YES(O(1),O(n^2))
172.96/60.03 YES(O(1),O(n^2))
172.96/60.03
172.96/60.03 We are left with following problem, upon which TcT provides the
172.96/60.03 certificate YES(O(1),O(n^2)).
172.96/60.03
172.96/60.03 Strict Trs:
172.96/60.03 { a__f(X) -> f(X)
172.96/60.03 , a__f(f(a())) -> c(f(g(f(a()))))
172.96/60.03 , mark(f(X)) -> a__f(mark(X))
172.96/60.03 , mark(a()) -> a()
172.96/60.03 , mark(c(X)) -> c(X)
172.96/60.03 , mark(g(X)) -> g(mark(X)) }
172.96/60.03 Obligation:
172.96/60.03 derivational complexity
172.96/60.03 Answer:
172.96/60.03 YES(O(1),O(n^2))
172.96/60.03
172.96/60.03 The weightgap principle applies (using the following nonconstant
172.96/60.03 growth matrix-interpretation)
172.96/60.03
172.96/60.03 TcT has computed the following triangular matrix interpretation.
172.96/60.03 Note that the diagonal of the component-wise maxima of
172.96/60.03 interpretation-entries contains no more than 1 non-zero entries.
172.96/60.03
172.96/60.03 [a__f](x1) = [1] x1 + [1]
172.96/60.03
172.96/60.03 [f](x1) = [1] x1 + [0]
172.96/60.03
172.96/60.03 [a] = [0]
172.96/60.03
172.96/60.03 [c](x1) = [1] x1 + [0]
172.96/60.03
172.96/60.03 [g](x1) = [1] x1 + [0]
172.96/60.03
172.96/60.03 [mark](x1) = [1] x1 + [0]
172.96/60.03
172.96/60.03 The order satisfies the following ordering constraints:
172.96/60.03
172.96/60.03 [a__f(X)] = [1] X + [1]
172.96/60.03 > [1] X + [0]
172.96/60.03 = [f(X)]
172.96/60.03
172.96/60.03 [a__f(f(a()))] = [1]
172.96/60.03 > [0]
172.96/60.03 = [c(f(g(f(a()))))]
172.96/60.03
172.96/60.03 [mark(f(X))] = [1] X + [0]
172.96/60.03 ? [1] X + [1]
172.96/60.03 = [a__f(mark(X))]
172.96/60.03
172.96/60.03 [mark(a())] = [0]
172.96/60.03 >= [0]
172.96/60.03 = [a()]
172.96/60.03
172.96/60.03 [mark(c(X))] = [1] X + [0]
172.96/60.03 >= [1] X + [0]
172.96/60.03 = [c(X)]
172.96/60.03
172.96/60.03 [mark(g(X))] = [1] X + [0]
172.96/60.03 >= [1] X + [0]
172.96/60.03 = [g(mark(X))]
172.96/60.03
172.96/60.03
172.96/60.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition.
172.96/60.03
172.96/60.03 We are left with following problem, upon which TcT provides the
172.96/60.03 certificate YES(O(1),O(n^2)).
172.96/60.03
172.96/60.03 Strict Trs:
172.96/60.03 { mark(f(X)) -> a__f(mark(X))
172.96/60.03 , mark(a()) -> a()
172.96/60.03 , mark(c(X)) -> c(X)
172.96/60.03 , mark(g(X)) -> g(mark(X)) }
172.96/60.03 Weak Trs:
172.96/60.03 { a__f(X) -> f(X)
172.96/60.03 , a__f(f(a())) -> c(f(g(f(a())))) }
172.96/60.03 Obligation:
172.96/60.03 derivational complexity
172.96/60.03 Answer:
172.96/60.03 YES(O(1),O(n^2))
172.96/60.03
172.96/60.03 We use the processor 'matrix interpretation of dimension 1' to
172.96/60.03 orient following rules strictly.
172.96/60.03
172.96/60.03 Trs:
172.96/60.03 { mark(a()) -> a()
172.96/60.03 , mark(c(X)) -> c(X) }
172.96/60.03
172.96/60.03 The induced complexity on above rules (modulo remaining rules) is
172.96/60.03 YES(?,O(n^1)) . These rules are moved into the corresponding weak
172.96/60.03 component(s).
172.96/60.03
172.96/60.03 Sub-proof:
172.96/60.03 ----------
172.96/60.03 TcT has computed the following triangular matrix interpretation.
