*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        bits(0()) -> 0()
        bits(s(x)) -> s(bits(half(s(x))))
        half(0()) -> 0()
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {bits/1,half/1} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {bits,half}/{0,s}
    Applied Processor:
      DependencyPairs {dpKind_ = WIDP}
    Proof:
      We add the following weak innermost dependency pairs:
      
      Strict DPs
        bits#(0()) -> c_1()
        bits#(s(x)) -> c_2(bits#(half(s(x))))
        half#(0()) -> c_3()
        half#(s(0())) -> c_4()
        half#(s(s(x))) -> c_5(half#(x))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        bits#(0()) -> c_1()
        bits#(s(x)) -> c_2(bits#(half(s(x))))
        half#(0()) -> c_3()
        half#(s(0())) -> c_4()
        half#(s(s(x))) -> c_5(half#(x))
      Strict TRS Rules:
        bits(0()) -> 0()
        bits(s(x)) -> s(bits(half(s(x))))
        half(0()) -> 0()
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
      Obligation:
        Innermost
        basic terms: {bits#,half#}/{0,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        half(0()) -> 0()
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
        bits#(0()) -> c_1()
        bits#(s(x)) -> c_2(bits#(half(s(x))))
        half#(0()) -> c_3()
        half#(s(0())) -> c_4()
        half#(s(s(x))) -> c_5(half#(x))
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        bits#(0()) -> c_1()
        bits#(s(x)) -> c_2(bits#(half(s(x))))
        half#(0()) -> c_3()
        half#(s(0())) -> c_4()
        half#(s(s(x))) -> c_5(half#(x))
      Strict TRS Rules:
        half(0()) -> 0()
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
      Obligation:
        Innermost
        basic terms: {bits#,half#}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    Proof:
      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(s) = {1},
          uargs(bits#) = {1},
          uargs(c_2) = {1},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
              p(0) = [0]          
           p(bits) = [0]          
           p(half) = [1] x1 + [1] 
              p(s) = [1] x1 + [10]
          p(bits#) = [1] x1 + [0] 
          p(half#) = [0]          
            p(c_1) = [0]          
            p(c_2) = [1] x1 + [0] 
            p(c_3) = [0]          
            p(c_4) = [0]          
            p(c_5) = [1] x1 + [0] 
        
        Following rules are strictly oriented:
            half(0()) = [1]         
                      > [0]         
                      = 0()         
        
         half(s(0())) = [11]        
                      > [0]         
                      = 0()         
        
        half(s(s(x))) = [1] x + [21]
                      > [1] x + [11]
                      = s(half(x))  
        
        
        Following rules are (at-least) weakly oriented:
            bits#(0()) =  [0]                   
                       >= [0]                   
                       =  c_1()                 
        
           bits#(s(x)) =  [1] x + [10]          
                       >= [1] x + [11]          
                       =  c_2(bits#(half(s(x))))
        
            half#(0()) =  [0]                   
                       >= [0]                   
                       =  c_3()                 
        
         half#(s(0())) =  [0]                   
                       >= [0]                   
                       =  c_4()                 
        
        half#(s(s(x))) =  [0]                   
                       >= [0]                   
                       =  c_5(half#(x))         
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        bits#(0()) -> c_1()
        bits#(s(x)) -> c_2(bits#(half(s(x))))
        half#(0()) -> c_3()
        half#(s(0())) -> c_4()
        half#(s(s(x))) -> c_5(half#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        half(0()) -> 0()
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
      Signature:
        {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
      Obligation:
        Innermost
        basic terms: {bits#,half#}/{0,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,3,4}
      by application of
        Pre({1,3,4}) = {2,5}.
      Here rules are labelled as follows:
        1: bits#(0()) -> c_1()            
        2: bits#(s(x)) ->                 
             c_2(bits#(half(s(x))))       
        3: half#(0()) -> c_3()            
        4: half#(s(0())) -> c_4()         
        5: half#(s(s(x))) -> c_5(half#(x))
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        bits#(s(x)) -> c_2(bits#(half(s(x))))
        half#(s(s(x))) -> c_5(half#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        bits#(0()) -> c_1()
        half#(0()) -> c_3()
        half#(s(0())) -> c_4()
      Weak TRS Rules:
        half(0()) -> 0()
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
      Signature:
        {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
      Obligation:
        Innermost
        basic terms: {bits#,half#}/{0,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:bits#(s(x)) -> c_2(bits#(half(s(x))))
           -->_1 bits#(0()) -> c_1():3
           -->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):1
        
        2:S:half#(s(s(x))) -> c_5(half#(x))
           -->_1 half#(s(0())) -> c_4():5
           -->_1 half#(0()) -> c_3():4
           -->_1 half#(s(s(x))) -> c_5(half#(x)):2
        
