/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: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1} / {cons/2,empty/0}
        - Obligation:
             derivational complexity wrt. signature {cons,empty,r1,rev}
    + 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(cons) = [1] x1 + [1] x2 + [12]
          p(empty) = [8]                   
             p(r1) = [1] x1 + [1] x2 + [4] 
            p(rev) = [1] x1 + [12]         
        
        Following rules are strictly oriented:
        r1(empty(),a) = [1] a + [12]
                      > [1] a + [0] 
                      = a           
        
        
        Following rules are (at-least) weakly oriented:
        r1(cons(x,k),a) =  [1] a + [1] k + [1] x + [16]
                        >= [1] a + [1] k + [1] x + [16]
                        =  r1(k,cons(x,a))             
        
                rev(ls) =  [1] ls + [12]               
                        >= [1] ls + [12]               
                        =  r1(ls,empty())              
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            rev(ls) -> r1(ls,empty())
        - Weak TRS:
            r1(empty(),a) -> a
        - Signature:
            {r1/2,rev/1} / {cons/2,empty/0}
        - Obligation:
             derivational complexity wrt. signature {cons,empty,r1,rev}
    + 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(cons) = [1] x1 + [1] x2 + [0]
          p(empty) = [0]                  
             p(r1) = [1] x1 + [1] x2 + [0]
            p(rev) = [1] x1 + [1]         
        
        Following rules are strictly oriented:
        rev(ls) = [1] ls + [1]  
                > [1] ls + [0]  
                = r1(ls,empty())
        
        
        Following rules are (at-least) weakly oriented:
        r1(cons(x,k),a) =  [1] a + [1] k + [1] x + [0]
                        >= [1] a + [1] k + [1] x + [0]
                        =  r1(k,cons(x,a))            
        
          r1(empty(),a) =  [1] a + [0]                
                        >= [1] a + [0]                
                        =  a                          
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
        - Weak TRS:
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1} / {cons/2,empty/0}
        - Obligation:
             derivational complexity wrt. signature {cons,empty,r1,rev}
    + 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(cons) = [1 1] x1 + [1 0] x2 + [0]
                     [0 1]      [0 1]      [1]
          p(empty) = [3]                      
                     [3]                      
             p(r1) = [1 1] x1 + [1 0] x2 + [0]
                     [0 1]      [0 1]      [0]
            p(rev) = [1 2] x1 + [3]           
                     [0 1]      [4]           
        
        Following rules are strictly oriented:
        r1(cons(x,k),a) = [1 0] a + [1 1] k + [1 2] x + [1]
                          [0 1]     [0 1]     [0 1]     [1]
                        > [1 0] a + [1 1] k + [1 1] x + [0]
                          [0 1]     [0 1]     [0 1]     [1]
                        = r1(k,cons(x,a))                  
        
        
        Following rules are (at-least) weakly oriented:
        r1(empty(),a) =  [1 0] a + [6] 
                         [0 1]     [3] 
                      >= [1 0] a + [0] 
                         [0 1]     [0] 
                      =  a             
        
              rev(ls) =  [1 2] ls + [3]
                         [0 1]      [4]
                      >= [1 1] ls + [3]
                         [0 1]      [3]
                      =  r1(ls,empty())
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            r1(cons(x,k),a) -> r1(k,cons(x,a))
            r1(empty(),a) -> a
            rev(ls) -> r1(ls,empty())
        - Signature:
            {r1/2,rev/1} / {cons/2,empty/0}
        - Obligation:
             derivational complexity wrt. signature {cons,empty,r1,rev}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))