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:
  { 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(1),O(n^1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { not^#(true()) -> c_1()
  , not^#(false()) -> c_2()
  , evenodd^#(x, 0()) -> c_3(not^#(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(1),O(n^1)).

Strict DPs:
  { not^#(true()) -> c_1()
  , not^#(false()) -> c_2()
  , evenodd^#(x, 0()) -> c_3(not^#(evenodd(x, s(0()))))
  , evenodd^#(0(), s(0())) -> c_4()
  , evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
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(1),O(n^1))

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

The following argument positions are usable:
  Uargs(not) = {1}, Uargs(not^#) = {1}, Uargs(c_3) = {1},
  Uargs(c_5) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

            [not](x1) = [1 1] x1 + [0]           
                        [0 0]      [1]           
                                                 
               [true] = [0]                      
                        [1]                      
                                                 
              [false] = [0]                      
                        [1]                      
                                                 
    [evenodd](x1, x2) = [2 1] x1 + [0 2] x2 + [0]
                        [0 0]      [0 0]      [1]
                                                 
                  [0] = [0]                      
                        [1]                      
                                                 
              [s](x1) = [1 1] x1 + [2]           
                        [0 0]      [0]           
                                                 
          [not^#](x1) = [1 0] x1 + [0]           
                        [0 0]      [0]           
                                                 
                [c_1] = [1]                      
                        [0]                      
                                                 
                [c_2] = [1]                      
                        [0]                      
                                                 
  [evenodd^#](x1, x2) = [2 1] x1 + [0]           
                        [0 0]      [0]           
                                                 
            [c_3](x1) = [1 0] x1 + [2]           
                        [0 1]      [2]           
                                                 
                [c_4] = [0]                      
                        [0]                      
                                                 
            [c_5](x1) = [1 0] x1 + [2]           
                        [0 1]      [2]           

The following symbols are considered usable

  {not, evenodd, not^#, evenodd^#}

The order satisfies the following ordering constraints:

              [not(true())] = [1]                             
                              [1]                             
                            > [0]                             
                              [1]                             
                            = [false()]                       
                                                              
             [not(false())] = [1]                             
                              [1]                             
                            > [0]                             
                              [1]                             
                            = [true()]                        
                                                              
          [evenodd(x, 0())] = [2 1] x + [2]                   
                              [0 0]     [1]                   
                            > [2 1] x + [1]                   
                              [0 0]     [1]                   
                            = [not(evenodd(x, s(0())))]       
                                                              
     [evenodd(0(), s(0()))] = [1]                             
                              [1]                             
                            > [0]                             
                              [1]                             
                            = [false()]                       
                                                              
    [evenodd(s(x), s(0()))] = [2 2] x + [4]                   
                              [0 0]     [1]                   
                            > [2 1] x + [2]                   
                              [0 0]     [1]                   
                            = [evenodd(x, 0())]               
                                                              
            [not^#(true())] = [0]                             
                              [0]                             
                            ? [1]                             
                              [0]                             
                            = [c_1()]                         
                                                              
           [not^#(false())] = [0]                             
                              [0]                             
                            ? [1]                             
                              [0]                             
                            = [c_2()]                         
                                                              
        [evenodd^#(x, 0())] = [2 1] x + [0]                   
                              [0 0]     [0]                   
                            ? [2 1] x + [2]                   
                              [0 0]     [2]                   
                            = [c_3(not^#(evenodd(x, s(0()))))]
                                                              
   [evenodd^#(0(), s(0()))] = [1]                             
                              [0]                             
                            > [0]                             
                              [0]                             
                            = [c_4()]                         
                                                              
  [evenodd^#(s(x), s(0()))] = [2 2] x + [4]                   
                              [0 0]     [0]                   
                            ? [2 1] x + [2]                   
                              [0 0]     [2]                   
                            = [c_5(evenodd^#(x, 0()))]        
                                                              

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:
  { not^#(true()) -> c_1()
  , not^#(false()) -> c_2()
  , evenodd^#(x, 0()) -> c_3(not^#(evenodd(x, s(0()))))
  , evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
Weak DPs: { 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(1),O(1))

We estimate the number of application of {1,2} by applications of
Pre({1,2}) = {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()))))
    , 4: evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0()))
    , 5: evenodd^#(0(), s(0())) -> c_4() }

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

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

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

Strict DPs: { evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
Weak DPs:
  { not^#(true()) -> c_1()
  , not^#(false()) -> c_2()
  , evenodd^#(x, 0()) -> c_3(not^#(evenodd(x, s(0()))))
  , 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(1),O(1))

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

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

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

Weak DPs:
  { not^#(true()) -> c_1()
  , not^#(false()) -> c_2()
  , evenodd^#(x, 0()) -> c_3(not^#(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(1),O(1))

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^#(x, 0()) -> c_3(not^#(evenodd(x, s(0()))))
, evenodd^#(0(), s(0())) -> c_4()
, evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }

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

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(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))