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


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WORST_CASE(?,O(n^2))
* Step 1: WeightGap.  WORST_CASE(?,O(n^2))
    + 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 derivational complexity wrt. signature {0,half,log,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind triangular matrix interpretation:
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [7]          
            p(half) = [1] x1 + [11]
             p(log) = [1] x1 + [4] 
               p(s) = [1] x1 + [0] 
          
          Following rules are strictly oriented:
            half(0()) = [18]
                      > [7] 
                      = 0() 
          
          log(s(0())) = [11]
                      > [7] 
                      = 0() 
          
          
          Following rules are (at-least) weakly oriented:
          half(s(s(x))) =  [1] x + [11]      
                        >= [1] x + [11]      
                        =  s(half(x))        
          
           log(s(s(x))) =  [1] x + [4]       
                        >= [1] x + [15]      
                        =  s(log(s(half(x))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + 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 derivational complexity wrt. signature {0,half,log,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
             p(0) = [2]         
          p(half) = [1] x1 + [0]
           p(log) = [1] x1 + [0]
             p(s) = [1] x1 + [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()) =  [2]               
                     >= [2]               
                     =  0()               
        
         log(s(0())) =  [10]              
                     >= [2]               
                     =  0()               
        
        log(s(s(x))) =  [1] x + [16]      
                     >= [1] x + [16]      
                     =  s(log(s(half(x))))
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + 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 derivational complexity wrt. signature {0,half,log,s}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
             p(0) = [0]           
                    [0]           
          p(half) = [1 0] x1 + [0]
                    [0 1]      [0]
           p(log) = [1 4] x1 + [2]
                    [0 1]      [6]
             p(s) = [1 0] x1 + [2]
                    [0 1]      [1]
        
        Following rules are strictly oriented:
        log(s(s(x))) = [1 4] x + [14]    
                       [0 1]     [8]     
                     > [1 4] x + [10]    
                       [0 1]     [8]     
                     = s(log(s(half(x))))
        
        
        Following rules are (at-least) weakly oriented:
            half(0()) =  [0]          
                         [0]          
                      >= [0]          
                         [0]          
                      =  0()          
        
        half(s(s(x))) =  [1 0] x + [4]
                         [0 1]     [2]
                      >= [1 0] x + [2]
                         [0 1]     [1]
                      =  s(half(x))   
        
          log(s(0())) =  [8]          
                         [7]          
                      >= [0]          
                         [0]          
                      =  0()          
        
* Step 4: 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 derivational complexity wrt. signature {0,half,log,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))