YES(O(1),O(n^2))
181.73/60.03 YES(O(1),O(n^2))
181.73/60.03
181.73/60.03 We are left with following problem, upon which TcT provides the
181.73/60.03 certificate YES(O(1),O(n^2)).
181.73/60.03
181.73/60.03 Strict Trs:
181.73/60.03 { f(a(), f(b(), x)) -> f(b(), f(a(), x))
181.73/60.03 , f(b(), f(c(), x)) -> f(c(), f(b(), x))
181.73/60.03 , f(c(), f(a(), x)) -> f(a(), f(c(), x)) }
181.73/60.03 Obligation:
181.73/60.03 derivational complexity
181.73/60.03 Answer:
181.73/60.03 YES(O(1),O(n^2))
181.73/60.03
181.73/60.03 We uncurry the input using the following uncurry rules.
181.73/60.03
181.73/60.03 { f(a(), x_1) -> a_1(x_1)
181.73/60.03 , f(b(), x_1) -> b_1(x_1)
181.73/60.03 , f(c(), x_1) -> c_1(x_1) }
181.73/60.03
181.73/60.03 We are left with following problem, upon which TcT provides the
181.73/60.03 certificate YES(O(1),O(n^2)).
181.73/60.03
181.73/60.03 Strict Trs:
181.73/60.03 { a_1(b_1(x)) -> b_1(a_1(x))
181.73/60.03 , b_1(c_1(x)) -> c_1(b_1(x))
181.73/60.03 , c_1(a_1(x)) -> a_1(c_1(x)) }
181.73/60.03 Weak Trs:
181.73/60.03 { f(a(), x_1) -> a_1(x_1)
181.73/60.03 , f(b(), x_1) -> b_1(x_1)
181.73/60.03 , f(c(), x_1) -> c_1(x_1) }
181.73/60.03 Obligation:
181.73/60.03 derivational complexity
181.73/60.03 Answer:
181.73/60.03 YES(O(1),O(n^2))
181.73/60.03
181.73/60.03 We use the processor 'matrix interpretation of dimension 2' to
181.73/60.03 orient following rules strictly.
181.73/60.03
181.73/60.03 Trs: { b_1(c_1(x)) -> c_1(b_1(x)) }
181.73/60.03
181.73/60.03 The induced complexity on above rules (modulo remaining rules) is
181.73/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak
181.73/60.03 component(s).
181.73/60.03
181.73/60.03 Sub-proof:
181.73/60.03 ----------
181.73/60.03 TcT has computed the following triangular matrix interpretation.
181.73/60.03
181.73/60.03 [f](x1, x2) = [1 0] x1 + [1 2] x2 + [0]
181.73/60.03 [0 0] [0 1] [1]
181.73/60.03
181.73/60.03 [a_1](x1) = [1 0] x1 + [0]
181.73/60.03 [0 1] [0]
181.73/60.03
181.73/60.03 [b_1](x1) = [1 2] x1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03
181.73/60.03 [c_1](x1) = [1 1] x1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03
181.73/60.03 [a] = [0]
181.73/60.03 [0]
181.73/60.03
181.73/60.03 [b] = [0]
181.73/60.03 [0]
181.73/60.03
181.73/60.03 [c] = [0]
181.73/60.03 [0]
181.73/60.03
181.73/60.03 The order satisfies the following ordering constraints:
181.73/60.03
181.73/60.03 [f(a(), x_1)] = [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 0] x_1 + [0]
181.73/60.03 [0 1] [0]
181.73/60.03 = [a_1(x_1)]
181.73/60.03
181.73/60.03 [f(b(), x_1)] = [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 = [b_1(x_1)]
181.73/60.03
181.73/60.03 [f(c(), x_1)] = [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 1] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 = [c_1(x_1)]
181.73/60.03
181.73/60.03 [a_1(b_1(x))] = [1 2] x + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 2] x + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 = [b_1(a_1(x))]
181.73/60.03
181.73/60.03 [b_1(c_1(x))] = [1 3] x + [2]
181.73/60.03 [0 1] [2]
181.73/60.03 > [1 3] x + [1]
181.73/60.03 [0 1] [2]
181.73/60.03 = [c_1(b_1(x))]
181.73/60.03
181.73/60.03 [c_1(a_1(x))] = [1 1] x + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 1] x + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 = [a_1(c_1(x))]
181.73/60.03
181.73/60.03
181.73/60.03 We return to the main proof.
