MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            f(0()) -> 0()
            f(s(x)) -> -(s(x),g(f(x)))
            g(0()) -> s(0())
            g(s(x)) -> -(s(x),f(g(x)))
        - Signature:
            {-/2,f/1,g/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-,f,g} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          -#(x,0()) -> c_1()
          -#(0(),s(y)) -> c_2()
          -#(s(x),s(y)) -> c_3(-#(x,y))
          f#(0()) -> c_4()
          f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
          g#(0()) -> c_6()
          g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1()
            -#(0(),s(y)) -> c_2()
            -#(s(x),s(y)) -> c_3(-#(x,y))
            f#(0()) -> c_4()
            f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
            g#(0()) -> c_6()
            g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
        - Weak TRS:
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            f(0()) -> 0()
            f(s(x)) -> -(s(x),g(f(x)))
            g(0()) -> s(0())
            g(s(x)) -> -(s(x),f(g(x)))
        - Signature:
            {-/2,f/1,g/1,-#/2,f#/1,g#/1} / {0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/3,c_6/0,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,f#,g#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,4,6}
        by application of
          Pre({1,2,4,6}) = {3,5,7}.
        Here rules are labelled as follows:
          1: -#(x,0()) -> c_1()
          2: -#(0(),s(y)) -> c_2()
          3: -#(s(x),s(y)) -> c_3(-#(x,y))
          4: f#(0()) -> c_4()
          5: f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
          6: g#(0()) -> c_6()
          7: g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
* Step 3: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_3(-#(x,y))
            f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
            g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
        - Weak DPs:
            -#(x,0()) -> c_1()
            -#(0(),s(y)) -> c_2()
            f#(0()) -> c_4()
            g#(0()) -> c_6()
        - Weak TRS:
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            f(0()) -> 0()
            f(s(x)) -> -(s(x),g(f(x)))
            g(0()) -> s(0())
            g(s(x)) -> -(s(x),f(g(x)))
        - Signature:
            {-/2,f/1,g/1,-#/2,f#/1,g#/1} / {0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/3,c_6/0,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,f#,g#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:-#(s(x),s(y)) -> c_3(-#(x,y))
             -->_1 -#(0(),s(y)) -> c_2():5
             -->_1 -#(x,0()) -> c_1():4
             -->_1 -#(s(x),s(y)) -> c_3(-#(x,y)):1
          
          2:S:f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
             -->_2 g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x)):3
             -->_2 g#(0()) -> c_6():7
             -->_3 f#(0()) -> c_4():6
             -->_1 -#(x,0()) -> c_1():4
             -->_3 f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x)):2
             -->_1 -#(s(x),s(y)) -> c_3(-#(x,y)):1
          
          3:S:g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
             -->_3 g#(0()) -> c_6():7
             -->_2 f#(0()) -> c_4():6
             -->_1 -#(x,0()) -> c_1():4
             -->_3 g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x)):3
             -->_2 f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x)):2
             -->_1 -#(s(x),s(y)) -> c_3(-#(x,y)):1
          
          4:W:-#(x,0()) -> c_1()
             
          
          5:W:-#(0(),s(y)) -> c_2()
             
          
          6:W:f#(0()) -> c_4()
             
          
          7:W:g#(0()) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: f#(0()) -> c_4()
          7: g#(0()) -> c_6()
          4: -#(x,0()) -> c_1()
          5: -#(0(),s(y)) -> c_2()
* Step 4: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_3(-#(x,y))
            f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
            g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
        - Weak TRS:
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            f(0()) -> 0()
            f(s(x)) -> -(s(x),g(f(x)))
            g(0()) -> s(0())
            g(s(x)) -> -(s(x),f(g(x)))
        - Signature:
            {-/2,f/1,g/1,-#/2,f#/1,g#/1} / {0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/3,c_6/0,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,f#,g#} and constructors {0,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              -#(s(x),s(y)) -> c_3(-#(x,y))
          - Weak DPs:
              f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
              g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
          - Weak TRS:
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              f(0()) -> 0()
              f(s(x)) -> -(s(x),g(f(x)))
              g(0()) -> s(0())
              g(s(x)) -> -(s(x),f(g(x)))
          - Signature:
              {-/2,f/1,g/1,-#/2,f#/1,g#/1} / {0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/3,c_6/0,c_7/3}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {-#,f#,g#} and constructors {0,s}
        
