*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a__b() -> a()
        a__b() -> b()
        a__f(X,X) -> a__f(a(),b())
        a__f(X1,X2) -> f(X1,X2)
        mark(a()) -> a()
        mark(b()) -> a__b()
        mark(f(X1,X2)) -> a__f(mark(X1),X2)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2}
      Obligation:
        Full
        basic terms: {a__b,a__f,mark}/{a,b,f}
    Applied Processor:
      ToInnermost
    Proof:
      switch to innermost, as the system is overlay and right linear and does not contain weak rules
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a__b() -> a()
        a__b() -> b()
        a__f(X,X) -> a__f(a(),b())
        a__f(X1,X2) -> f(X1,X2)
        mark(a()) -> a()
        mark(b()) -> a__b()
        mark(f(X1,X2)) -> a__f(mark(X1),X2)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2}
      Obligation:
        Innermost
        basic terms: {a__b,a__f,mark}/{a,b,f}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        a__b#() -> c_1()
        a__b#() -> c_2()
        a__f#(X,X) -> c_3(a__f#(a(),b()))
        a__f#(X1,X2) -> c_4()
        mark#(a()) -> c_5()
        mark#(b()) -> c_6(a__b#())
        mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        a__b#() -> c_1()
        a__b#() -> c_2()
        a__f#(X,X) -> c_3(a__f#(a(),b()))
        a__f#(X1,X2) -> c_4()
        mark#(a()) -> c_5()
        mark#(b()) -> c_6(a__b#())
        mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a__b() -> a()
        a__b() -> b()
        a__f(X,X) -> a__f(a(),b())
        a__f(X1,X2) -> f(X1,X2)
        mark(a()) -> a()
        mark(b()) -> a__b()
        mark(f(X1,X2)) -> a__f(mark(X1),X2)
      Signature:
        {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
      Obligation:
        Innermost
        basic terms: {a__b#,a__f#,mark#}/{a,b,f}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,2,4,5}
      by application of
        Pre({1,2,4,5}) = {3,6,7}.
      Here rules are labelled as follows:
        1: a__b#() -> c_1()             
        2: a__b#() -> c_2()             
        3: a__f#(X,X) -> c_3(a__f#(a()  
                                  ,b()))
        4: a__f#(X1,X2) -> c_4()        
        5: mark#(a()) -> c_5()          
        6: mark#(b()) -> c_6(a__b#())   
        7: mark#(f(X1,X2)) ->           
             c_7(a__f#(mark(X1),X2)     
                ,mark#(X1))             
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        a__f#(X,X) -> c_3(a__f#(a(),b()))
        mark#(b()) -> c_6(a__b#())
        mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
      Strict TRS Rules:
        
      Weak DP Rules:
        a__b#() -> c_1()
        a__b#() -> c_2()
        a__f#(X1,X2) -> c_4()
        mark#(a()) -> c_5()
      Weak TRS Rules:
        a__b() -> a()
        a__b() -> b()
        a__f(X,X) -> a__f(a(),b())
        a__f(X1,X2) -> f(X1,X2)
        mark(a()) -> a()
        mark(b()) -> a__b()
        mark(f(X1,X2)) -> a__f(mark(X1),X2)
      Signature:
        {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
      Obligation:
        Innermost
        basic terms: {a__b#,a__f#,mark#}/{a,b,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#(X,X) -> c_3(a__f#(a()  
                                  ,b()))
        2: mark#(b()) -> c_6(a__b#())   
        3: mark#(f(X1,X2)) ->           
             c_7(a__f#(mark(X1),X2)     
                ,mark#(X1))             
        4: a__b#() -> c_1()             
        5: a__b#() -> c_2()             
        6: a__f#(X1,X2) -> c_4()        
        7: mark#(a()) -> c_5()          
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
      Strict TRS Rules:
        
      Weak DP Rules:
        a__b#() -> c_1()
        a__b#() -> c_2()
        a__f#(X,X) -> c_3(a__f#(a(),b()))
        a__f#(X1,X2) -> c_4()
        mark#(a()) -> c_5()
        mark#(b()) -> c_6(a__b#())
      Weak TRS Rules:
        a__b() -> a()
        a__b() -> b()
        a__f(X,X) -> a__f(a(),b())
        a__f(X1,X2) -> f(X1,X2)
        mark(a()) -> a()
        mark(b()) -> a__b()
        mark(f(X1,X2)) -> a__f(mark(X1),X2)
      Signature:
        {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
      Obligation:
        Innermost
        basic terms: {a__b#,a__f#,mark#}/{a,b,f}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
           -->_2 mark#(b()) -> c_6(a__b#()):7
           -->_1 a__f#(X,X) -> c_3(a__f#(a(),b())):4
           -->_2 mark#(a()) -> c_5():6
           -->_1 a__f#(X1,X2) -> c_4():5
           -->_2 mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)):1
        
