YES(O(1),O(n^2))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs:
  { g(x, y) -> x
  , g(x, y) -> y
  , f(x, y, s(z)) -> s(f(0(), 1(), z))
  , f(0(), 1(), x) -> f(s(x), x, x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We add following weak dependency pairs:

Strict DPs:
  { g^#(x, y) -> c_1()
  , g^#(x, y) -> c_2()
  , f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { g^#(x, y) -> c_1()
  , g^#(x, y) -> c_2()
  , f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
Strict Trs:
  { g(x, y) -> x
  , g(x, y) -> y
  , f(x, y, s(z)) -> s(f(0(), 1(), z))
  , f(0(), 1(), x) -> f(s(x), x, x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

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(n^2)).

Strict DPs:
  { g^#(x, y) -> c_1()
  , g^#(x, y) -> c_2()
  , f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

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

The following argument positions are usable:
  Uargs(c_3) = {1}, Uargs(c_4) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

                [0] = [1]         
                                  
                [1] = [1]         
                                  
            [s](x1) = [1] x1 + [1]
                                  
      [g^#](x1, x2) = [1]         
                                  
              [c_1] = [0]         
                                  
              [c_2] = [0]         
                                  
  [f^#](x1, x2, x3) = [1] x3 + [2]
                                  
          [c_3](x1) = [1] x1 + [2]
                                  
          [c_4](x1) = [1] x1 + [2]

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(n^2)).

Strict DPs:
  { f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
Weak DPs:
  { g^#(x, y) -> c_1()
  , g^#(x, y) -> c_2() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

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

{ g^#(x, y) -> c_1()
, g^#(x, y) -> c_2() }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^2)).

Strict DPs:
  { f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

The following argument positions are usable:
  Uargs(c_3) = {1}, Uargs(c_4) = {1}

TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).

                [0] = [2]                      
                      [0]                      
                                               
                [1] = [0]                      
                      [0]                      
                                               
            [s](x1) = [0 0] x1 + [1]           
                      [1 1]      [2]           
                                               
  [f^#](x1, x2, x3) = [2 0] x1 + [1 2] x3 + [0]
                      [1 0]      [0 0]      [0]
                                               
          [c_3](x1) = [1 0] x1 + [0]           
                      [0 0]      [0]           
                                               
          [c_4](x1) = [1 1] x1 + [0]           
                      [0 0]      [1]           

This order satisfies following ordering constraints:

   [f^#(x, y, s(z))] = [2 0] x + [2 2] z + [5]
                       [1 0]     [0 0]     [0]
                     > [1 2] z + [4]          
                       [0 0]     [0]          
                     = [c_3(f^#(0(), 1(), z))]
                                              
  [f^#(0(), 1(), x)] = [1 2] x + [4]          
                       [0 0]     [2]          
                     > [1 2] x + [3]          
                       [0 0]     [1]          
                     = [c_4(f^#(s(x), x, x))] 
                                              

Hurray, we answered YES(O(1),O(n^2))