* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(half(x))))
        - Signature:
            {half/1,log/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          half#(0()) -> c_1()
          half#(s(s(x))) -> c_2(half#(x))
          log#(s(0())) -> c_3()
          log#(s(s(x))) -> c_4(log#(s(half(x))))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(0()) -> c_1()
            half#(s(s(x))) -> c_2(half#(x))
            log#(s(0())) -> c_3()
            log#(s(s(x))) -> c_4(log#(s(half(x))))
        - Strict TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(half(x))))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          half(0()) -> 0()
          half(s(s(x))) -> s(half(x))
          half#(0()) -> c_1()
          half#(s(s(x))) -> c_2(half#(x))
          log#(s(0())) -> c_3()
          log#(s(s(x))) -> c_4(log#(s(half(x))))
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(0()) -> c_1()
            half#(s(s(x))) -> c_2(half#(x))
            log#(s(0())) -> c_3()
            log#(s(s(x))) -> c_4(log#(s(half(x))))
        - Strict TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} 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(log#) = {1},
            uargs(c_2) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [0]         
             p(half) = [1] x1 + [5]
              p(log) = [0]         
                p(s) = [1] x1 + [1]
            p(half#) = [8] x1 + [0]
             p(log#) = [1] x1 + [0]
              p(c_1) = [0]         
              p(c_2) = [1] x1 + [0]
              p(c_3) = [0]         
              p(c_4) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          half#(s(s(x))) = [8] x + [16] 
                         > [8] x + [0]  
                         = c_2(half#(x))
          
            log#(s(0())) = [1]          
                         > [0]          
                         = c_3()        
          
               half(0()) = [5]          
                         > [0]          
                         = 0()          
          
           half(s(s(x))) = [1] x + [7]  
                         > [1] x + [6]  
                         = s(half(x))   
          
          
          Following rules are (at-least) weakly oriented:
             half#(0()) =  [0]                  
                        >= [0]                  
                        =  c_1()                
          
          log#(s(s(x))) =  [1] x + [2]          
                        >= [1] x + [6]          
                        =  c_4(log#(s(half(x))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(0()) -> c_1()
            log#(s(s(x))) -> c_4(log#(s(half(x))))
        - Weak DPs:
            half#(s(s(x))) -> c_2(half#(x))
            log#(s(0())) -> c_3()
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:half#(0()) -> c_1()
             
          
          2:S:log#(s(s(x))) -> c_4(log#(s(half(x))))
             -->_1 log#(s(0())) -> c_3():4
             -->_1 log#(s(s(x))) -> c_4(log#(s(half(x)))):2
          
          3:W:half#(s(s(x))) -> c_2(half#(x))
             -->_1 half#(s(s(x))) -> c_2(half#(x)):3
             -->_1 half#(0()) -> c_1():1
          
          4:W:log#(s(0())) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: log#(s(0())) -> c_3()
* Step 5: Decompose WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(0()) -> c_1()
            log#(s(s(x))) -> c_4(log#(s(half(x))))
        - Weak DPs:
            half#(s(s(x))) -> c_2(half#(x))
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} 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:
              half#(0()) -> c_1()
          - Weak DPs:
              half#(s(s(x))) -> c_2(half#(x))
              log#(s(s(x))) -> c_4(log#(s(half(x))))
          - Weak TRS:
              half(0()) -> 0()
              half(s(s(x))) -> s(half(x))
          - Signature:
              {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
        
        Problem (S)
          - Strict DPs:
              log#(s(s(x))) -> c_4(log#(s(half(x))))
          - Weak DPs:
              half#(0()) -> c_1()
              half#(s(s(x))) -> c_2(half#(x))
          - Weak TRS:
              half(0()) -> 0()
              half(s(s(x))) -> s(half(x))
          - Signature:
              {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            half#(0()) -> c_1()
        - Weak DPs:
            half#(s(s(x))) -> c_2(half#(x))
            log#(s(s(x))) -> c_4(log#(s(half(x))))
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:half#(0()) -> c_1()
             
          
          2:W:log#(s(s(x))) -> c_4(log#(s(half(x))))
             -->_1 log#(s(s(x))) -> c_4(log#(s(half(x)))):2
          
