YES(?,O(n^2))

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

Strict Trs:
  { not(true()) -> false()
  , not(false()) -> true()
  , evenodd(x, 0()) -> not(evenodd(x, s(0())))
  , evenodd(0(), s(0())) -> false()
  , evenodd(s(x), s(0())) -> evenodd(x, 0()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

We add following dependency tuples:

Strict DPs:
  { not^#(true()) -> c_1()
  , not^#(false()) -> c_2()
  , evenodd^#(x, 0()) ->
    c_3(not^#(evenodd(x, s(0()))), evenodd^#(x, s(0())))
  , evenodd^#(0(), s(0())) -> c_4()
  , evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }

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:
  { not^#(true()) -> c_1()
  , not^#(false()) -> c_2()
  , evenodd^#(x, 0()) ->
    c_3(not^#(evenodd(x, s(0()))), evenodd^#(x, s(0())))
  , evenodd^#(0(), s(0())) -> c_4()
  , evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
Weak Trs:
  { not(true()) -> false()
  , not(false()) -> true()
  , evenodd(x, 0()) -> not(evenodd(x, s(0())))
  , evenodd(0(), s(0())) -> false()
  , evenodd(s(x), s(0())) -> evenodd(x, 0()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

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

  DPs:
    { 1: not^#(true()) -> c_1()
    , 2: not^#(false()) -> c_2()
    , 3: evenodd^#(x, 0()) ->
         c_3(not^#(evenodd(x, s(0()))), evenodd^#(x, s(0())))
    , 4: evenodd^#(0(), s(0())) -> c_4()
    , 5: evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }

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

Strict DPs:
  { evenodd^#(x, 0()) ->
    c_3(not^#(evenodd(x, s(0()))), evenodd^#(x, s(0())))
  , evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
Weak DPs:
  { not^#(true()) -> c_1()
  , not^#(false()) -> c_2()
  , evenodd^#(0(), s(0())) -> c_4() }
Weak Trs:
  { not(true()) -> false()
  , not(false()) -> true()
  , evenodd(x, 0()) -> not(evenodd(x, s(0())))
  , evenodd(0(), s(0())) -> false()
  , evenodd(s(x), s(0())) -> evenodd(x, 0()) }
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.

{ not^#(true()) -> c_1()
, not^#(false()) -> c_2()
, evenodd^#(0(), s(0())) -> c_4() }

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

Strict DPs:
  { evenodd^#(x, 0()) ->
    c_3(not^#(evenodd(x, s(0()))), evenodd^#(x, s(0())))
  , evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
Weak Trs:
  { not(true()) -> false()
  , not(false()) -> true()
  , evenodd(x, 0()) -> not(evenodd(x, s(0())))
  , evenodd(0(), s(0())) -> false()
  , evenodd(s(x), s(0())) -> evenodd(x, 0()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { evenodd^#(x, 0()) ->
    c_3(not^#(evenodd(x, s(0()))), evenodd^#(x, s(0()))) }

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

Strict DPs:
  { evenodd^#(x, 0()) -> c_1(evenodd^#(x, s(0())))
  , evenodd^#(s(x), s(0())) -> c_2(evenodd^#(x, 0())) }
Weak Trs:
  { not(true()) -> false()
  , not(false()) -> true()
  , evenodd(x, 0()) -> not(evenodd(x, s(0())))
  , evenodd(0(), s(0())) -> false()
  , evenodd(s(x), s(0())) -> evenodd(x, 0()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,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(n^2)).

Strict DPs:
  { evenodd^#(x, 0()) -> c_1(evenodd^#(x, s(0())))
  , evenodd^#(s(x), s(0())) -> c_2(evenodd^#(x, 0())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

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).

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

This order satisfies following ordering constraints:

        [evenodd^#(x, 0())] = [0 2] x + [1]              
                              [0 0]     [1]              
                            > [0 2] x + [0]              
                              [0 0]     [1]              
                            = [c_1(evenodd^#(x, s(0())))]
                                                         
  [evenodd^#(s(x), s(0()))] = [0 2] x + [4]              
                              [0 0]     [0]              
                            > [0 2] x + [3]              
                              [0 0]     [0]              
                            = [c_2(evenodd^#(x, 0()))]   
                                                         

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