* Step 1: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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]         
              p(fac) = [1] x1 + [0]
                p(p) = [1] x1 + [3]
                p(s) = [1] x1 + [0]
            p(times) = [1] x2 + [0]
          
          Following rules are strictly oriented:
          p(s(x)) = [1] x + [3]
                  > [1] x + [0]
                  = x          
          
          
          Following rules are (at-least) weakly oriented:
           fac(0()) =  [0]                     
                    >= [0]                     
                    =  s(0())                  
          
          fac(s(x)) =  [1] x + [0]             
                    >= [1] x + [3]             
                    =  times(s(x),fac(p(s(x))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
        - Weak TRS:
            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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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]          
              p(fac) = [1] x1 + [2] 
                p(p) = [1] x1 + [0] 
                p(s) = [1] x1 + [0] 
            p(times) = [1] x2 + [13]
          
          Following rules are strictly oriented:
          fac(0()) = [2]   
                   > [0]   
                   = s(0())
          
          
          Following rules are (at-least) weakly oriented:
          fac(s(x)) =  [1] x + [2]             
                    >= [1] x + [15]            
                    =  times(s(x),fac(p(s(x))))
          
            p(s(x)) =  [1] x + [0]             
                    >= [1] x + [0]             
                    =  x                       
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI 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:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))):
        
        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]              
                     [4]              
                     [0]              
            p(fac) = [1 0 1]       [4]
                     [2 0 0] x_1 + [0]
                     [0 0 0]       [4]
              p(p) = [1 0 0]       [2]
                     [2 0 4] x_1 + [3]
                     [0 1 0]       [0]
              p(s) = [1 1 4]       [0]
                     [0 0 1] x_1 + [0]
                     [0 0 1]       [3]
          p(times) = [1 0 0]       [0]
                     [0 0 0] x_2 + [0]
                     [0 0 0]       [4]
        
        Following rules are strictly oriented:
        fac(s(x)) = [1 1 5]     [7]         
                    [2 2 8] x + [0]         
                    [0 0 0]     [4]         
                  > [1 1 5]     [6]         
                    [0 0 0] x + [0]         
                    [0 0 0]     [4]         
                  = times(s(x),fac(p(s(x))))
        
        
        Following rules are (at-least) weakly oriented:
        fac(0()) =  [4]              
                    [0]              
                    [4]              
                 >= [4]              
                    [0]              
                    [3]              
                 =  s(0())           
        
         p(s(x)) =  [1 1  4]     [2] 
                    [2 2 12] x + [15]
                    [0 0  1]     [0] 
                 >= [1 0 0]     [0]  
                    [0 1 0] x + [0]  
                    [0 0 1]     [0]  
                 =  x                
        
* Step 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(?,O(n^2))