172.96/60.03
172.96/60.03 [a__f](x1) = [1] x1 + [0]
172.96/60.03
172.96/60.03 [f](x1) = [1] x1 + [0]
172.96/60.03
172.96/60.03 [a] = [0]
172.96/60.03
172.96/60.03 [c](x1) = [1] x1 + [0]
172.96/60.03
172.96/60.03 [g](x1) = [1] x1 + [0]
172.96/60.03
172.96/60.03 [mark](x1) = [1] x1 + [1]
172.96/60.03
172.96/60.03 The order satisfies the following ordering constraints:
172.96/60.03
172.96/60.03 [a__f(X)] = [1] X + [0]
172.96/60.03 >= [1] X + [0]
172.96/60.03 = [f(X)]
172.96/60.03
172.96/60.03 [a__f(f(a()))] = [0]
172.96/60.03 >= [0]
172.96/60.03 = [c(f(g(f(a()))))]
172.96/60.03
172.96/60.03 [mark(f(X))] = [1] X + [1]
172.96/60.03 >= [1] X + [1]
172.96/60.03 = [a__f(mark(X))]
172.96/60.03
172.96/60.03 [mark(a())] = [1]
172.96/60.03 > [0]
172.96/60.03 = [a()]
172.96/60.03
172.96/60.03 [mark(c(X))] = [1] X + [1]
172.96/60.03 > [1] X + [0]
172.96/60.03 = [c(X)]
172.96/60.03
172.96/60.03 [mark(g(X))] = [1] X + [1]
172.96/60.03 >= [1] X + [1]
172.96/60.03 = [g(mark(X))]
172.96/60.03
172.96/60.03
172.96/60.03 We return to the main proof.
172.96/60.03
172.96/60.03 We are left with following problem, upon which TcT provides the
172.96/60.03 certificate YES(O(1),O(n^2)).
172.96/60.03
172.96/60.03 Strict Trs:
172.96/60.03 { mark(f(X)) -> a__f(mark(X))
172.96/60.03 , mark(g(X)) -> g(mark(X)) }
172.96/60.03 Weak Trs:
172.96/60.03 { a__f(X) -> f(X)
172.96/60.03 , a__f(f(a())) -> c(f(g(f(a()))))
172.96/60.03 , mark(a()) -> a()
172.96/60.03 , mark(c(X)) -> c(X) }
172.96/60.03 Obligation:
172.96/60.03 derivational complexity
172.96/60.03 Answer:
172.96/60.03 YES(O(1),O(n^2))
172.96/60.03
172.96/60.03 We use the processor 'matrix interpretation of dimension 2' to
172.96/60.03 orient following rules strictly.
172.96/60.03
172.96/60.03 Trs: { mark(g(X)) -> g(mark(X)) }
172.96/60.03
172.96/60.03 The induced complexity on above rules (modulo remaining rules) is
172.96/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak
172.96/60.03 component(s).
172.96/60.03
172.96/60.03 Sub-proof:
172.96/60.03 ----------
172.96/60.03 TcT has computed the following triangular matrix interpretation.
172.96/60.03
172.96/60.03 [a__f](x1) = [1 0] x1 + [0]
172.96/60.03 [0 1] [0]
172.96/60.03
172.96/60.03 [f](x1) = [1 0] x1 + [0]
172.96/60.03 [0 1] [0]
172.96/60.03
172.96/60.03 [a] = [0]
172.96/60.03 [0]
172.96/60.03
172.96/60.03 [c](x1) = [1 0] x1 + [0]
172.96/60.03 [0 0] [0]
172.96/60.03
172.96/60.03 [g](x1) = [1 0] x1 + [0]
172.96/60.03 [0 1] [2]
172.96/60.03
172.96/60.03 [mark](x1) = [1 1] x1 + [0]
172.96/60.03 [0 1] [0]
172.96/60.03
172.96/60.03 The order satisfies the following ordering constraints:
172.96/60.03
172.96/60.03 [a__f(X)] = [1 0] X + [0]
172.96/60.03 [0 1] [0]
172.96/60.03 >= [1 0] X + [0]
172.96/60.03 [0 1] [0]
172.96/60.03 = [f(X)]
172.96/60.03
172.96/60.03 [a__f(f(a()))] = [0]
172.96/60.03 [0]
172.96/60.03 >= [0]
172.96/60.03 [0]
172.96/60.03 = [c(f(g(f(a()))))]
172.96/60.03
172.96/60.03 [mark(f(X))] = [1 1] X + [0]
172.96/60.03 [0 1] [0]
172.96/60.03 >= [1 1] X + [0]
172.96/60.03 [0 1] [0]
172.96/60.03 = [a__f(mark(X))]
172.96/60.03
172.96/60.03 [mark(a())] = [0]
172.96/60.03 [0]
172.96/60.03 >= [0]
172.96/60.03 [0]
172.96/60.03 = [a()]
172.96/60.03
172.96/60.03 [mark(c(X))] = [1 0] X + [0]
172.96/60.03 [0 0] [0]
172.96/60.03 >= [1 0] X + [0]
172.96/60.03 [0 0] [0]
172.96/60.03 = [c(X)]
172.96/60.03
172.96/60.03 [mark(g(X))] = [1 1] X + [2]
172.96/60.03 [0 1] [2]
172.96/60.03 > [1 1] X + [0]
172.96/60.03 [0 1] [2]
172.96/60.03 = [g(mark(X))]
172.96/60.03
172.96/60.03
172.96/60.03 We return to the main proof.
