/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1} / {0/0,s/1,times/2}
        - Obligation:
             runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1} / {0/0,s/1,times/2}
        - Obligation:
             runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          fac(x){x -> s(x)} =
            fac(s(x)) ->^+ times(s(x),fac(x))
              = C[fac(x) = fac(x){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1} / {0/0,s/1,times/2}
        - Obligation:
             runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(fac) = {1},
          uargs(times) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
              p(0) = 2       
            p(fac) = 8 + 8*x1
              p(p) = x1      
              p(s) = x1      
          p(times) = x2      
        
        Following rules are strictly oriented:
        fac(0()) = 24    
                 > 2     
                 = s(0())
        
        
        Following rules are (at-least) weakly oriented:
        fac(s(x)) =  8 + 8*x                 
                  >= 8 + 8*x                 
                  =  times(s(x),fac(p(s(x))))
        
          p(s(x)) =  x                       
                  >= x                       
                  =  x                       
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Weak TRS:
            fac(0()) -> s(0())
        - Signature:
            {fac/1,p/1} / {0/0,s/1,times/2}
        - Obligation:
             runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(fac) = {1},
          uargs(times) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
              p(0) = 1       
            p(fac) = 1 + 4*x1
              p(p) = x1      
              p(s) = 4 + x1  
          p(times) = x2      
        
        Following rules are strictly oriented:
        p(s(x)) = 4 + x
                > x    
                = x    
        
        
        Following rules are (at-least) weakly oriented:
         fac(0()) =  5                       
                  >= 5                       
                  =  s(0())                  
        
        fac(s(x)) =  17 + 4*x                
                  >= 17 + 4*x                
                  =  times(s(x),fac(p(s(x))))
        
** Step 1.b:3: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            fac(s(x)) -> times(s(x),fac(p(s(x))))
        - Weak TRS:
            fac(0()) -> s(0())
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1} / {0/0,s/1,times/2}
        - Obligation:
             runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(fac) = {1},
          uargs(times) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
              p(0) = [0]                          
                     [0]                          
                     [4]                          
            p(fac) = [2 0 2]      [0]             
                     [4 0 2] x1 + [4]             
                     [0 1 0]      [6]             
              p(p) = [1 0 0]      [0]             
                     [3 0 2] x1 + [0]             
                     [0 1 0]      [0]             
              p(s) = [1 1 2]      [0]             
                     [0 0 1] x1 + [0]             
                     [0 0 1]      [2]             
          p(times) = [0 0 0]      [1 0 0]      [1]
                     [0 0 1] x1 + [0 0 0] x2 + [1]
                     [0 0 1]      [0 0 0]      [4]
        
        Following rules are strictly oriented:
        fac(s(x)) = [2 2  6]     [4]        
                    [4 4 10] x + [8]        
                    [0 0  1]     [6]        
                  > [2 2 6]     [1]         
                    [0 0 1] x + [3]         
                    [0 0 1]     [6]         
                  = times(s(x),fac(p(s(x))))
        
        
        Following rules are (at-least) weakly oriented:
        fac(0()) =  [8]            
                    [12]           
                    [6]            
                 >= [8]            
                    [4]            
                    [6]            
                 =  s(0())         
        
         p(s(x)) =  [1 1 2]     [0]
                    [3 3 8] x + [4]
                    [0 0 1]     [0]
                 >= [1 0 0]     [0]
                    [0 1 0] x + [0]
                    [0 0 1]     [0]
                 =  x              
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1} / {0/0,s/1,times/2}
        - Obligation:
             runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))