YES(?,O(1))

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Strict Trs:
  { f(X) -> n__f(X)
  , f(n__f(n__a())) -> f(n__g(f(n__a())))
  , a() -> n__a()
  , g(X) -> n__g(X)
  , activate(X) -> X
  , activate(n__f(X)) -> f(X)
  , activate(n__a()) -> a()
  , activate(n__g(X)) -> g(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

We add following dependency tuples:

Strict DPs:
  { f^#(X) -> c_1()
  , f^#(n__f(n__a())) -> c_2(f^#(n__g(f(n__a()))), f^#(n__a()))
  , a^#() -> c_3()
  , g^#(X) -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__f(X)) -> c_6(f^#(X))
  , activate^#(n__a()) -> c_7(a^#())
  , activate^#(n__g(X)) -> c_8(g^#(X)) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Strict DPs:
  { f^#(X) -> c_1()
  , f^#(n__f(n__a())) -> c_2(f^#(n__g(f(n__a()))), f^#(n__a()))
  , a^#() -> c_3()
  , g^#(X) -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__f(X)) -> c_6(f^#(X))
  , activate^#(n__a()) -> c_7(a^#())
  , activate^#(n__g(X)) -> c_8(g^#(X)) }
Weak Trs:
  { f(X) -> n__f(X)
  , f(n__f(n__a())) -> f(n__g(f(n__a())))
  , a() -> n__a()
  , g(X) -> n__g(X)
  , activate(X) -> X
  , activate(n__f(X)) -> f(X)
  , activate(n__a()) -> a()
  , activate(n__g(X)) -> g(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

We estimate the number of application of {1,3,4,5} by applications
of Pre({1,3,4,5}) = {2,6,7,8}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(X) -> c_1()
    , 2: f^#(n__f(n__a())) -> c_2(f^#(n__g(f(n__a()))), f^#(n__a()))
    , 3: a^#() -> c_3()
    , 4: g^#(X) -> c_4()
    , 5: activate^#(X) -> c_5()
    , 6: activate^#(n__f(X)) -> c_6(f^#(X))
    , 7: activate^#(n__a()) -> c_7(a^#())
    , 8: activate^#(n__g(X)) -> c_8(g^#(X)) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Strict DPs:
  { f^#(n__f(n__a())) -> c_2(f^#(n__g(f(n__a()))), f^#(n__a()))
  , activate^#(n__f(X)) -> c_6(f^#(X))
  , activate^#(n__a()) -> c_7(a^#())
  , activate^#(n__g(X)) -> c_8(g^#(X)) }
Weak DPs:
  { f^#(X) -> c_1()
  , a^#() -> c_3()
  , g^#(X) -> c_4()
  , activate^#(X) -> c_5() }
Weak Trs:
  { f(X) -> n__f(X)
  , f(n__f(n__a())) -> f(n__g(f(n__a())))
  , a() -> n__a()
  , g(X) -> n__g(X)
  , activate(X) -> X
  , activate(n__f(X)) -> f(X)
  , activate(n__a()) -> a()
  , activate(n__g(X)) -> g(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

We estimate the number of application of {1,3,4} by applications of
Pre({1,3,4}) = {2}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(n__f(n__a())) -> c_2(f^#(n__g(f(n__a()))), f^#(n__a()))
    , 2: activate^#(n__f(X)) -> c_6(f^#(X))
    , 3: activate^#(n__a()) -> c_7(a^#())
    , 4: activate^#(n__g(X)) -> c_8(g^#(X))
    , 5: f^#(X) -> c_1()
    , 6: a^#() -> c_3()
    , 7: g^#(X) -> c_4()
    , 8: activate^#(X) -> c_5() }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Strict DPs: { activate^#(n__f(X)) -> c_6(f^#(X)) }
Weak DPs:
  { f^#(X) -> c_1()
  , f^#(n__f(n__a())) -> c_2(f^#(n__g(f(n__a()))), f^#(n__a()))
  , a^#() -> c_3()
  , g^#(X) -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__a()) -> c_7(a^#())
  , activate^#(n__g(X)) -> c_8(g^#(X)) }
Weak Trs:
  { f(X) -> n__f(X)
  , f(n__f(n__a())) -> f(n__g(f(n__a())))
  , a() -> n__a()
  , g(X) -> n__g(X)
  , activate(X) -> X
  , activate(n__f(X)) -> f(X)
  , activate(n__a()) -> a()
  , activate(n__g(X)) -> g(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

We estimate the number of application of {1} by applications of
Pre({1}) = {}. Here rules are labeled as follows:

  DPs:
    { 1: activate^#(n__f(X)) -> c_6(f^#(X))
    , 2: f^#(X) -> c_1()
    , 3: f^#(n__f(n__a())) -> c_2(f^#(n__g(f(n__a()))), f^#(n__a()))
    , 4: a^#() -> c_3()
    , 5: g^#(X) -> c_4()
    , 6: activate^#(X) -> c_5()
    , 7: activate^#(n__a()) -> c_7(a^#())
    , 8: activate^#(n__g(X)) -> c_8(g^#(X)) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Weak DPs:
  { f^#(X) -> c_1()
  , f^#(n__f(n__a())) -> c_2(f^#(n__g(f(n__a()))), f^#(n__a()))
  , a^#() -> c_3()
  , g^#(X) -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__f(X)) -> c_6(f^#(X))
  , activate^#(n__a()) -> c_7(a^#())
  , activate^#(n__g(X)) -> c_8(g^#(X)) }
Weak Trs:
  { f(X) -> n__f(X)
  , f(n__f(n__a())) -> f(n__g(f(n__a())))
  , a() -> n__a()
  , g(X) -> n__g(X)
  , activate(X) -> X
  , activate(n__f(X)) -> f(X)
  , activate(n__a()) -> a()
  , activate(n__g(X)) -> g(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ f^#(X) -> c_1()
, f^#(n__f(n__a())) -> c_2(f^#(n__g(f(n__a()))), f^#(n__a()))
, a^#() -> c_3()
, g^#(X) -> c_4()
, activate^#(X) -> c_5()
, activate^#(n__f(X)) -> c_6(f^#(X))
, activate^#(n__a()) -> c_7(a^#())
, activate^#(n__g(X)) -> c_8(g^#(X)) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Weak Trs:
  { f(X) -> n__f(X)
  , f(n__f(n__a())) -> f(n__g(f(n__a())))
  , a() -> n__a()
  , g(X) -> n__g(X)
  , activate(X) -> X
  , activate(n__f(X)) -> f(X)
  , activate(n__a()) -> a()
  , activate(n__g(X)) -> g(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The input was oriented with the instance of 'Small Polynomial Path
Order (PS)' as induced by the safe mapping


and precedence

 empty .

Following symbols are considered recursive:

 {}

The recursion depth is 0.

Further, following argument filtering is employed:

 empty

Usable defined function symbols are a subset of:

 {}

For your convenience, here are the satisfied ordering constraints:


Hurray, we answered YES(?,O(1))