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

We add following weak dependency pairs:

Strict DPs:
  { average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))
  , average^#(s(x), y) -> c_2(average^#(x, s(y)))
  , average^#(0(), s(s(0()))) -> c_3()
  , average^#(0(), s(0())) -> c_4()
  , average^#(0(), 0()) -> c_5() }

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:
  { average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))
  , average^#(s(x), y) -> c_2(average^#(x, s(y)))
  , average^#(0(), s(s(0()))) -> c_3()
  , average^#(0(), s(0())) -> c_4()
  , average^#(0(), 0()) -> c_5() }
Strict Trs:
  { average(x, s(s(s(y)))) -> s(average(s(x), y))
  , average(s(x), y) -> average(x, s(y))
  , average(0(), s(s(0()))) -> s(0())
  , average(0(), s(0())) -> 0()
  , average(0(), 0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^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(n^1)).

Strict DPs:
  { average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))
  , average^#(s(x), y) -> c_2(average^#(x, s(y)))
  , average^#(0(), s(s(0()))) -> c_3()
  , average^#(0(), s(0())) -> c_4()
  , average^#(0(), 0()) -> c_5() }
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(c_1) = {1}, Uargs(c_2) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

              [s](x1) = [1] x1 + [0]
                                    
                  [0] = [0]         
                                    
  [average^#](x1, x2) = [1]         
                                    
            [c_1](x1) = [1] x1 + [0]
                                    
            [c_2](x1) = [1] x1 + [0]
                                    
                [c_3] = [0]         
                                    
                [c_4] = [0]         
                                    
                [c_5] = [0]         

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^1)).

Strict DPs:
  { average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))
  , average^#(s(x), y) -> c_2(average^#(x, s(y))) }
Weak DPs:
  { average^#(0(), s(s(0()))) -> c_3()
  , average^#(0(), s(0())) -> c_4()
  , average^#(0(), 0()) -> c_5() }
Obligation:
  innermost 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.

{ average^#(0(), s(s(0()))) -> c_3()
, average^#(0(), s(0())) -> c_4()
, average^#(0(), 0()) -> c_5() }

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

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

The following argument positions are usable:
  Uargs(c_1) = {1}, Uargs(c_2) = {1}

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

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

This order satisfies following ordering constraints:

  [average^#(x, s(s(s(y))))] = [2] x + [1] y + [6]      
                             > [2] x + [1] y + [5]      
                             = [c_1(average^#(s(x), y))]
                                                        
        [average^#(s(x), y)] = [2] x + [1] y + [4]      
                             > [2] x + [1] y + [3]      
                             = [c_2(average^#(x, s(y)))]
                                                        

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