/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?,O(n^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a(),a()) -> f(a(),b())
            f(a(),b()) -> f(s(a()),c())
            f(c(),c()) -> f(a(),a())
            f(s(X),c()) -> f(X,c())
        - Signature:
            {f/2} / {a/0,b/0,c/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,f,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(a) = [0]                  
            p(b) = [0]                  
            p(c) = [0]                  
            p(f) = [1] x1 + [1] x2 + [0]
            p(s) = [1] x1 + [5]         
          
          Following rules are strictly oriented:
          f(s(X),c()) = [1] X + [5]
                      > [1] X + [0]
                      = f(X,c())   
          
          
          Following rules are (at-least) weakly oriented:
          f(a(),a()) =  [0]          
                     >= [0]          
                     =  f(a(),b())   
          
          f(a(),b()) =  [0]          
                     >= [5]          
                     =  f(s(a()),c())
          
          f(c(),c()) =  [0]          
                     >= [0]          
                     =  f(a(),a())   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a(),a()) -> f(a(),b())
            f(a(),b()) -> f(s(a()),c())
            f(c(),c()) -> f(a(),a())
        - Weak TRS:
            f(s(X),c()) -> f(X,c())
        - Signature:
            {f/2} / {a/0,b/0,c/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,f,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(a) = [12]                 
            p(b) = [2]                  
            p(c) = [7]                  
            p(f) = [1] x1 + [1] x2 + [2]
            p(s) = [1] x1 + [6]         
          
          Following rules are strictly oriented:
          f(a(),a()) = [26]      
                     > [16]      
                     = f(a(),b())
          
          
          Following rules are (at-least) weakly oriented:
           f(a(),b()) =  [16]         
                      >= [27]         
                      =  f(s(a()),c())
          
           f(c(),c()) =  [16]         
                      >= [26]         
                      =  f(a(),a())   
          
          f(s(X),c()) =  [1] X + [15] 
                      >= [1] X + [9]  
                      =  f(X,c())     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a(),b()) -> f(s(a()),c())
            f(c(),c()) -> f(a(),a())
        - Weak TRS:
            f(a(),a()) -> f(a(),b())
            f(s(X),c()) -> f(X,c())
        - Signature:
            {f/2} / {a/0,b/0,c/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,f,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(a) = [3]                  
            p(b) = [3]                  
            p(c) = [13]                 
            p(f) = [1] x1 + [1] x2 + [4]
            p(s) = [1] x1 + [7]         
          
          Following rules are strictly oriented:
          f(c(),c()) = [30]      
                     > [10]      
                     = f(a(),a())
          
          
          Following rules are (at-least) weakly oriented:
           f(a(),a()) =  [10]         
                      >= [10]         
                      =  f(a(),b())   
          
           f(a(),b()) =  [10]         
                      >= [27]         
                      =  f(s(a()),c())
          
          f(s(X),c()) =  [1] X + [24] 
                      >= [1] X + [17] 
                      =  f(X,c())     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a(),b()) -> f(s(a()),c())
        - Weak TRS:
            f(a(),a()) -> f(a(),b())
            f(c(),c()) -> f(a(),a())
            f(s(X),c()) -> f(X,c())
        - Signature:
            {f/2} / {a/0,b/0,c/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,f,s}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [7]                      
                 [1]                      
          p(b) = [0]                      
                 [8]                      
          p(c) = [0]                      
                 [4]                      
          p(f) = [1 4] x1 + [1 1] x2 + [1]
                 [0 0]      [0 0]      [4]
          p(s) = [1 4] x1 + [1]           
                 [0 0]      [0]           
        
        Following rules are strictly oriented:
        f(a(),b()) = [20]         
                     [4]          
                   > [17]         
                     [4]          
                   = f(s(a()),c())
        
        
        Following rules are (at-least) weakly oriented:
         f(a(),a()) =  [20]         
                       [4]          
                    >= [20]         
                       [4]          
                    =  f(a(),b())   
        
         f(c(),c()) =  [21]         
                       [4]          
                    >= [20]         
                       [4]          
                    =  f(a(),a())   
        
        f(s(X),c()) =  [1 4] X + [6]
                       [0 0]     [4]
                    >= [1 4] X + [5]
                       [0 0]     [4]
                    =  f(X,c())     
        
* Step 5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(a(),a()) -> f(a(),b())
            f(a(),b()) -> f(s(a()),c())
            f(c(),c()) -> f(a(),a())
            f(s(X),c()) -> f(X,c())
        - Signature:
            {f/2} / {a/0,b/0,c/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,f,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))