        Problem (S)
          - Strict DPs:
              f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
              g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
          - Weak DPs:
              -#(s(x),s(y)) -> c_3(-#(x,y))
          - Weak TRS:
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              f(0()) -> 0()
              f(s(x)) -> -(s(x),g(f(x)))
              g(0()) -> s(0())
              g(s(x)) -> -(s(x),f(g(x)))
          - Signature:
              {-/2,f/1,g/1,-#/2,f#/1,g#/1} / {0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/3,c_6/0,c_7/3}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {-#,f#,g#} and constructors {0,s}
** Step 4.a:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          -#(s(x),s(y)) -> c_3(-#(x,y))
      - Weak DPs:
          f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
          g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
      - Weak TRS:
          -(x,0()) -> x
          -(0(),s(y)) -> 0()
          -(s(x),s(y)) -> -(x,y)
          f(0()) -> 0()
          f(s(x)) -> -(s(x),g(f(x)))
          g(0()) -> s(0())
          g(s(x)) -> -(s(x),f(g(x)))
      - Signature:
          {-/2,f/1,g/1,-#/2,f#/1,g#/1} / {0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/3,c_6/0,c_7/3}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {-#,f#,g#} and constructors {0,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 4.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
            g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
        - Weak DPs:
            -#(s(x),s(y)) -> c_3(-#(x,y))
        - Weak TRS:
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            f(0()) -> 0()
            f(s(x)) -> -(s(x),g(f(x)))
            g(0()) -> s(0())
            g(s(x)) -> -(s(x),f(g(x)))
        - Signature:
            {-/2,f/1,g/1,-#/2,f#/1,g#/1} / {0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/3,c_6/0,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,f#,g#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
             -->_1 -#(s(x),s(y)) -> c_3(-#(x,y)):3
             -->_2 g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x)):2
             -->_3 f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x)):1
          
          2:S:g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
             -->_1 -#(s(x),s(y)) -> c_3(-#(x,y)):3
             -->_3 g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x)):2
             -->_2 f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x)):1
          
          3:W:-#(s(x),s(y)) -> c_3(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_3(-#(x,y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: -#(s(x),s(y)) -> c_3(-#(x,y))
** Step 4.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
            g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
        - Weak TRS:
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            f(0()) -> 0()
            f(s(x)) -> -(s(x),g(f(x)))
            g(0()) -> s(0())
            g(s(x)) -> -(s(x),f(g(x)))
        - Signature:
            {-/2,f/1,g/1,-#/2,f#/1,g#/1} / {0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/3,c_6/0,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,f#,g#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x))
             -->_2 g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x)):2
             -->_3 f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x)):1
          
          2:S:g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x))
             -->_3 g#(s(x)) -> c_7(-#(s(x),f(g(x))),f#(g(x)),g#(x)):2
             -->_2 f#(s(x)) -> c_5(-#(s(x),g(f(x))),g#(f(x)),f#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(s(x)) -> c_5(g#(f(x)),f#(x))
          g#(s(x)) -> c_7(f#(g(x)),g#(x))
** Step 4.b:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          f#(s(x)) -> c_5(g#(f(x)),f#(x))
          g#(s(x)) -> c_7(f#(g(x)),g#(x))
      - Weak TRS:
          -(x,0()) -> x
          -(0(),s(y)) -> 0()
          -(s(x),s(y)) -> -(x,y)
          f(0()) -> 0()
          f(s(x)) -> -(s(x),g(f(x)))
          g(0()) -> s(0())
          g(s(x)) -> -(s(x),f(g(x)))
      - Signature:
          {-/2,f/1,g/1,-#/2,f#/1,g#/1} / {0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {-#,f#,g#} and constructors {0,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE