WORST_CASE(?,O(n^2))
* Step 1: Sum 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:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: NaturalMI 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:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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:
          {fac,p}
        TcT has computed the following interpretation:
              p(0) = [0]         
            p(fac) = [1] x1 + [4]
              p(p) = [1] x1 + [0]
              p(s) = [1] x1 + [0]
          p(times) = [1] x2 + [0]
        
        Following rules are strictly oriented:
        fac(0()) = [4]   
                 > [0]   
                 = s(0())
        
        
        Following rules are (at-least) weakly oriented:
        fac(s(x)) =  [1] x + [4]             
                  >= [1] x + [4]             
                  =  times(s(x),fac(p(s(x))))
        
          p(s(x)) =  [1] x + [0]             
                  >= [1] x + [0]             
                  =  x                       
        
* Step 3: NaturalMI 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:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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:
          {fac,p}
        TcT has computed the following interpretation:
              p(0) = [4]         
            p(fac) = [1] x1 + [4]
              p(p) = [1] x1 + [0]
              p(s) = [1] x1 + [4]
          p(times) = [1] x2 + [0]
        
        Following rules are strictly oriented:
        p(s(x)) = [1] x + [4]
                > [1] x + [0]
                = x          
        
        
        Following rules are (at-least) weakly oriented:
         fac(0()) =  [8]                     
                  >= [8]                     
                  =  s(0())                  
        
        fac(s(x)) =  [1] x + [8]             
                  >= [1] x + [8]             
                  =  times(s(x),fac(p(s(x))))
        
* Step 4: 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:
            innermost 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:
          {fac,p}
        TcT has computed the following interpretation:
              p(0) = [0]                          
                     [3]                          
                     [2]                          
            p(fac) = [2 0 4]      [0]             
                     [2 0 1] x1 + [0]             
                     [2 0 0]      [4]             
              p(p) = [1 0 0]      [0]             
                     [4 0 7] 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]      [7]
                     [0 0 0] x1 + [0 0 0] x2 + [0]
                     [0 0 1]      [0 0 0]      [0]
        
        Following rules are strictly oriented:
        fac(s(x)) = [2 2 8]     [8]         
                    [2 2 5] x + [2]         
                    [2 2 4]     [4]         
                  > [2 2 8]     [7]         
                    [0 0 0] x + [0]         
                    [0 0 1]     [2]         
                  = times(s(x),fac(p(s(x))))
        
        
        Following rules are (at-least) weakly oriented:
        fac(0()) =  [8]              
                    [2]              
                    [4]              
                 >= [7]              
                    [2]              
                    [4]              
                 =  s(0())           
        
         p(s(x)) =  [1 1  2]     [0] 
                    [4 4 15] x + [14]
                    [0 0  1]     [0] 
                 >= [1 0 0]     [0]  
                    [0 1 0] x + [0]  
                    [0 0 1]     [0]  
                 =  x                
        
* Step 5: 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:
            innermost 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(?,O(n^2))