/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            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))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            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))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            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))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          half(x){x -> s(s(x))} =
            half(s(s(x))) ->^+ s(half(x))
              = C[half(x) = half(x){}]

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
** Step 1.b:2: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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))
        - 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_ = WIDP}
    + Details:
        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.
** Step 1.b:3: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - 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))
        - Strict TRS:
            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))
        - 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 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#(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))
** Step 1.b:4: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - 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))
        - Strict 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_2/1,c_3/0,c_4/0,c_5/1}
        - 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 = WgOnTrs}
    + 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_2) = {1},
            uargs(c_5) = {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 + [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.
** Step 1.b:5: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - 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 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_2/1,c_3/0,c_4/0,c_5/1}
        - 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
          {1,3,4}
        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))
** Step 1.b:6: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bits#(s(x)) -> c_2(bits#(half(s(x))))
            half#(s(s(x))) -> c_5(half#(x))
        - Weak DPs:
            bits#(0()) -> c_1()
            half#(0()) -> c_3()
            half#(s(0())) -> c_4()
        - 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_2/1,c_3/0,c_4/0,c_5/1}
        - 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(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()
** Step 1.b:7: Decompose.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bits#(s(x)) -> c_2(bits#(half(s(x))))
            half#(s(s(x))) -> c_5(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_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        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 DPs:
              bits#(s(x)) -> c_2(bits#(half(s(x))))
          - Weak DPs:
              half#(s(s(x))) -> c_5(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_2/1,c_3/0,c_4/0,c_5/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
        
        Problem (S)
          - Strict DPs:
              half#(s(s(x))) -> c_5(half#(x))
          - Weak DPs:
              bits#(s(x)) -> c_2(bits#(half(s(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_2/1,c_3/0,c_4/0,c_5/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
*** Step 1.b:7.a:1: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bits#(s(x)) -> c_2(bits#(half(s(x))))
        - Weak DPs:
            half#(s(s(x))) -> c_5(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_2/1,c_3/0,c_4/0,c_5/1}
        - 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(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))
*** Step 1.b:7.a:2: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bits#(s(x)) -> c_2(bits#(half(s(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_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} 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.
**** Step 1.b:7.a:2.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bits#(s(x)) -> c_2(bits#(half(s(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_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        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) = [0]           
                     [1]           
           p(bits) = [0 1] x1 + [0]
                     [1 1]      [0]
           p(half) = [0 1] x1 + [0]
                     [0 1]      [0]
              p(s) = [0 1] x1 + [5]
                     [0 1]      [4]
          p(bits#) = [2 0] x1 + [0]
                     [0 2]      [0]
          p(half#) = [1]           
                     [0]           
            p(c_1) = [2]           
                     [1]           
            p(c_2) = [1 0] x1 + [0]
                     [0 0]      [0]
            p(c_3) = [2]           
                     [4]           
            p(c_4) = [1]           
                     [0]           
            p(c_5) = [0 1] x1 + [0]
                     [0 1]      [1]
        
        Following rules are strictly oriented:
        bits#(s(x)) = [0 2] x + [10]        
                      [0 2]     [8]         
                    > [0 2] x + [8]         
                      [0 0]     [0]         
                    = c_2(bits#(half(s(x))))
        
        
        Following rules are (at-least) weakly oriented:
            half(0()) =  [1]          
                         [1]          
                      >= [0]          
                         [1]          
                      =  0()          
        
         half(s(0())) =  [5]          
                         [5]          
                      >= [0]          
                         [1]          
                      =  0()          
        
        half(s(s(x))) =  [0 1] x + [8]
                         [0 1]     [8]
                      >= [0 1] x + [5]
                         [0 1]     [4]
                      =  s(half(x))   
        
**** Step 1.b:7.a:2.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            bits#(s(x)) -> c_2(bits#(half(s(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_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

**** Step 1.b:7.a:2.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            bits#(s(x)) -> c_2(bits#(half(s(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_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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))))
**** Step 1.b:7.a:2.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - 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_2/1,c_3/0,c_4/0,c_5/1}
        - 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 1.b:7.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(s(s(x))) -> c_5(half#(x))
        - Weak DPs:
            bits#(s(x)) -> c_2(bits#(half(s(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_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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))))
*** Step 1.b:7.b:2: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(s(s(x))) -> c_5(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_2/1,c_3/0,c_4/0,c_5/1}
        - 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#(s(s(x))) -> c_5(half#(x))
*** Step 1.b:7.b:3: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(s(s(x))) -> c_5(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 runtime complexity wrt. defined symbols {bits#,half#} and constructors {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}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: half#(s(s(x))) -> c_5(half#(x))
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:7.b:3.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(s(s(x))) -> c_5(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 runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        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) = [4]          
           p(bits) = [0]          
           p(half) = [0]          
              p(s) = [1] x1 + [1] 
          p(bits#) = [0]          
          p(half#) = [8] x1 + [2] 
            p(c_1) = [1]          
            p(c_2) = [2] x1 + [2] 
            p(c_3) = [1]          
            p(c_4) = [1]          
            p(c_5) = [1] x1 + [14]
        
        Following rules are strictly oriented:
        half#(s(s(x))) = [8] x + [18] 
                       > [8] x + [16] 
                       = c_5(half#(x))
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 1.b:7.b:3.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            half#(s(s(x))) -> c_5(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 runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

**** Step 1.b:7.b:3.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            half#(s(s(x))) -> c_5(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 runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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))
**** Step 1.b:7.b:3.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - 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 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(Omega(n^1),O(n^1))