MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            a__c() -> a()
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),b(),X) -> a__f(X,X,mark(X))
            mark(a()) -> a()
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,mark(X3))
        - Signature:
            {a__c/0,a__f/3,mark/1} / {a/0,b/0,c/0,f/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {a,b,c,f}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a__c#() -> c_1()
          a__c#() -> c_2()
          a__c#() -> c_3()
          a__f#(X1,X2,X3) -> c_4()
          a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X))
          mark#(a()) -> c_6()
          mark#(b()) -> c_7()
          mark#(c()) -> c_8(a__c#())
          mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            a__c#() -> c_1()
            a__c#() -> c_2()
            a__c#() -> c_3()
            a__f#(X1,X2,X3) -> c_4()
            a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X))
            mark#(a()) -> c_6()
            mark#(b()) -> c_7()
            mark#(c()) -> c_8(a__c#())
            mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
        - Weak TRS:
            a__c() -> a()
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),b(),X) -> a__f(X,X,mark(X))
            mark(a()) -> a()
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,mark(X3))
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {a/0,b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0
            ,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {a,b,c,f}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,4,6,7}
        by application of
          Pre({1,2,3,4,6,7}) = {5,8,9}.
        Here rules are labelled as follows:
          1: a__c#() -> c_1()
          2: a__c#() -> c_2()
          3: a__c#() -> c_3()
          4: a__f#(X1,X2,X3) -> c_4()
          5: a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X))
          6: mark#(a()) -> c_6()
          7: mark#(b()) -> c_7()
          8: mark#(c()) -> c_8(a__c#())
          9: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X))
            mark#(c()) -> c_8(a__c#())
            mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
        - Weak DPs:
            a__c#() -> c_1()
            a__c#() -> c_2()
            a__c#() -> c_3()
            a__f#(X1,X2,X3) -> c_4()
            mark#(a()) -> c_6()
            mark#(b()) -> c_7()
        - Weak TRS:
            a__c() -> a()
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),b(),X) -> a__f(X,X,mark(X))
            mark(a()) -> a()
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,mark(X3))
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {a/0,b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0
            ,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {a,b,c,f}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1,3}.
        Here rules are labelled as follows:
          1: a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X))
          2: mark#(c()) -> c_8(a__c#())
          3: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
          4: a__c#() -> c_1()
          5: a__c#() -> c_2()
          6: a__c#() -> c_3()
          7: a__f#(X1,X2,X3) -> c_4()
          8: mark#(a()) -> c_6()
          9: mark#(b()) -> c_7()
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X))
            mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
        - Weak DPs:
            a__c#() -> c_1()
            a__c#() -> c_2()
            a__c#() -> c_3()
            a__f#(X1,X2,X3) -> c_4()
            mark#(a()) -> c_6()
            mark#(b()) -> c_7()
            mark#(c()) -> c_8(a__c#())
        - Weak TRS:
            a__c() -> a()
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),b(),X) -> a__f(X,X,mark(X))
            mark(a()) -> a()
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,mark(X3))
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {a/0,b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0
            ,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {a,b,c,f}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X))
             -->_2 mark#(c()) -> c_8(a__c#()):9
             -->_2 mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3)):2
             -->_2 mark#(b()) -> c_7():8
             -->_2 mark#(a()) -> c_6():7
             -->_1 a__f#(X1,X2,X3) -> c_4():6
          
          2:S:mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
             -->_2 mark#(c()) -> c_8(a__c#()):9
             -->_2 mark#(b()) -> c_7():8
             -->_2 mark#(a()) -> c_6():7
             -->_1 a__f#(X1,X2,X3) -> c_4():6
             -->_2 mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3)):2
             -->_1 a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X)):1
          
          3:W:a__c#() -> c_1()
             
          
          4:W:a__c#() -> c_2()
             
          
          5:W:a__c#() -> c_3()
             
          
          6:W:a__f#(X1,X2,X3) -> c_4()
             
          
          7:W:mark#(a()) -> c_6()
             
          
          8:W:mark#(b()) -> c_7()
             