181.73/60.03
181.73/60.03 We are left with following problem, upon which TcT provides the
181.73/60.03 certificate YES(O(1),O(n^2)).
181.73/60.03
181.73/60.03 Strict Trs:
181.73/60.03 { a_1(b_1(x)) -> b_1(a_1(x))
181.73/60.03 , c_1(a_1(x)) -> a_1(c_1(x)) }
181.73/60.03 Weak Trs:
181.73/60.03 { f(a(), x_1) -> a_1(x_1)
181.73/60.03 , f(b(), x_1) -> b_1(x_1)
181.73/60.03 , f(c(), x_1) -> c_1(x_1)
181.73/60.03 , b_1(c_1(x)) -> c_1(b_1(x)) }
181.73/60.03 Obligation:
181.73/60.03 derivational complexity
181.73/60.03 Answer:
181.73/60.03 YES(O(1),O(n^2))
181.73/60.03
181.73/60.03 We use the processor 'matrix interpretation of dimension 2' to
181.73/60.03 orient following rules strictly.
181.73/60.03
181.73/60.03 Trs: { a_1(b_1(x)) -> b_1(a_1(x)) }
181.73/60.03
181.73/60.03 The induced complexity on above rules (modulo remaining rules) is
181.73/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak
181.73/60.03 component(s).
181.73/60.03
181.73/60.03 Sub-proof:
181.73/60.03 ----------
181.73/60.03 TcT has computed the following triangular matrix interpretation.
181.73/60.03
181.73/60.03 [f](x1, x2) = [1 0] x1 + [1 2] x2 + [0]
181.73/60.03 [0 0] [0 1] [1]
181.73/60.03
181.73/60.03 [a_1](x1) = [1 2] x1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03
181.73/60.03 [b_1](x1) = [1 1] x1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03
181.73/60.03 [c_1](x1) = [1 0] x1 + [0]
181.73/60.03 [0 1] [0]
181.73/60.03
181.73/60.03 [a] = [0]
181.73/60.03 [0]
181.73/60.03
181.73/60.03 [b] = [0]
181.73/60.03 [0]
181.73/60.03
181.73/60.03 [c] = [0]
181.73/60.03 [0]
181.73/60.03
181.73/60.03 The order satisfies the following ordering constraints:
181.73/60.03
181.73/60.03 [f(a(), x_1)] = [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 = [a_1(x_1)]
181.73/60.03
181.73/60.03 [f(b(), x_1)] = [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 1] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 = [b_1(x_1)]
181.73/60.03
181.73/60.03 [f(c(), x_1)] = [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 0] x_1 + [0]
181.73/60.03 [0 1] [0]
181.73/60.03 = [c_1(x_1)]
181.73/60.03
181.73/60.03 [a_1(b_1(x))] = [1 3] x + [2]
181.73/60.03 [0 1] [2]
181.73/60.03 > [1 3] x + [1]
181.73/60.03 [0 1] [2]
181.73/60.03 = [b_1(a_1(x))]
181.73/60.03
181.73/60.03 [b_1(c_1(x))] = [1 1] x + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 1] x + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 = [c_1(b_1(x))]
181.73/60.03
181.73/60.03 [c_1(a_1(x))] = [1 2] x + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 2] x + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 = [a_1(c_1(x))]
181.73/60.03
181.73/60.03
181.73/60.03 We return to the main proof.
181.73/60.03
181.73/60.03 We are left with following problem, upon which TcT provides the
181.73/60.03 certificate YES(O(1),O(n^2)).
181.73/60.03
181.73/60.03 Strict Trs: { c_1(a_1(x)) -> a_1(c_1(x)) }
181.73/60.03 Weak Trs:
181.73/60.03 { f(a(), x_1) -> a_1(x_1)
181.73/60.03 , f(b(), x_1) -> b_1(x_1)
181.73/60.03 , f(c(), x_1) -> c_1(x_1)
181.73/60.03 , a_1(b_1(x)) -> b_1(a_1(x))
181.73/60.03 , b_1(c_1(x)) -> c_1(b_1(x)) }
181.73/60.03 Obligation:
181.73/60.03 derivational complexity
181.73/60.03 Answer:
181.73/60.03 YES(O(1),O(n^2))
181.73/60.03
181.73/60.03 We use the processor 'matrix interpretation of dimension 2' to
181.73/60.03 orient following rules strictly.