        2:W:a__b#() -> c_1()
           
        
        3:W:a__b#() -> c_2()
           
        
        4:W:a__f#(X,X) -> c_3(a__f#(a(),b()))
           -->_1 a__f#(X1,X2) -> c_4():5
        
        5:W:a__f#(X1,X2) -> c_4()
           
        
        6:W:mark#(a()) -> c_5()
           
        
        7:W:mark#(b()) -> c_6(a__b#())
           -->_1 a__b#() -> c_2():3
           -->_1 a__b#() -> 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#(a()) -> c_5()          
        4: a__f#(X,X) -> c_3(a__f#(a()  
                                  ,b()))
        5: a__f#(X1,X2) -> c_4()        
        7: mark#(b()) -> c_6(a__b#())   
        2: a__b#() -> c_1()             
        3: a__b#() -> c_2()             
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a__b() -> a()
        a__b() -> b()
        a__f(X,X) -> a__f(a(),b())
        a__f(X1,X2) -> f(X1,X2)
        mark(a()) -> a()
        mark(b()) -> a__b()
        mark(f(X1,X2)) -> a__f(mark(X1),X2)
      Signature:
        {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
      Obligation:
        Innermost
        basic terms: {a__b#,a__f#,mark#}/{a,b,f}
    Applied Processor:
      SimplifyRHS
    Proof:
      Consider the dependency graph
        1:S:mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
           -->_2 mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)):1
        
      Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
        mark#(f(X1,X2)) -> c_7(mark#(X1))
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        mark#(f(X1,X2)) -> c_7(mark#(X1))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a__b() -> a()
        a__b() -> b()
        a__f(X,X) -> a__f(a(),b())
        a__f(X1,X2) -> f(X1,X2)
        mark(a()) -> a()
        mark(b()) -> a__b()
        mark(f(X1,X2)) -> a__f(mark(X1),X2)
      Signature:
        {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
      Obligation:
        Innermost
        basic terms: {a__b#,a__f#,mark#}/{a,b,f}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        mark#(f(X1,X2)) -> c_7(mark#(X1))
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        mark#(f(X1,X2)) -> c_7(mark#(X1))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
      Obligation:
        Innermost
        basic terms: {a__b#,a__f#,mark#}/{a,b,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)) ->
             c_7(mark#(X1))  
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          mark#(f(X1,X2)) -> c_7(mark#(X1))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
        Obligation:
          Innermost
          basic terms: {a__b#,a__f#,mark#}/{a,b,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__b#,a__f#,mark#}
        TcT has computed the following interpretation:
              p(a) = [8]          
           p(a__b) = [2]          
           p(a__f) = [4]          
              p(b) = [8]          
              p(f) = [1] x1 + [7] 
           p(mark) = [2] x1 + [0] 
          p(a__b#) = [1]          
          p(a__f#) = [1] x2 + [0] 
          p(mark#) = [4] x1 + [0] 
            p(c_1) = [1]          
            p(c_2) = [0]          
            p(c_3) = [2]          
            p(c_4) = [4]          
            p(c_5) = [8]          
            p(c_6) = [1]          
            p(c_7) = [1] x1 + [14]
        
        Following rules are strictly oriented:
        mark#(f(X1,X2)) = [4] X1 + [28] 
                        > [4] X1 + [14] 
                        = c_7(mark#(X1))
        
        
        Following rules are (at-least) weakly oriented:
        
  *** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          mark#(f(X1,X2)) -> c_7(mark#(X1))
        Weak TRS Rules:
          
        Signature:
          {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
        Obligation:
          Innermost
          basic terms: {a__b#,a__f#,mark#}/{a,b,f}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          mark#(f(X1,X2)) -> c_7(mark#(X1))
        Weak TRS Rules:
          
        Signature:
          {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
        Obligation:
          Innermost
          basic terms: {a__b#,a__f#,mark#}/{a,b,f}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:W:mark#(f(X1,X2)) -> c_7(mark#(X1))
             -->_1 mark#(f(X1,X2)) -> c_7(mark#(X1)):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)) ->
               c_7(mark#(X1))  
  *** 1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
        Obligation:
          Innermost
          basic terms: {a__b#,a__f#,mark#}/{a,b,f}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).