        3:W:bits#(0()) -> c_1()
           
        
        4:W:half#(0()) -> c_3()
           
        
        5:W:half#(s(0())) -> c_4()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        4: half#(0()) -> c_3()   
        5: half#(s(0())) -> c_4()
        3: bits#(0()) -> c_1()   
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        bits#(s(x)) -> c_2(bits#(half(s(x))))
        half#(s(s(x))) -> c_5(half#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        half(0()) -> 0()
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
      Signature:
        {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
      Obligation:
        Innermost
        basic terms: {bits#,half#}/{0,s}
    Applied Processor:
      Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    Proof:
      We analyse the complexity of following sub-problems (R) and (S).
      Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
      
      Problem (R)
        Strict DP Rules:
          bits#(s(x)) -> c_2(bits#(half(s(x))))
        Strict TRS Rules:
          
        Weak DP Rules:
          half#(s(s(x))) -> c_5(half#(x))
        Weak TRS Rules:
          half(0()) -> 0()
          half(s(0())) -> 0()
          half(s(s(x))) -> s(half(x))
        Signature:
          {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {bits#,half#}/{0,s}
      
      Problem (S)
        Strict DP Rules:
          half#(s(s(x))) -> c_5(half#(x))
        Strict TRS Rules:
          
        Weak DP Rules:
          bits#(s(x)) -> c_2(bits#(half(s(x))))
        Weak TRS Rules:
          half(0()) -> 0()
          half(s(0())) -> 0()
          half(s(s(x))) -> s(half(x))
        Signature:
          {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {bits#,half#}/{0,s}
  *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          bits#(s(x)) -> c_2(bits#(half(s(x))))
        Strict TRS Rules:
          
        Weak DP Rules:
          half#(s(s(x))) -> c_5(half#(x))
        Weak TRS Rules:
          half(0()) -> 0()
          half(s(0())) -> 0()
          half(s(s(x))) -> s(half(x))
        Signature:
          {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {bits#,half#}/{0,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:S:bits#(s(x)) -> c_2(bits#(half(s(x))))
             -->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):1
          
          2:W:half#(s(s(x))) -> c_5(half#(x))
             -->_1 half#(s(s(x))) -> c_5(half#(x)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: half#(s(s(x))) -> c_5(half#(x))
  *** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          bits#(s(x)) -> c_2(bits#(half(s(x))))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          half(0()) -> 0()
          half(s(0())) -> 0()
          half(s(s(x))) -> s(half(x))
        Signature:
          {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {bits#,half#}/{0,s}
      Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
      Proof:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
          1: bits#(s(x)) ->          
               c_2(bits#(half(s(x))))
          
        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:
            bits#(s(x)) -> c_2(bits#(half(s(x))))
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
          Signature:
            {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
          Obligation:
            Innermost
            basic terms: {bits#,half#}/{0,s}
        Applied Processor:
          NaturalMI {miDimension = 3, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            {half,bits#,half#}
          TcT has computed the following interpretation:
                p(0) = [1]             
                       [2]             
                       [2]             
             p(bits) = [0]             
                       [1]             
                       [0]             
             p(half) = [0 1 0]      [0]
                       [0 1 0] x1 + [1]
                       [1 0 0]      [2]
                p(s) = [0 1 0]      [3]
                       [0 1 0] x1 + [2]
                       [0 0 0]      [0]
            p(bits#) = [1 0 0]      [2]
                       [0 0 0] x1 + [0]
                       [0 0 0]      [0]
            p(half#) = [2 0 0]      [0]
                       [0 2 2] x1 + [0]
                       [0 0 0]      [0]
              p(c_1) = [2]             
                       [1]             
                       [1]             
              p(c_2) = [1 0 0]      [0]
                       [0 2 2] x1 + [0]
                       [0 2 1]      [0]
              p(c_3) = [0]             
                       [0]             
                       [2]             
              p(c_4) = [0]             
                       [1]             
                       [0]             
              p(c_5) = [0 0 0]      [1]
                       [0 1 1] x1 + [0]
                       [0 0 2]      [0]
          
          Following rules are strictly oriented:
          bits#(s(x)) = [0 1 0]     [5]       
                        [0 0 0] x + [0]       
                        [0 0 0]     [0]       
                      > [0 1 0]     [4]       
                        [0 0 0] x + [0]       
                        [0 0 0]     [0]       
                      = c_2(bits#(half(s(x))))
          
          
          Following rules are (at-least) weakly oriented:
              half(0()) =  [2]            
                           [3]            
                           [3]            
                        >= [1]            
                           [2]            
                           [2]            
                        =  0()            
          
           half(s(0())) =  [4]            
                           [5]            
                           [7]            
                        >= [1]            
                           [2]            
                           [2]            
                        =  0()            
          
          half(s(s(x))) =  [0 1 0]     [4]
                           [0 1 0] x + [5]
                           [0 1 0]     [7]
                        >= [0 1 0]     [4]
                           [0 1 0] x + [3]
                           [0 0 0]     [0]
                        =  s(half(x))     
          