          3:W:half#(s(s(x))) -> c_2(half#(x))
             -->_1 half#(0()) -> c_1():1
             -->_1 half#(s(s(x))) -> c_2(half#(x)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: log#(s(s(x))) -> c_4(log#(s(half(x))))
** Step 5.a:2: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            half#(0()) -> c_1()
        - Weak DPs:
            half#(s(s(x))) -> c_2(half#(x))
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          half#(0()) -> c_1()
          half#(s(s(x))) -> c_2(half#(x))
** Step 5.a:3: PredecessorEstimationCP WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            half#(0()) -> c_1()
        - Weak DPs:
            half#(s(s(x))) -> c_2(half#(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: half#(0()) -> c_1()
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.a:3.a:1: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            half#(0()) -> c_1()
        - Weak DPs:
            half#(s(s(x))) -> c_2(half#(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, 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 0 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#,log#}
        TcT has computed the following interpretation:
              p(0) = [0]         
           p(half) = [0]         
            p(log) = [0]         
              p(s) = [6]         
          p(half#) = [5]         
           p(log#) = [4] x1 + [0]
            p(c_1) = [0]         
            p(c_2) = [1] x1 + [0]
            p(c_3) = [1]         
            p(c_4) = [4] x1 + [0]
        
        Following rules are strictly oriented:
        half#(0()) = [5]  
                   > [0]  
                   = c_1()
        
        
        Following rules are (at-least) weakly oriented:
        half#(s(s(x))) =  [5]          
                       >= [5]          
                       =  c_2(half#(x))
        
*** Step 5.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            half#(0()) -> c_1()
            half#(s(s(x))) -> c_2(half#(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            half#(0()) -> c_1()
            half#(s(s(x))) -> c_2(half#(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:half#(0()) -> c_1()
             
          
          2:W:half#(s(s(x))) -> c_2(half#(x))
             -->_1 half#(s(s(x))) -> c_2(half#(x)):2
             -->_1 half#(0()) -> c_1():1
          
        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_2(half#(x))
          1: half#(0()) -> c_1()
*** Step 5.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_4(log#(s(half(x))))
        - Weak DPs:
            half#(0()) -> c_1()
            half#(s(s(x))) -> c_2(half#(x))
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:log#(s(s(x))) -> c_4(log#(s(half(x))))
             -->_1 log#(s(s(x))) -> c_4(log#(s(half(x)))):1
          
          2:W:half#(0()) -> c_1()
             
          
          3:W:half#(s(s(x))) -> c_2(half#(x))
             -->_1 half#(s(s(x))) -> c_2(half#(x)):3
             -->_1 half#(0()) -> c_1():2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: half#(s(s(x))) -> c_2(half#(x))
          2: half#(0()) -> c_1()
** Step 5.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_4(log#(s(half(x))))
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} 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: log#(s(s(x))) -> c_4(log#(s(half(x))))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_4(log#(s(half(x))))
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} 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_4) = {1}
        
        Following symbols are considered usable:
          {half,half#,log#}
        TcT has computed the following interpretation:
              p(0) = [1]         
           p(half) = [1] x1 + [0]
            p(log) = [1] x1 + [1]
              p(s) = [1] x1 + [2]
          p(half#) = [2] x1 + [2]
           p(log#) = [4] x1 + [0]
            p(c_1) = [2]         
            p(c_2) = [4] x1 + [8]
            p(c_3) = [2]         
            p(c_4) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        log#(s(s(x))) = [4] x + [16]         
                      > [4] x + [8]          
                      = c_4(log#(s(half(x))))
        
        
        Following rules are (at-least) weakly oriented:
            half(0()) =  [1]        
                      >= [1]        
                      =  0()        
        
        half(s(s(x))) =  [1] x + [4]
                      >= [1] x + [2]
                      =  s(half(x)) 
        
*** Step 5.b:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            log#(s(s(x))) -> c_4(log#(s(half(x))))
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            log#(s(s(x))) -> c_4(log#(s(half(x))))
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:log#(s(s(x))) -> c_4(log#(s(half(x))))
             -->_1 log#(s(s(x))) -> c_4(log#(s(half(x)))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: log#(s(s(x))) -> c_4(log#(s(half(x))))
*** Step 5.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {half/1,log/1,half#/1,log#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half#,log#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))