* Step 1: ToInnermost 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:
             runtime complexity wrt. defined symbols {half,log} 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 2: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(log) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [1]         
            p(half) = [8]         
             p(log) = [1] x1 + [3]
               p(s) = [1] x1 + [0]
          
          Following rules are strictly oriented:
            half(0()) = [8]
                      > [1]
                      = 0()
          
          log(s(0())) = [4]
                      > [1]
                      = 0()
          
          
          Following rules are (at-least) weakly oriented:
          half(s(s(x))) =  [8]               
                        >= [8]               
                        =  s(half(x))        
          
           log(s(s(x))) =  [1] x + [3]       
                        >= [11]              
                        =  s(log(s(half(x))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            half(s(s(x))) -> s(half(x))
            log(s(s(x))) -> s(log(s(half(x))))
        - Weak TRS:
            half(0()) -> 0()
            log(s(0())) -> 0()
        - Signature:
            {half/1,log/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
        
        The following argument positions are considered usable:
          uargs(log) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {half,log}
        TcT has computed the following interpretation:
             p(0) = [0]          
          p(half) = [1] x_1 + [0]
           p(log) = [1] x_1 + [6]
             p(s) = [1] x_1 + [8]
        
        Following rules are strictly oriented:
        half(s(s(x))) = [1] x + [16]
                      > [1] x + [8] 
                      = s(half(x))  
        
        
        Following rules are (at-least) weakly oriented:
           half(0()) =  [0]               
                     >= [0]               
                     =  0()               
        
         log(s(0())) =  [14]              
                     >= [0]               
                     =  0()               
        
        log(s(s(x))) =  [1] x + [22]      
                     >= [1] x + [22]      
                     =  s(log(s(half(x))))
        
* Step 4: MI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            log(s(s(x))) -> s(log(s(half(x))))
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
            log(s(0())) -> 0()
        - Signature:
            {half/1,log/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
        
        The following argument positions are considered usable:
          uargs(log) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {half,log}
        TcT has computed the following interpretation:
             p(0) = [2]           
          p(half) = [1] x_1 + [0] 
           p(log) = [4] x_1 + [13]
             p(s) = [1] x_1 + [2] 
        
        Following rules are strictly oriented:
        log(s(s(x))) = [4] x + [29]      
                     > [4] x + [23]      
                     = s(log(s(half(x))))
        
        
        Following rules are (at-least) weakly oriented:
            half(0()) =  [2]        
                      >= [2]        
                      =  0()        
        
        half(s(s(x))) =  [1] x + [4]
                      >= [1] x + [2]
                      =  s(half(x)) 
        
          log(s(0())) =  [29]       
                      >= [2]        
                      =  0()        
        
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak 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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))