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


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WORST_CASE(?,O(n^2))
* Step 1: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
            app(id(),x) -> x
            app(plus(),0()) -> id()
        - Signature:
            {app/2} / {0/0,id/0,plus/0,s/0}
        - Obligation:
             derivational complexity wrt. signature {0,app,id,plus,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) = [1]                  
             p(app) = [1] x1 + [1] x2 + [0]
              p(id) = [0]                  
            p(plus) = [0]                  
               p(s) = [0]                  
          
          Following rules are strictly oriented:
          app(plus(),0()) = [1] 
                          > [0] 
                          = id()
          
          
          Following rules are (at-least) weakly oriented:
          app(app(plus(),app(s(),x)),y) =  [1] x + [1] y + [0]          
                                        >= [1] x + [1] y + [0]          
                                        =  app(s(),app(app(plus(),x),y))
          
                            app(id(),x) =  [1] x + [0]                  
                                        >= [1] x + [0]                  
                                        =  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:
            app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
            app(id(),x) -> x
        - Weak TRS:
            app(plus(),0()) -> id()
        - Signature:
            {app/2} / {0/0,id/0,plus/0,s/0}
        - Obligation:
             derivational complexity wrt. signature {0,app,id,plus,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) = [12]                 
           p(app) = [1] x1 + [1] x2 + [0]
            p(id) = [11]                 
          p(plus) = [4]                  
             p(s) = [1]                  
        
        Following rules are strictly oriented:
        app(id(),x) = [1] x + [11]
                    > [1] x + [0] 
                    = x           
        
        
        Following rules are (at-least) weakly oriented:
        app(app(plus(),app(s(),x)),y) =  [1] x + [1] y + [5]          
                                      >= [1] x + [1] y + [5]          
                                      =  app(s(),app(app(plus(),x),y))
        
                      app(plus(),0()) =  [16]                         
                                      >= [11]                         
                                      =  id()                         
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
        - Weak TRS:
            app(id(),x) -> x
            app(plus(),0()) -> id()
        - Signature:
            {app/2} / {0/0,id/0,plus/0,s/0}
        - Obligation:
             derivational complexity wrt. signature {0,app,id,plus,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) = [1]                      
                    [2]                      
           p(app) = [1 2] x1 + [1 0] x2 + [0]
                    [0 1]      [0 1]      [0]
            p(id) = [3]                      
                    [2]                      
          p(plus) = [2]                      
                    [0]                      
             p(s) = [0]                      
                    [2]                      
        
        Following rules are strictly oriented:
        app(app(plus(),app(s(),x)),y) = [1 2] x + [1 0] y + [10]     
                                        [0 1]     [0 1]     [2]      
                                      > [1 2] x + [1 0] y + [6]      
                                        [0 1]     [0 1]     [2]      
                                      = app(s(),app(app(plus(),x),y))
        
        
        Following rules are (at-least) weakly oriented:
            app(id(),x) =  [1 0] x + [7]
                           [0 1]     [2]
                        >= [1 0] x + [0]
                           [0 1]     [0]
                        =  x            
        
        app(plus(),0()) =  [3]          
                           [2]          
                        >= [3]          
                           [2]          
                        =  id()         
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
            app(id(),x) -> x
            app(plus(),0()) -> id()
        - Signature:
            {app/2} / {0/0,id/0,plus/0,s/0}
        - Obligation:
             derivational complexity wrt. signature {0,app,id,plus,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))