          
          9:W:mark#(c()) -> c_8(a__c#())
             -->_1 a__c#() -> c_3():5
             -->_1 a__c#() -> c_2():4
             -->_1 a__c#() -> c_1():3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: a__f#(X1,X2,X3) -> c_4()
          7: mark#(a()) -> c_6()
          8: mark#(b()) -> c_7()
          9: mark#(c()) -> c_8(a__c#())
          3: a__c#() -> c_1()
          4: a__c#() -> c_2()
          5: a__c#() -> c_3()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X))
            mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
        - Weak TRS:
            a__c() -> a()
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),b(),X) -> a__f(X,X,mark(X))
            mark(a()) -> a()
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,mark(X3))
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {a/0,b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0
            ,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {a,b,c,f}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X))
             -->_2 mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3)):2
          
          2:S:mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
             -->_2 mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3)):2
             -->_1 a__f#(a(),b(),X) -> c_5(a__f#(X,X,mark(X)),mark#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          a__f#(a(),b(),X) -> c_5(mark#(X))
* Step 6: WeightGap MAYBE
    + Considered Problem:
        - Strict DPs:
            a__f#(a(),b(),X) -> c_5(mark#(X))
            mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
        - Weak TRS:
            a__c() -> a()
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),b(),X) -> a__f(X,X,mark(X))
            mark(a()) -> a()
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,mark(X3))
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {a/0,b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0
            ,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {a,b,c,f}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            uargs(a__f) = {3},
            uargs(a__f#) = {3},
            uargs(c_5) = {1},
            uargs(c_9) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(a) = [0]                  
             p(a__c) = [0]                  
             p(a__f) = [1] x3 + [0]         
                p(b) = [0]                  
                p(c) = [0]                  
                p(f) = [0]                  
             p(mark) = [0]                  
            p(a__c#) = [2]                  
            p(a__f#) = [1] x3 + [6]         
            p(mark#) = [0]                  
              p(c_1) = [0]                  
              p(c_2) = [0]                  
              p(c_3) = [0]                  
              p(c_4) = [0]                  
              p(c_5) = [1] x1 + [0]         
              p(c_6) = [0]                  
              p(c_7) = [0]                  
              p(c_8) = [0]                  
              p(c_9) = [1] x1 + [1] x2 + [5]
          
          Following rules are strictly oriented:
          a__f#(a(),b(),X) = [1] X + [6]  
                           > [0]          
                           = c_5(mark#(X))
          
          
          Following rules are (at-least) weakly oriented:
          mark#(f(X1,X2,X3)) =  [0]                                 
                             >= [11]                                
                             =  c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
          
                      a__c() =  [0]                                 
                             >= [0]                                 
                             =  a()                                 
          
                      a__c() =  [0]                                 
                             >= [0]                                 
                             =  b()                                 
          
                      a__c() =  [0]                                 
                             >= [0]                                 
                             =  c()                                 
          
              a__f(X1,X2,X3) =  [1] X3 + [0]                        
                             >= [0]                                 
                             =  f(X1,X2,X3)                         
          
             a__f(a(),b(),X) =  [1] X + [0]                         
                             >= [0]                                 
                             =  a__f(X,X,mark(X))                   
          
                   mark(a()) =  [0]                                 
                             >= [0]                                 
                             =  a()                                 
          
                   mark(b()) =  [0]                                 
                             >= [0]                                 
                             =  b()                                 
          
                   mark(c()) =  [0]                                 
                             >= [0]                                 
                             =  a__c()                              
          
           mark(f(X1,X2,X3)) =  [0]                                 
                             >= [0]                                 
                             =  a__f(X1,X2,mark(X3))                
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,mark(X3)),mark#(X3))
      - Weak DPs:
          a__f#(a(),b(),X) -> c_5(mark#(X))
      - Weak TRS:
          a__c() -> a()
          a__c() -> b()
          a__c() -> c()
          a__f(X1,X2,X3) -> f(X1,X2,X3)
          a__f(a(),b(),X) -> a__f(X,X,mark(X))
          mark(a()) -> a()
          mark(b()) -> b()
          mark(c()) -> a__c()
          mark(f(X1,X2,X3)) -> a__f(X1,X2,mark(X3))
      - Signature:
          {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {a/0,b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0
          ,c_8/1,c_9/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {a,b,c,f}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE