MAYBE
* Step 1: InnermostRuleRemoval MAYBE
    + Considered Problem:
        - Strict TRS:
            f(x,x) -> f(i(x),g(g(x)))
            f(x,y) -> x
            f(x,i(x)) -> f(x,x)
            f(i(x),i(g(x))) -> a()
            g(x) -> i(x)
        - Signature:
            {f/2,g/1} / {a/0,i/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {a,i}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          f(i(x),i(g(x))) -> a()
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            f(x,x) -> f(i(x),g(g(x)))
            f(x,y) -> x
            f(x,i(x)) -> f(x,x)
            g(x) -> i(x)
        - Signature:
            {f/2,g/1} / {a/0,i/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {a,i}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x))
          f#(x,y) -> c_2()
          f#(x,i(x)) -> c_3(f#(x,x))
          g#(x) -> c_4()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x))
            f#(x,y) -> c_2()
            f#(x,i(x)) -> c_3(f#(x,x))
            g#(x) -> c_4()
        - Weak TRS:
            f(x,x) -> f(i(x),g(g(x)))
            f(x,y) -> x
            f(x,i(x)) -> f(x,x)
            g(x) -> i(x)
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {a/0,i/1,c_1/3,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,i}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          g(x) -> i(x)
          f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x))
          f#(x,y) -> c_2()
          f#(x,i(x)) -> c_3(f#(x,x))
          g#(x) -> c_4()
* Step 4: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x))
            f#(x,y) -> c_2()
            f#(x,i(x)) -> c_3(f#(x,x))
            g#(x) -> c_4()
        - Weak TRS:
            g(x) -> i(x)
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {a/0,i/1,c_1/3,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,i}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4}
        by application of
          Pre({2,4}) = {1,3}.
        Here rules are labelled as follows:
          1: f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x))
          2: f#(x,y) -> c_2()
          3: f#(x,i(x)) -> c_3(f#(x,x))
          4: g#(x) -> c_4()
* Step 5: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x))
            f#(x,i(x)) -> c_3(f#(x,x))
        - Weak DPs:
            f#(x,y) -> c_2()
            g#(x) -> c_4()
        - Weak TRS:
            g(x) -> i(x)
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {a/0,i/1,c_1/3,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,i}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x))
             -->_1 f#(x,i(x)) -> c_3(f#(x,x)):2
             -->_3 g#(x) -> c_4():4
             -->_2 g#(x) -> c_4():4
             -->_1 f#(x,y) -> c_2():3
             -->_1 f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x)):1
          
          2:S:f#(x,i(x)) -> c_3(f#(x,x))
             -->_1 f#(x,y) -> c_2():3
             -->_1 f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x)):1
          
          3:W:f#(x,y) -> c_2()
             
          
          4:W:g#(x) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: g#(x) -> c_4()
          3: f#(x,y) -> c_2()
* Step 6: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x))
            f#(x,i(x)) -> c_3(f#(x,x))
        - Weak TRS:
            g(x) -> i(x)
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {a/0,i/1,c_1/3,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,i}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x))
             -->_1 f#(x,i(x)) -> c_3(f#(x,x)):2
             -->_1 f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x)):1
          
          2:S:f#(x,i(x)) -> c_3(f#(x,x))
             -->_1 f#(x,x) -> c_1(f#(i(x),g(g(x))),g#(g(x)),g#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(x,x) -> c_1(f#(i(x),g(g(x))))
* Step 7: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          f#(x,x) -> c_1(f#(i(x),g(g(x))))
          f#(x,i(x)) -> c_3(f#(x,x))
      - Weak TRS:
          g(x) -> i(x)
      - Signature:
          {f/2,g/1,f#/2,g#/1} / {a/0,i/1,c_1/1,c_2/0,c_3/1,c_4/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,i}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE