*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a__c() -> b()
        a__c() -> c()
        a__f(X1,X2,X3) -> f(X1,X2,X3)
        a__f(b(),X,c()) -> a__f(X,a__c(),X)
        mark(b()) -> b()
        mark(c()) -> a__c()
        mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
      Obligation:
        Innermost
        basic terms: {a__c,a__f,mark}/{b,c,f}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        a__c#() -> c_1()
        a__c#() -> c_2()
        a__f#(X1,X2,X3) -> c_3()
        a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
        mark#(b()) -> c_5()
        mark#(c()) -> c_6(a__c#())
        mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        a__c#() -> c_1()
        a__c#() -> c_2()
        a__f#(X1,X2,X3) -> c_3()
        a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
        mark#(b()) -> c_5()
        mark#(c()) -> c_6(a__c#())
        mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a__c() -> b()
        a__c() -> c()
        a__f(X1,X2,X3) -> f(X1,X2,X3)
        a__f(b(),X,c()) -> a__f(X,a__c(),X)
        mark(b()) -> b()
        mark(c()) -> a__c()
        mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
      Signature:
        {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
      Obligation:
        Innermost
        basic terms: {a__c#,a__f#,mark#}/{b,c,f}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,2,3,5}
      by application of
        Pre({1,2,3,5}) = {4,6,7}.
      Here rules are labelled as follows:
        1: a__c#() -> c_1()                    
        2: a__c#() -> c_2()                    
        3: a__f#(X1,X2,X3) -> c_3()            
        4: a__f#(b(),X,c()) -> c_4(a__f#(X     
                                        ,a__c()
                                        ,X)    
                                  ,a__c#())    
        5: mark#(b()) -> c_5()                 
        6: mark#(c()) -> c_6(a__c#())          
        7: mark#(f(X1,X2,X3)) ->               
             c_7(a__f#(X1,mark(X2),X3)         
                ,mark#(X2))                    
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
        mark#(c()) -> c_6(a__c#())
        mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
      Strict TRS Rules:
        
      Weak DP Rules:
        a__c#() -> c_1()
        a__c#() -> c_2()
        a__f#(X1,X2,X3) -> c_3()
        mark#(b()) -> c_5()
      Weak TRS Rules:
        a__c() -> b()
        a__c() -> c()
        a__f(X1,X2,X3) -> f(X1,X2,X3)
        a__f(b(),X,c()) -> a__f(X,a__c(),X)
        mark(b()) -> b()
        mark(c()) -> a__c()
        mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
      Signature:
        {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
      Obligation:
        Innermost
        basic terms: {a__c#,a__f#,mark#}/{b,c,f}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,2}
      by application of
        Pre({1,2}) = {3}.
      Here rules are labelled as follows:
        1: a__f#(b(),X,c()) -> c_4(a__f#(X     
                                        ,a__c()
                                        ,X)    
                                  ,a__c#())    
        2: mark#(c()) -> c_6(a__c#())          
        3: mark#(f(X1,X2,X3)) ->               
             c_7(a__f#(X1,mark(X2),X3)         
                ,mark#(X2))                    
        4: a__c#() -> c_1()                    
        5: a__c#() -> c_2()                    
        6: a__f#(X1,X2,X3) -> c_3()            
        7: mark#(b()) -> c_5()                 
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
      Strict TRS Rules:
        
      Weak DP Rules:
        a__c#() -> c_1()
        a__c#() -> c_2()
        a__f#(X1,X2,X3) -> c_3()
        a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
        mark#(b()) -> c_5()
        mark#(c()) -> c_6(a__c#())
      Weak TRS Rules:
        a__c() -> b()
        a__c() -> c()
        a__f(X1,X2,X3) -> f(X1,X2,X3)
        a__f(b(),X,c()) -> a__f(X,a__c(),X)
        mark(b()) -> b()
        mark(c()) -> a__c()
        mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
      Signature:
        {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
      Obligation:
        Innermost
        basic terms: {a__c#,a__f#,mark#}/{b,c,f}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
           -->_2 mark#(c()) -> c_6(a__c#()):7
           -->_1 a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#()):5
           -->_2 mark#(b()) -> c_5():6
           -->_1 a__f#(X1,X2,X3) -> c_3():4
           -->_2 mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2)):1
        
