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

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

Arguments of following rules are not normal-forms:

{ g(0(), 1()) -> s(0()) }

All above mentioned rules can be savely removed.

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

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

We add the following weak dependency pairs:

Strict DPs:
  { f^#(s(x)) -> c_1(f^#(g(x, x)))
  , 0^#() -> c_2() }

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^#(s(x)) -> c_1(f^#(g(x, x)))
  , 0^#() -> c_2() }
Strict Trs:
  { f(s(x)) -> f(g(x, x))
  , 0() -> 1() }
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^#(s(x)) -> c_1(f^#(g(x, x)))
  , 0^#() -> c_2() }
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 the following constructor-restricted matrix
interpretation.

      [s](x1) = [0]
                [0]
                   
  [g](x1, x2) = [0]
                [0]
                   
    [f^#](x1) = [0]
                [0]
                   
    [c_1](x1) = [0]
                [0]
                   
        [0^#] = [1]
                [0]
                   
        [c_2] = [0]
                [0]

The order satisfies the following ordering constraints:

  [f^#(s(x))] =  [0]                
                 [0]                
              >= [0]                
                 [0]                
              =  [c_1(f^#(g(x, x)))]
                                    
      [0^#()] =  [1]                
                 [0]                
              >  [0]                
                 [0]                
              =  [c_2()]            
                                    

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),O(1)).

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

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

  DPs:
    { 1: f^#(s(x)) -> c_1(f^#(g(x, x)))
    , 2: 0^#() -> c_2() }

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

Weak DPs:
  { f^#(s(x)) -> c_1(f^#(g(x, x)))
  , 0^#() -> c_2() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

{ f^#(s(x)) -> c_1(f^#(g(x, x)))
, 0^#() -> c_2() }

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

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

Empty rules are trivially bounded

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