/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: 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
            app(plus(),0()) -> id()
        - Signature:
            {app/2} / {0/0,id/0,plus/0,s/0}
        - Obligation:
            innermost 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) = [11]                 
           p(app) = [1] x1 + [1] x2 + [0]
            p(id) = [8]                  
          p(plus) = [4]                  
             p(s) = [5]                  
        
        Following rules are strictly oriented:
            app(id(),x) = [1] x + [8]
                        > [1] x + [0]
                        = x          
        
        app(plus(),0()) = [15]       
                        > [8]        
                        = id()       
        
        
        Following rules are (at-least) weakly oriented:
        app(app(plus(),app(s(),x)),y) =  [1] x + [1] y + [9]          
                                      >= [1] x + [1] y + [9]          
                                      =  app(s(),app(app(plus(),x),y))
        
* 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))
        - Weak TRS:
            app(id(),x) -> x
            app(plus(),0()) -> id()
        - Signature:
            {app/2} / {0/0,id/0,plus/0,s/0}
        - Obligation:
            innermost 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 3: 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:
            innermost 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))