 163.64/60.07 { s(s(x)) -> x 163.64/60.07 , s(a()) -> a() 163.64/60.07 , f(x, a()) -> x 163.64/60.07 , f(a(), y) -> y 163.64/60.07 , f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v)) 163.64/60.07 , g(a(), a()) -> a() } 163.64/60.07 Obligation: 163.64/60.07 derivational complexity 163.64/60.07 Answer: 163.64/60.07 YES(O(1),O(n^2)) 163.64/60.07 163.64/60.07 We use the processor 'matrix interpretation of dimension 2' to 163.64/60.07 orient following rules strictly. 163.64/60.07 163.64/60.07 Trs: 163.64/60.07 { s(f(x, y)) -> f(s(y), s(x)) 163.64/60.07 , s(g(x, y)) -> g(s(x), s(y)) } 163.64/60.07 163.64/60.07 The induced complexity on above rules (modulo remaining rules) is 163.64/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 163.64/60.07 component(s). 163.64/60.07 163.64/60.07 Sub-proof: 163.64/60.07 ---------- 163.64/60.07 TcT has computed the following triangular matrix interpretation. 163.64/60.07 163.64/60.07 [s](x1) = [1 1] x1 + [0] 163.64/60.07 [0 1] [0] 163.64/60.07 163.64/60.07 [a] = [0] 163.64/60.07 [0] 163.64/60.07 163.64/60.07 [f](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 163.64/60.07 [0 1] [0 1] [2] 163.64/60.07 163.64/60.07 [g](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 163.64/60.07 [0 1] [0 1] [2] 163.64/60.07 163.64/60.07 The order satisfies the following ordering constraints: 163.64/60.07 163.64/60.07 [s(s(x))] = [1 2] x + [0] 163.64/60.07 [0 1] [0] 163.64/60.07 >= [1 0] x + [0] 163.64/60.07 [0 1] [0] 163.64/60.07 = [x] 163.64/60.07 163.64/60.07 [s(a())] = [0] 163.64/60.07 [0] 163.64/60.07 >= [0] 163.64/60.07 [0] 163.64/60.07 = [a()] 163.64/60.07 163.64/60.07 [s(f(x, y))] = [1 1] x + [1 1] y + [4] 163.64/60.07 [0 1] [0 1] [2] 163.64/60.07 > [1 1] x + [1 1] y + [2] 163.64/60.07 [0 1] [0 1] [2] 163.64/60.07 = [f(s(y), s(x))] 163.64/60.07 163.64/60.07 [s(g(x, y))] = [1 1] x + [1 1] y + [4] 163.64/60.07 [0 1] [0 1] [2] 163.64/60.07 > [1 1] x + [1 1] y + [2] 163.64/60.07 [0 1] [0 1] [2] 163.64/60.07 = [g(s(x), s(y))] 163.64/60.07 163.64/60.07 [f(x, a())] = [1 0] x + [2] 163.64/60.07 [0 1] [2] 163.64/60.07 > [1 0] x + [0] 163.64/60.07 [0 1] [0] 163.64/60.07 = [x] 163.64/60.07 163.64/60.07 [f(a(), y)] = [1 0] y + [2] 163.64/60.07 [0 1] [2] 163.64/60.07 > [1 0] y + [0] 163.64/60.07 [0 1] [0] 163.64/60.07 = [y] 163.64/60.07 163.64/60.07 [f(g(x, y), g(u, v))] = [1 0] x + [1 0] y + [1 0] u + [1 0] v + [6] 163.64/60.07 [0 1] [0 1] [0 1] [0 1] [6] 163.64/60.07 >= [1 0] x + [1 0] y + [1 0] u + [1 0] v + [6] 163.64/60.07 [0 1] [0 1] [0 1] [0 1] [6] 163.64/60.07 = [g(f(x, u), f(y, v))] 163.64/60.07 163.64/60.07 [g(a(), a())] = [2] 163.64/60.07 [2] 163.64/60.07 > [0] 163.64/60.07 [0] 163.64/60.07 = [a()] 163.64/60.07 163.64/60.07 163.64/60.07 We return to the main proof. 163.64/60.07 163.64/60.07 We are left with following problem, upon which TcT provides the 163.64/60.07 certificate YES(O(1),O(1)). 163.64/60.07 163.64/60.07 Weak Trs: 163.64/60.07 { s(s(x)) -> x 163.64/60.07 , s(a()) -> a() 163.64/60.07 , s(f(x, y)) -> f(s(y), s(x)) 163.64/60.07 , s(g(x, y)) -> g(s(x), s(y)) 163.64/60.07 , f(x, a()) -> x 163.64/60.07 , f(a(), y) -> y 163.64/60.07 , f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v)) 163.64/60.07 , g(a(), a()) -> a() } 163.64/60.07 Obligation: 163.64/60.07 derivational complexity 163.64/60.07 Answer: 163.64/60.07 YES(O(1),O(1)) 163.64/60.07 163.64/60.07 Empty rules are trivially bounded 163.64/60.07 163.64/60.07 Hurray, we answered YES(O(1),O(n^2)) 163.64/60.08 EOF