        2:W:a__c#() -> c_1()
           
        
        3:W:a__c#() -> c_2()
           
        
        4:W:a__f#(X1,X2,X3) -> c_3()
           
        
        5:W:a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
           -->_1 a__f#(X1,X2,X3) -> c_3():4
           -->_2 a__c#() -> c_2():3
           -->_2 a__c#() -> c_1():2
        
        6:W:mark#(b()) -> c_5()
           
        
        7:W:mark#(c()) -> c_6(a__c#())
           -->_1 a__c#() -> c_2():3
           -->_1 a__c#() -> c_1():2
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        6: mark#(b()) -> c_5()                 
        5: a__f#(b(),X,c()) -> c_4(a__f#(X     
                                        ,a__c()
                                        ,X)    
                                  ,a__c#())    
        4: a__f#(X1,X2,X3) -> c_3()            
        7: mark#(c()) -> c_6(a__c#())          
        2: a__c#() -> c_1()                    
        3: a__c#() -> c_2()                    
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a__c() -> b()
        a__c() -> c()
        a__f(X1,X2,X3) -> f(X1,X2,X3)
        a__f(b(),X,c()) -> a__f(X,a__c(),X)
        mark(b()) -> b()
        mark(c()) -> a__c()
        mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
      Signature:
        {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
      Obligation:
        Innermost
        basic terms: {a__c#,a__f#,mark#}/{b,c,f}
    Applied Processor:
      SimplifyRHS
    Proof:
      Consider the dependency graph
        1:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
           -->_2 mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2)):1
        
      Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
        mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a__c() -> b()
        a__c() -> c()
        a__f(X1,X2,X3) -> f(X1,X2,X3)
        a__f(b(),X,c()) -> a__f(X,a__c(),X)
        mark(b()) -> b()
        mark(c()) -> a__c()
        mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
      Signature:
        {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
      Obligation:
        Innermost
        basic terms: {a__c#,a__f#,mark#}/{b,c,f}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
      Obligation:
        Innermost
        basic terms: {a__c#,a__f#,mark#}/{b,c,f}
    Applied Processor:
      PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
    Proof:
      We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
        1: mark#(f(X1,X2,X3)) ->
             c_7(mark#(X2))     
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
        Obligation:
          Innermost
          basic terms: {a__c#,a__f#,mark#}/{b,c,f}
      Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
      Proof:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_7) = {1}
        
        Following symbols are considered usable:
          {a__c#,a__f#,mark#}
        TcT has computed the following interpretation:
           p(a__c) = [1]                  
           p(a__f) = [1] x1 + [1] x3 + [8]
              p(b) = [0]                  
              p(c) = [1]                  
              p(f) = [1] x2 + [6]         
           p(mark) = [1]                  
          p(a__c#) = [8]                  
          p(a__f#) = [2] x3 + [1]         
          p(mark#) = [3] x1 + [8]         
            p(c_1) = [1]                  
            p(c_2) = [1]                  
            p(c_3) = [0]                  
            p(c_4) = [1]                  
            p(c_5) = [0]                  
            p(c_6) = [1] x1 + [1]         
            p(c_7) = [1] x1 + [10]        
        
        Following rules are strictly oriented:
        mark#(f(X1,X2,X3)) = [3] X2 + [26] 
                           > [3] X2 + [18] 
                           = c_7(mark#(X2))
        
        
        Following rules are (at-least) weakly oriented:
        
  *** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
        Weak TRS Rules:
          
        Signature:
          {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
        Obligation:
          Innermost
          basic terms: {a__c#,a__f#,mark#}/{b,c,f}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
        Weak TRS Rules:
          
        Signature:
          {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
        Obligation:
          Innermost
          basic terms: {a__c#,a__f#,mark#}/{b,c,f}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:W:mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
             -->_1 mark#(f(X1,X2,X3)) -> c_7(mark#(X2)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: mark#(f(X1,X2,X3)) ->
               c_7(mark#(X2))     
  *** 1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
        Obligation:
          Innermost
          basic terms: {a__c#,a__f#,mark#}/{b,c,f}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).