YES(?,O(n^2))

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

Strict Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sum(x), s(x))
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

We add following dependency tuples:

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

and mark the set of starting terms.

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

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

We estimate the number of application of {1,3} by applications of
Pre({1,3}) = {2,4}. Here rules are labeled as follows:

  DPs:
    { 1: sum^#(0()) -> c_1()
    , 2: sum^#(s(x)) -> c_2(+^#(sum(x), s(x)), sum^#(x))
    , 3: +^#(x, 0()) -> c_3()
    , 4: +^#(x, s(y)) -> c_4(+^#(x, y)) }

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

Strict DPs:
  { sum^#(s(x)) -> c_2(+^#(sum(x), s(x)), sum^#(x))
  , +^#(x, s(y)) -> c_4(+^#(x, y)) }
Weak DPs:
  { sum^#(0()) -> c_1()
  , +^#(x, 0()) -> c_3() }
Weak Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sum(x), s(x))
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
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.

{ sum^#(0()) -> c_1()
, +^#(x, 0()) -> c_3() }

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

Strict DPs:
  { sum^#(s(x)) -> c_2(+^#(sum(x), s(x)), sum^#(x))
  , +^#(x, s(y)) -> c_4(+^#(x, y)) }
Weak Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sum(x), s(x))
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

The following argument positions are usable:
  Uargs(c_2) = {1, 2}, Uargs(c_4) = {1}

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

      [sum](x1) = [0]                      
                  [0]                      
                                           
            [0] = [0]                      
                  [0]                      
                                           
        [s](x1) = [1 3] x1 + [3]           
                  [0 1]      [1]           
                                           
    [+](x1, x2) = [0]                      
                  [0]                      
                                           
    [sum^#](x1) = [1 1] x1 + [0]           
                  [0 1]      [0]           
                                           
  [c_2](x1, x2) = [1 0] x1 + [1 2] x2 + [2]
                  [0 0]      [0 0]      [1]
                                           
  [+^#](x1, x2) = [0 1] x2 + [0]           
                  [0 0]      [0]           
                                           
      [c_4](x1) = [1 0] x1 + [0]           
                  [0 0]      [0]           

This order satisfies following ordering constraints:

   [sum^#(s(x))] = [1 4] x + [4]                     
                   [0 1]     [1]                     
                 > [1 4] x + [3]                     
                   [0 0]     [1]                     
                 = [c_2(+^#(sum(x), s(x)), sum^#(x))]
                                                     
  [+^#(x, s(y))] = [0 1] y + [1]                     
                   [0 0]     [0]                     
                 > [0 1] y + [0]                     
                   [0 0]     [0]                     
                 = [c_4(+^#(x, y))]                  
                                                     

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