/export/starexec/sandbox/solver/bin/starexec_run_tct_dc /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:
            ++(x,nil()) -> x
            ++(++(x,y),z) -> ++(x,++(y,z))
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
        - Signature:
            {++/2} / {./2,nil/0}
        - Obligation:
             derivational complexity wrt. signature {++,.,nil}
    + 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(++) = [1] x1 + [1] x2 + [2]
              p(.) = [1] x1 + [1] x2 + [0]
            p(nil) = [0]                  
          
          Following rules are strictly oriented:
          ++(x,nil()) = [1] x + [2]
                      > [1] x + [0]
                      = x          
          
          ++(nil(),y) = [1] y + [2]
                      > [1] y + [0]
                      = y          
          
          
          Following rules are (at-least) weakly oriented:
          ++(++(x,y),z) =  [1] x + [1] y + [1] z + [4]
                        >= [1] x + [1] y + [1] z + [4]
                        =  ++(x,++(y,z))              
          
           ++(.(x,y),z) =  [1] x + [1] y + [1] z + [2]
                        >= [1] x + [1] y + [1] z + [2]
                        =  .(x,++(y,z))               
          
        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:
            ++(++(x,y),z) -> ++(x,++(y,z))
            ++(.(x,y),z) -> .(x,++(y,z))
        - Weak TRS:
            ++(x,nil()) -> x
            ++(nil(),y) -> y
        - Signature:
            {++/2} / {./2,nil/0}
        - Obligation:
             derivational complexity wrt. signature {++,.,nil}
    + 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(++) = [1 6] x1 + [1 0] x2 + [0]
                   [0 1]      [0 1]      [2]
            p(.) = [1 0] x1 + [1 0] x2 + [1]
                   [0 0]      [0 1]      [0]
          p(nil) = [7]                      
                   [0]                      
        
        Following rules are strictly oriented:
        ++(++(x,y),z) = [1 12] x + [1 6] y + [1 0] z + [12]
                        [0  1]     [0 1]     [0 1]     [4] 
                      > [1 6] x + [1 6] y + [1 0] z + [0]  
                        [0 1]     [0 1]     [0 1]     [4]  
                      = ++(x,++(y,z))                      
        
        
        Following rules are (at-least) weakly oriented:
         ++(x,nil()) =  [1 6] x + [7]                    
                        [0 1]     [2]                    
                     >= [1 0] x + [0]                    
                        [0 1]     [0]                    
                     =  x                                
        
        ++(.(x,y),z) =  [1 0] x + [1 6] y + [1 0] z + [1]
                        [0 0]     [0 1]     [0 1]     [2]
                     >= [1 0] x + [1 6] y + [1 0] z + [1]
                        [0 0]     [0 1]     [0 1]     [2]
                     =  .(x,++(y,z))                     
        
         ++(nil(),y) =  [1 0] y + [7]                    
                        [0 1]     [2]                    
                     >= [1 0] y + [0]                    
                        [0 1]     [0]                    
                     =  y                                
        
* Step 3: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
        - Weak TRS:
            ++(x,nil()) -> x
            ++(++(x,y),z) -> ++(x,++(y,z))
            ++(nil(),y) -> y
        - Signature:
            {++/2} / {./2,nil/0}
        - Obligation:
             derivational complexity wrt. signature {++,.,nil}
    + 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(++) = [1 1] x1 + [1 0] x2 + [4]
                   [0 1]      [0 1]      [0]
            p(.) = [1 2] x1 + [1 0] x2 + [2]
                   [0 1]      [0 1]      [4]
          p(nil) = [3]                      
                   [4]                      
        
        Following rules are strictly oriented:
        ++(.(x,y),z) = [1 3] x + [1 1] y + [1 0] z + [10]
                       [0 1]     [0 1]     [0 1]     [4] 
                     > [1 2] x + [1 1] y + [1 0] z + [6] 
                       [0 1]     [0 1]     [0 1]     [4] 
                     = .(x,++(y,z))                      
        
        
        Following rules are (at-least) weakly oriented:
          ++(x,nil()) =  [1 1] x + [7]                    
                         [0 1]     [4]                    
                      >= [1 0] x + [0]                    
                         [0 1]     [0]                    
                      =  x                                
        
        ++(++(x,y),z) =  [1 2] x + [1 1] y + [1 0] z + [8]
                         [0 1]     [0 1]     [0 1]     [0]
                      >= [1 1] x + [1 1] y + [1 0] z + [8]
                         [0 1]     [0 1]     [0 1]     [0]
                      =  ++(x,++(y,z))                    
        
          ++(nil(),y) =  [1 0] y + [11]                   
                         [0 1]     [4]                    
                      >= [1 0] y + [0]                    
                         [0 1]     [0]                    
                      =  y                                
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            ++(x,nil()) -> x
            ++(++(x,y),z) -> ++(x,++(y,z))
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
        - Signature:
            {++/2} / {./2,nil/0}
        - Obligation:
             derivational complexity wrt. signature {++,.,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))