    *** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            bits#(s(x)) -> c_2(bits#(half(s(x))))
          Weak TRS Rules:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
          Signature:
            {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
          Obligation:
            Innermost
            basic terms: {bits#,half#}/{0,s}
        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:
            bits#(s(x)) -> c_2(bits#(half(s(x))))
          Weak TRS Rules:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
          Signature:
            {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
          Obligation:
            Innermost
            basic terms: {bits#,half#}/{0,s}
        Applied Processor:
          RemoveWeakSuffixes
        Proof:
          Consider the dependency graph
            1:W:bits#(s(x)) -> c_2(bits#(half(s(x))))
               -->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):1
            
          The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
            1: bits#(s(x)) ->          
                 c_2(bits#(half(s(x))))
    *** 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:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
          Signature:
            {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
          Obligation:
            Innermost
            basic terms: {bits#,half#}/{0,s}
        Applied Processor:
          EmptyProcessor
        Proof:
          The problem is already closed. The intended complexity is O(1).
    
  *** 1.1.1.1.1.1.2 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          half#(s(s(x))) -> c_5(half#(x))
        Strict TRS Rules:
          
        Weak DP Rules:
          bits#(s(x)) -> c_2(bits#(half(s(x))))
        Weak TRS Rules:
          half(0()) -> 0()
          half(s(0())) -> 0()
          half(s(s(x))) -> s(half(x))
        Signature:
          {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {bits#,half#}/{0,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:S:half#(s(s(x))) -> c_5(half#(x))
             -->_1 half#(s(s(x))) -> c_5(half#(x)):1
          
          2:W:bits#(s(x)) -> c_2(bits#(half(s(x))))
             -->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: bits#(s(x)) ->          
               c_2(bits#(half(s(x))))
  *** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          half#(s(s(x))) -> c_5(half#(x))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          half(0()) -> 0()
          half(s(0())) -> 0()
          half(s(s(x))) -> s(half(x))
        Signature:
          {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {bits#,half#}/{0,s}
      Applied Processor:
        UsableRules
      Proof:
        We replace rewrite rules by usable rules:
          half#(s(s(x))) -> c_5(half#(x))
  *** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          half#(s(s(x))) -> c_5(half#(x))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {bits#,half#}/{0,s}
      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: half#(s(s(x))) -> c_5(half#(x))
          
        The strictly oriented rules are moved into the weak component.
    *** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            half#(s(s(x))) -> c_5(half#(x))
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            
          Signature:
            {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
          Obligation:
            Innermost
            basic terms: {bits#,half#}/{0,s}
        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_5) = {1}
          
          Following symbols are considered usable:
            {bits#,half#}
          TcT has computed the following interpretation:
                p(0) = [1]         
             p(bits) = [1] x1 + [1]
             p(half) = [1]         
                p(s) = [1] x1 + [1]
            p(bits#) = [2] x1 + [0]
            p(half#) = [1] x1 + [0]
              p(c_1) = [1]         
              p(c_2) = [1] x1 + [0]
              p(c_3) = [0]         
              p(c_4) = [4]         
              p(c_5) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          half#(s(s(x))) = [1] x + [2]  
                         > [1] x + [1]  
                         = c_5(half#(x))
          
          
          Following rules are (at-least) weakly oriented:
          
    *** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            half#(s(s(x))) -> c_5(half#(x))
          Weak TRS Rules:
            
          Signature:
            {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
          Obligation:
            Innermost
            basic terms: {bits#,half#}/{0,s}
        Applied Processor:
          Assumption
        Proof:
          ()
    
    *** 1.1.1.1.1.1.2.1.1.2 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            half#(s(s(x))) -> c_5(half#(x))
          Weak TRS Rules:
            
          Signature:
            {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
          Obligation:
            Innermost
            basic terms: {bits#,half#}/{0,s}
        Applied Processor:
          RemoveWeakSuffixes
        Proof:
          Consider the dependency graph
            1:W:half#(s(s(x))) -> c_5(half#(x))
               -->_1 half#(s(s(x))) -> c_5(half#(x)):1
            
          The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
            1: half#(s(s(x))) -> c_5(half#(x))
    *** 1.1.1.1.1.1.2.1.1.2.1 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            
          Signature:
            {bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
          Obligation:
            Innermost
            basic terms: {bits#,half#}/{0,s}
        Applied Processor:
          EmptyProcessor
        Proof:
          The problem is already closed. The intended complexity is O(1).