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

We add the following innermost 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:
  innermost 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 + [2 2] x2 + [0]
                  [0 0]      [2 2]      [0]
                                           
          [c_1] = [1]                      
                  [2]                      
                                           
      [c_2](x1) = [1 0] x1 + [1]           
                  [0 1]      [2]           

The following symbols are considered usable

  {f, f^#}

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)] = [2 2] y + [1]          
                   [2 2]     [0]          
                 ? [1]                    
                   [2]                    
                 = [c_1()]                
                                          
  [f^#(s(x), y)] = [2 2] y + [1 1] x + [4]
                   [2 2]     [0 0]     [0]
                 ? [2 2] y + [1 1] x + [2]
                   [2 2]     [0 0]     [2]
                 = [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(1),O(1)).

Strict 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:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

  DPs:
    { 1: f^#(0(), y) -> c_1()
    , 2: 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(1),O(1)).

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

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

Weak Trs:
  { f(0(), y) -> 0()
  , f(s(x), y) -> f(f(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)).

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

Empty rules are trivially bounded

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