/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^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,cons,filter,nats,s,sieve,zprimes}
    + 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) = [9]                           
               p(cons) = [1] x1 + [0]                  
             p(filter) = [1] x1 + [1] x2 + [1] x3 + [0]
               p(nats) = [1] x1 + [3]                  
                  p(s) = [1] x1 + [0]                  
              p(sieve) = [1] x1 + [2]                  
            p(zprimes) = [15]                          
          
          Following rules are strictly oriented:
                    nats(N) = [1] N + [3]           
                            > [1] N + [0]           
                            = cons(N)               
          
           sieve(cons(0())) = [11]                  
                            > [9]                   
                            = cons(0())             
          
          sieve(cons(s(N))) = [1] N + [2]           
                            > [1] N + [0]           
                            = cons(s(N))            
          
                  zprimes() = [15]                  
                            > [14]                  
                            = sieve(nats(s(s(0()))))
          
          
          Following rules are (at-least) weakly oriented:
           filter(cons(X),0(),M) =  [1] M + [1] X + [9]        
                                 >= [9]                        
                                 =  cons(0())                  
          
          filter(cons(X),s(N),M) =  [1] M + [1] N + [1] X + [0]
                                 >= [1] X + [0]                
                                 =  cons(X)                    
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
        - Weak TRS:
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,cons,filter,nats,s,sieve,zprimes}
    + 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(cons) = [1] x1 + [0]                   
             p(filter) = [1] x1 + [1] x2 + [1] x3 + [10]
               p(nats) = [1] x1 + [7]                   
                  p(s) = [1] x1 + [0]                   
              p(sieve) = [1] x1 + [0]                   
            p(zprimes) = [8]                            
          
          Following rules are strictly oriented:
           filter(cons(X),0(),M) = [1] M + [1] X + [11]        
                                 > [1]                         
                                 = cons(0())                   
          
          filter(cons(X),s(N),M) = [1] M + [1] N + [1] X + [10]
                                 > [1] X + [0]                 
                                 = cons(X)                     
          
          
          Following rules are (at-least) weakly oriented:
                    nats(N) =  [1] N + [7]           
                            >= [1] N + [0]           
                            =  cons(N)               
          
           sieve(cons(0())) =  [1]                   
                            >= [1]                   
                            =  cons(0())             
          
          sieve(cons(s(N))) =  [1] N + [0]           
                            >= [1] N + [0]           
                            =  cons(s(N))            
          
                  zprimes() =  [8]                   
                            >= [8]                   
                            =  sieve(nats(s(s(0()))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,cons,filter,nats,s,sieve,zprimes}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))