YES(O(1),O(n^1))

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

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

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

The following argument positions are usable:
  none

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

      [f](x1) = [1] x1 + [7]
                            
      [s](x1) = [1] x1 + [7]
                            
  [g](x1, x2) = [1] x1 + [7]
                            
          [0] = [7]         
                            
          [1] = [3]         

The following symbols are considered usable

  {f, 0}

The order satisfies the following ordering constraints:

  [f(s(x))] =  [1] x + [14]
            >= [1] x + [14]
            =  [f(g(x, x))]
                           
      [0()] =  [7]         
            >  [3]         
            =  [1()]       
                           

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(n^1)).

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

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

The following argument positions are usable:
  none

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

      [f](x1) = [1] x1 + [7]
                            
      [s](x1) = [1] x1 + [7]
                            
  [g](x1, x2) = [1] x2 + [6]
                            
          [0] = [7]         
                            
          [1] = [7]         

The following symbols are considered usable

  {f, 0}

The order satisfies the following ordering constraints:

  [f(s(x))] =  [1] x + [14]
            >  [1] x + [13]
            =  [f(g(x, x))]
                           
      [0()] =  [7]         
            >= [7]         
            =  [1()]       
                           

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)).

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

Empty rules are trivially bounded

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