YES(O(1),O(1))

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

Strict Trs:
  { f(0(), 1(), X) -> f(X, X, X)
  , g(X, Y) -> X
  , g(X, Y) -> Y }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add following weak dependency pairs:

Strict DPs:
  { f^#(0(), 1(), X) -> c_1(f^#(X, X, X))
  , g^#(X, Y) -> c_2()
  , g^#(X, Y) -> c_3() }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(0(), 1(), X) -> c_1(f^#(X, X, X))
  , g^#(X, Y) -> c_2()
  , g^#(X, Y) -> c_3() }
Strict Trs:
  { f(0(), 1(), X) -> f(X, X, X)
  , g(X, Y) -> X
  , g(X, Y) -> Y }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),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),O(1)).

Strict DPs:
  { f^#(0(), 1(), X) -> c_1(f^#(X, X, X))
  , g^#(X, Y) -> c_2()
  , g^#(X, Y) -> c_3() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  none

TcT has computed following constructor-restricted matrix
interpretation.

                [0] = [2]         
                                  
                [1] = [1]         
                                  
  [f^#](x1, x2, x3) = [2] x3 + [2]
                                  
          [c_1](x1) = [1] x1 + [2]
                                  
      [g^#](x1, x2) = [2]         
                                  
              [c_2] = [1]         
                                  
              [c_3] = [1]         

This order satisfies following ordering constraints:


Further, it can be verified that all rules not oriented are covered by the weightgap condition.

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

Strict DPs: { f^#(0(), 1(), X) -> c_1(f^#(X, X, X)) }
Weak DPs:
  { g^#(X, Y) -> c_2()
  , g^#(X, Y) -> c_3() }
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: f^#(0(), 1(), X) -> c_1(f^#(X, X, X))
    , 2: g^#(X, Y) -> c_2()
    , 3: g^#(X, Y) -> c_3() }

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

Weak DPs:
  { f^#(0(), 1(), X) -> c_1(f^#(X, X, X))
  , g^#(X, Y) -> c_2()
  , g^#(X, Y) -> c_3() }
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^#(0(), 1(), X) -> c_1(f^#(X, X, X))
, g^#(X, Y) -> c_2()
, g^#(X, Y) -> c_3() }

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),O(1))