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_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          f#(x,x) -> c_1(f#(i(x),g(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))))
            f#(x,y) -> c_2()
            f#(x,i(x)) -> c_3(f#(x,x))
            g#(x) -> c_4()
        - 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,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:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          g(x) -> i(x)
          f#(x,x) -> c_1(f#(i(x),g(g(x))))
          f#(x,y) -> c_2()
          f#(x,i(x)) -> c_3(f#(x,x))
          g#(x) -> c_4()
* Step 4: WeightGap MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(x,x) -> c_1(f#(i(x),g(g(x))))
            f#(x,y) -> c_2()
            f#(x,i(x)) -> c_3(f#(x,x))
            g#(x) -> c_4()
        - Strict 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(g) = {1},
            uargs(f#) = {2},
            uargs(c_1) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(a) = [0]          
              p(f) = [0]          
              p(g) = [1] x1 + [1] 
              p(i) = [1] x1 + [0] 
             p(f#) = [1] x2 + [10]
             p(g#) = [0]          
            p(c_1) = [1] x1 + [0] 
            p(c_2) = [0]          
            p(c_3) = [1] x1 + [0] 
            p(c_4) = [0]          
          
          Following rules are strictly oriented:
          f#(x,y) = [1] y + [10]
                  > [0]         
                  = c_2()       
          
             g(x) = [1] x + [1] 
                  > [1] x + [0] 
                  = i(x)        
          
          
          Following rules are (at-least) weakly oriented:
             f#(x,x) =  [1] x + [10]         
                     >= [1] x + [12]         
                     =  c_1(f#(i(x),g(g(x))))
          
          f#(x,i(x)) =  [1] x + [10]         
                     >= [1] x + [10]         
                     =  c_3(f#(x,x))         
          
               g#(x) =  [0]                  
                     >= [0]                  
                     =  c_4()                
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation 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))
            g#(x) -> c_4()
        - Weak DPs:
            f#(x,y) -> c_2()
        - 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:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3}
        by application of
          Pre({3}) = {}.
        Here rules are labelled as follows:
          1: f#(x,x) -> c_1(f#(i(x),g(g(x))))
          2: f#(x,i(x)) -> c_3(f#(x,x))
          3: g#(x) -> c_4()
          4: f#(x,y) -> c_2()
* Step 6: RemoveWeakSuffixes 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 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_1/1,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,i}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(x,x) -> c_1(f#(i(x),g(g(x))))
             -->_1 f#(x,i(x)) -> c_3(f#(x,x)):2
             -->_1 f#(x,y) -> c_2():3
             -->_1 f#(x,x) -> c_1(f#(i(x),g(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)))):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 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