172.96/60.03
172.96/60.03 We are left with following problem, upon which TcT provides the
172.96/60.03 certificate YES(O(1),O(n^2)).
172.96/60.03
172.96/60.03 Strict Trs: { mark(f(X)) -> a__f(mark(X)) }
172.96/60.03 Weak Trs:
172.96/60.03 { a__f(X) -> f(X)
172.96/60.03 , a__f(f(a())) -> c(f(g(f(a()))))
172.96/60.03 , mark(a()) -> a()
172.96/60.03 , mark(c(X)) -> c(X)
172.96/60.03 , mark(g(X)) -> g(mark(X)) }
172.96/60.03 Obligation:
172.96/60.03 derivational complexity
172.96/60.03 Answer:
172.96/60.03 YES(O(1),O(n^2))
172.96/60.03
172.96/60.03 We use the processor 'matrix interpretation of dimension 2' to
172.96/60.03 orient following rules strictly.
172.96/60.03
172.96/60.03 Trs: { mark(f(X)) -> a__f(mark(X)) }
172.96/60.03
172.96/60.03 The induced complexity on above rules (modulo remaining rules) is
172.96/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak
172.96/60.03 component(s).
172.96/60.03
172.96/60.03 Sub-proof:
172.96/60.03 ----------
172.96/60.03 TcT has computed the following triangular matrix interpretation.
172.96/60.03
172.96/60.03 [a__f](x1) = [1 0] x1 + [2]
172.96/60.03 [0 1] [2]
172.96/60.03
172.96/60.03 [f](x1) = [1 0] x1 + [2]
172.96/60.03 [0 1] [2]
172.96/60.03
172.96/60.03 [a] = [0]
172.96/60.03 [0]
172.96/60.03
172.96/60.03 [c](x1) = [1 0] x1 + [0]
172.96/60.03 [0 0] [0]
172.96/60.03
172.96/60.03 [g](x1) = [1 0] x1 + [0]
172.96/60.03 [0 1] [0]
172.96/60.03
172.96/60.03 [mark](x1) = [1 1] x1 + [0]
172.96/60.03 [0 1] [0]
172.96/60.03
172.96/60.03 The order satisfies the following ordering constraints:
172.96/60.03
172.96/60.03 [a__f(X)] = [1 0] X + [2]
172.96/60.03 [0 1] [2]
172.96/60.03 >= [1 0] X + [2]
172.96/60.03 [0 1] [2]
172.96/60.03 = [f(X)]
172.96/60.03
172.96/60.03 [a__f(f(a()))] = [4]
172.96/60.03 [4]
172.96/60.03 >= [4]
172.96/60.03 [0]
172.96/60.03 = [c(f(g(f(a()))))]
172.96/60.03
172.96/60.03 [mark(f(X))] = [1 1] X + [4]
172.96/60.03 [0 1] [2]
172.96/60.03 > [1 1] X + [2]
172.96/60.03 [0 1] [2]
172.96/60.03 = [a__f(mark(X))]
172.96/60.03
172.96/60.03 [mark(a())] = [0]
172.96/60.03 [0]
172.96/60.03 >= [0]
172.96/60.03 [0]
172.96/60.03 = [a()]
172.96/60.03
172.96/60.03 [mark(c(X))] = [1 0] X + [0]
172.96/60.03 [0 0] [0]
172.96/60.03 >= [1 0] X + [0]
172.96/60.03 [0 0] [0]
172.96/60.03 = [c(X)]
172.96/60.03
172.96/60.03 [mark(g(X))] = [1 1] X + [0]
172.96/60.03 [0 1] [0]
172.96/60.03 >= [1 1] X + [0]
172.96/60.03 [0 1] [0]
172.96/60.03 = [g(mark(X))]
172.96/60.03
172.96/60.03
172.96/60.03 We return to the main proof.
172.96/60.03
172.96/60.03 We are left with following problem, upon which TcT provides the
172.96/60.03 certificate YES(O(1),O(1)).
172.96/60.03
172.96/60.03 Weak Trs:
172.96/60.03 { a__f(X) -> f(X)
172.96/60.03 , a__f(f(a())) -> c(f(g(f(a()))))
172.96/60.03 , mark(f(X)) -> a__f(mark(X))
172.96/60.03 , mark(a()) -> a()
172.96/60.03 , mark(c(X)) -> c(X)
172.96/60.03 , mark(g(X)) -> g(mark(X)) }
172.96/60.03 Obligation:
172.96/60.03 derivational complexity
172.96/60.03 Answer:
172.96/60.03 YES(O(1),O(1))
172.96/60.03
172.96/60.03 Empty rules are trivially bounded
172.96/60.03
172.96/60.03 Hurray, we answered YES(O(1),O(n^2))
172.96/60.05 EOF