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

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

We add the following weak dependency pairs:

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

and mark the set of starting terms.

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

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

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

The following argument positions are usable:
  Uargs(f) = {1}, Uargs(f^#) = {1}, Uargs(c_2) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

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

The order satisfies the following ordering constraints:

     [f(0(), y)] = [1]                   
                   [1]                   
                 > [0]                   
                   [1]                   
                 = [0()]                 
                                         
    [f(s(x), y)] = [1 1] x + [4]         
                   [0 0]     [1]         
                 > [1 1] x + [1]         
                   [0 0]     [1]         
                 = [f(f(x, y), y)]       
                                         
   [f^#(0(), y)] = [1]                   
                   [0]                   
                 > [0]                   
                   [0]                   
                 = [c_1()]               
                                         
  [f^#(s(x), y)] = [1 1] x + [4]         
                   [0 0]     [0]         
                 > [1 1] x + [1]         
                   [0 0]     [0]         
                 = [c_2(f^#(f(x, y), y))]
                                         

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(n^1)).

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

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

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

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

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

We employ 'linear path analysis' using the following approximated
dependency graph:
empty


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