/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^1))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            first(0(),X) -> nil()
            first(s(X),cons(Y)) -> cons(Y)
            from(X) -> cons(X)
        - Signature:
            {first/2,from/1} / {0/0,cons/1,nil/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {0,cons,first,from,nil,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) = [8]                  
           p(cons) = [1] x1 + [1]         
          p(first) = [1] x1 + [1] x2 + [8]
           p(from) = [1] x1 + [1]         
            p(nil) = [0]                  
              p(s) = [1] x1 + [8]         
        
        Following rules are strictly oriented:
               first(0(),X) = [1] X + [16]        
                            > [0]                 
                            = nil()               
        
        first(s(X),cons(Y)) = [1] X + [1] Y + [17]
                            > [1] Y + [1]         
                            = cons(Y)             
        
        
        Following rules are (at-least) weakly oriented:
        from(X) =  [1] X + [1]
                >= [1] X + [1]
                =  cons(X)    
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            from(X) -> cons(X)
        - Weak TRS:
            first(0(),X) -> nil()
            first(s(X),cons(Y)) -> cons(Y)
        - Signature:
            {first/2,from/1} / {0/0,cons/1,nil/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {0,cons,first,from,nil,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) = [3]                  
           p(cons) = [1] x1 + [0]         
          p(first) = [1] x1 + [1] x2 + [0]
           p(from) = [1] x1 + [8]         
            p(nil) = [0]                  
              p(s) = [1] x1 + [8]         
        
        Following rules are strictly oriented:
        from(X) = [1] X + [8]
                > [1] X + [0]
                = cons(X)    
        
        
        Following rules are (at-least) weakly oriented:
               first(0(),X) =  [1] X + [3]        
                            >= [0]                
                            =  nil()              
        
        first(s(X),cons(Y)) =  [1] X + [1] Y + [8]
                            >= [1] Y + [0]        
                            =  cons(Y)            
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            first(0(),X) -> nil()
            first(s(X),cons(Y)) -> cons(Y)
            from(X) -> cons(X)
        - Signature:
            {first/2,from/1} / {0/0,cons/1,nil/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {0,cons,first,from,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))