WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            bits(0()) -> 0()
            bits(s(0())) -> s(0())
            bits(s(s(x))) -> s(bits(s(half(x))))
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          bits#(0()) -> c_1()
          bits#(s(0())) -> c_2()
          bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
          half#(0()) -> c_4()
          half#(s(0())) -> c_5()
          half#(s(s(x))) -> c_6(half#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bits#(0()) -> c_1()
            bits#(s(0())) -> c_2()
            bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
            half#(0()) -> c_4()
            half#(s(0())) -> c_5()
            half#(s(s(x))) -> c_6(half#(x))
        - Weak TRS:
            bits(0()) -> 0()
            bits(s(0())) -> s(0())
            bits(s(s(x))) -> s(bits(s(half(x))))
            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_2/0,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,4,5}
        by application of
          Pre({1,2,4,5}) = {3,6}.
        Here rules are labelled as follows:
          1: bits#(0()) -> c_1()
          2: bits#(s(0())) -> c_2()
          3: bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
          4: half#(0()) -> c_4()
          5: half#(s(0())) -> c_5()
          6: half#(s(s(x))) -> c_6(half#(x))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
            half#(s(s(x))) -> c_6(half#(x))
        - Weak DPs:
            bits#(0()) -> c_1()
            bits#(s(0())) -> c_2()
            half#(0()) -> c_4()
            half#(s(0())) -> c_5()
        - Weak TRS:
            bits(0()) -> 0()
            bits(s(0())) -> s(0())
            bits(s(s(x))) -> s(bits(s(half(x))))
            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_2/0,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
             -->_2 half#(s(s(x))) -> c_6(half#(x)):2
             -->_2 half#(s(0())) -> c_5():6
             -->_2 half#(0()) -> c_4():5
             -->_1 bits#(s(0())) -> c_2():4
             -->_1 bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x)):1
          
          2:S:half#(s(s(x))) -> c_6(half#(x))
             -->_1 half#(s(0())) -> c_5():6
             -->_1 half#(0()) -> c_4():5
             -->_1 half#(s(s(x))) -> c_6(half#(x)):2
          
          3:W:bits#(0()) -> c_1()
             
          
          4:W:bits#(s(0())) -> c_2()
             
          
          5:W:half#(0()) -> c_4()
             
          
          6:W:half#(s(0())) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: bits#(0()) -> c_1()
          4: bits#(s(0())) -> c_2()
          5: half#(0()) -> c_4()
          6: half#(s(0())) -> c_5()
* Step 4: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
            half#(s(s(x))) -> c_6(half#(x))
        - Weak TRS:
            bits(0()) -> 0()
            bits(s(0())) -> s(0())
            bits(s(s(x))) -> s(bits(s(half(x))))
            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_2/0,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          half(0()) -> 0()
          half(s(0())) -> 0()
          half(s(s(x))) -> s(half(x))
          bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
          half#(s(s(x))) -> c_6(half#(x))
* Step 5: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
            half#(s(s(x))) -> c_6(half#(x))
        - Weak TRS:
            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_2/0,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
        and a lower component
          half#(s(s(x))) -> c_6(half#(x))
        Further, following extension rules are added to the lower component.
          bits#(s(s(x))) -> bits#(s(half(x)))
          bits#(s(s(x))) -> half#(x)
** Step 5.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
        - Weak TRS:
            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_2/0,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x))
             -->_1 bits#(s(s(x))) -> c_3(bits#(s(half(x))),half#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          bits#(s(s(x))) -> c_3(bits#(s(half(x))))
** Step 5.a:2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bits#(s(s(x))) -> c_3(bits#(s(half(x))))
        - Weak TRS:
            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_2/0,c_3/1,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, 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:
          The following argument positions are considered usable:
            uargs(s) = {1},
            uargs(bits#) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [0]         
             p(bits) = [0]         
             p(half) = [1] x1 + [1]
                p(s) = [1] x1 + [9]
            p(bits#) = [1] x1 + [4]
            p(half#) = [2] x1 + [1]
              p(c_1) = [4]         
              p(c_2) = [1]         
              p(c_3) = [1] x1 + [1]
              p(c_4) = [1]         
              p(c_5) = [0]         
              p(c_6) = [1]         
          
          Following rules are strictly oriented:
          bits#(s(s(x))) = [1] x + [22]          
                         > [1] x + [15]          
                         = c_3(bits#(s(half(x))))
          
          
          Following rules are (at-least) weakly oriented:
              half(0()) =  [1]         
                        >= [0]         
                        =  0()         
          
           half(s(0())) =  [10]        
                        >= [0]         
                        =  0()         
          
          half(s(s(x))) =  [1] x + [19]
                        >= [1] x + [10]
                        =  s(half(x))  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 5.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            bits#(s(s(x))) -> c_3(bits#(s(half(x))))
        - Weak TRS:
            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_2/0,c_3/1,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(s(s(x))) -> c_6(half#(x))
        - Weak DPs:
            bits#(s(s(x))) -> bits#(s(half(x)))
            bits#(s(s(x))) -> half#(x)
        - Weak TRS:
            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_2/0,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, 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:
          The following argument positions are considered usable:
            uargs(s) = {1},
            uargs(bits#) = {1},
            uargs(c_6) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [0]         
             p(bits) = [0]         
             p(half) = [1] x1 + [3]
                p(s) = [1] x1 + [8]
            p(bits#) = [1] x1 + [5]
            p(half#) = [1] x1 + [4]
              p(c_1) = [0]         
              p(c_2) = [0]         
              p(c_3) = [0]         
              p(c_4) = [0]         
              p(c_5) = [0]         
              p(c_6) = [1] x1 + [2]
          
          Following rules are strictly oriented:
          half#(s(s(x))) = [1] x + [20] 
                         > [1] x + [6]  
                         = c_6(half#(x))
          
          
          Following rules are (at-least) weakly oriented:
          bits#(s(s(x))) =  [1] x + [21]     
                         >= [1] x + [16]     
                         =  bits#(s(half(x)))
          
          bits#(s(s(x))) =  [1] x + [21]     
                         >= [1] x + [4]      
                         =  half#(x)         
          
               half(0()) =  [3]              
                         >= [0]              
                         =  0()              
          
            half(s(0())) =  [11]             
                         >= [0]              
                         =  0()              
          
           half(s(s(x))) =  [1] x + [19]     
                         >= [1] x + [11]     
                         =  s(half(x))       
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            bits#(s(s(x))) -> bits#(s(half(x)))
            bits#(s(s(x))) -> half#(x)
            half#(s(s(x))) -> c_6(half#(x))
        - Weak TRS:
            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_2/0,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))