181.73/60.03
181.73/60.03 Trs: { c_1(a_1(x)) -> a_1(c_1(x)) }
181.73/60.03
181.73/60.03 The induced complexity on above rules (modulo remaining rules) is
181.73/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak
181.73/60.03 component(s).
181.73/60.03
181.73/60.03 Sub-proof:
181.73/60.03 ----------
181.73/60.03 TcT has computed the following triangular matrix interpretation.
181.73/60.03
181.73/60.03 [f](x1, x2) = [1 0] x1 + [1 2] x2 + [0]
181.73/60.03 [0 0] [0 1] [1]
181.73/60.03
181.73/60.03 [a_1](x1) = [1 0] x1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03
181.73/60.03 [b_1](x1) = [1 0] x1 + [0]
181.73/60.03 [0 0] [0]
181.73/60.03
181.73/60.03 [c_1](x1) = [1 2] x1 + [0]
181.73/60.03 [0 1] [0]
181.73/60.03
181.73/60.03 [a] = [0]
181.73/60.03 [0]
181.73/60.03
181.73/60.03 [b] = [0]
181.73/60.03 [0]
181.73/60.03
181.73/60.03 [c] = [0]
181.73/60.03 [0]
181.73/60.03
181.73/60.03 The order satisfies the following ordering constraints:
181.73/60.03
181.73/60.03 [f(a(), x_1)] = [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 0] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 = [a_1(x_1)]
181.73/60.03
181.73/60.03 [f(b(), x_1)] = [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 0] x_1 + [0]
181.73/60.03 [0 0] [0]
181.73/60.03 = [b_1(x_1)]
181.73/60.03
181.73/60.03 [f(c(), x_1)] = [1 2] x_1 + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 >= [1 2] x_1 + [0]
181.73/60.03 [0 1] [0]
181.73/60.03 = [c_1(x_1)]
181.73/60.03
181.73/60.03 [a_1(b_1(x))] = [1 0] x + [0]
181.73/60.03 [0 0] [1]
181.73/60.03 >= [1 0] x + [0]
181.73/60.03 [0 0] [0]
181.73/60.03 = [b_1(a_1(x))]
181.73/60.03
181.73/60.03 [b_1(c_1(x))] = [1 2] x + [0]
181.73/60.03 [0 0] [0]
181.73/60.03 >= [1 0] x + [0]
181.73/60.03 [0 0] [0]
181.73/60.03 = [c_1(b_1(x))]
181.73/60.03
181.73/60.03 [c_1(a_1(x))] = [1 2] x + [2]
181.73/60.03 [0 1] [1]
181.73/60.03 > [1 2] x + [0]
181.73/60.03 [0 1] [1]
181.73/60.03 = [a_1(c_1(x))]
181.73/60.03
181.73/60.03
181.73/60.03 We return to the main proof.
181.73/60.03
181.73/60.03 We are left with following problem, upon which TcT provides the
181.73/60.03 certificate YES(O(1),O(1)).
181.73/60.03
181.73/60.03 Weak Trs:
181.73/60.03 { f(a(), x_1) -> a_1(x_1)
181.73/60.03 , f(b(), x_1) -> b_1(x_1)
181.73/60.03 , f(c(), x_1) -> c_1(x_1)
181.73/60.03 , a_1(b_1(x)) -> b_1(a_1(x))
181.73/60.03 , b_1(c_1(x)) -> c_1(b_1(x))
181.73/60.03 , c_1(a_1(x)) -> a_1(c_1(x)) }
181.73/60.03 Obligation:
181.73/60.03 derivational complexity
181.73/60.03 Answer:
181.73/60.03 YES(O(1),O(1))
181.73/60.03
181.73/60.03 Empty rules are trivially bounded
181.73/60.03
181.73/60.03 Hurray, we answered YES(O(1),O(n^2))
181.73/60.03 EOF