WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            fac(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1} / {*/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {*,0,s}
    + 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(*) = {1},
            uargs(fac) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(*) = [1] x1 + [1] x2 + [12]
              p(0) = [1]                   
            p(fac) = [1] x1 + [8]          
              p(p) = [2]                   
              p(s) = [1] x1 + [0]          
          
          Following rules are strictly oriented:
          p(s(0())) = [2]
                    > [1]
                    = 0()
          
          
          Following rules are (at-least) weakly oriented:
           fac(s(x)) =  [1] x + [8]         
                     >= [1] x + [22]        
                     =  *(fac(p(s(x))),s(x))
          
          p(s(s(x))) =  [2]                 
                     >= [2]                 
                     =  s(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(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(s(x))) -> s(p(s(x)))
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {fac/1,p/1} / {*/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {*,0,s}
    + 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(*) = {1},
            uargs(fac) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(*) = [1] x1 + [0]
              p(0) = [0]         
            p(fac) = [1] x1 + [9]
              p(p) = [7]         
              p(s) = [1] x1 + [8]
          
          Following rules are strictly oriented:
          fac(s(x)) = [1] x + [17]        
                    > [16]                
                    = *(fac(p(s(x))),s(x))
          
          
          Following rules are (at-least) weakly oriented:
           p(s(0())) =  [7]       
                     >= [0]       
                     =  0()       
          
          p(s(s(x))) =  [7]       
                     >= [15]      
                     =  s(p(s(x)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            p(s(s(x))) -> s(p(s(x)))
        - Weak TRS:
            fac(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
        - Signature:
            {fac/1,p/1} / {*/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {*,0,s}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(*) = {1},
          uargs(fac) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {fac,p}
        TcT has computed the following interpretation:
            p(*) = [1 0 0]      [0]
                   [0 0 2] x1 + [1]
                   [0 0 1]      [0]
            p(0) = [0]             
                   [0]             
                   [2]             
          p(fac) = [1 0 0]      [0]
                   [0 2 2] x1 + [1]
                   [0 1 1]      [0]
            p(p) = [1 0 0]      [0]
                   [0 1 0] x1 + [2]
                   [0 1 0]      [0]
            p(s) = [1 0 2]      [0]
                   [0 0 1] x1 + [0]
                   [0 0 1]      [2]
        
        Following rules are strictly oriented:
        p(s(s(x))) = [1 0 4]     [4]
                     [0 0 1] x + [4]
                     [0 0 1]     [2]
                   > [1 0 4]     [0]
                     [0 0 1] x + [0]
                     [0 0 1]     [2]
                   = s(p(s(x)))     
        
        
        Following rules are (at-least) weakly oriented:
        fac(s(x)) =  [1 0 2]     [0]     
                     [0 0 4] x + [5]     
                     [0 0 2]     [2]     
                  >= [1 0 2]     [0]     
                     [0 0 4] x + [5]     
                     [0 0 2]     [2]     
                  =  *(fac(p(s(x))),s(x))
        
        p(s(0())) =  [4]                 
                     [4]                 
                     [2]                 
                  >= [0]                 
                     [0]                 
                     [2]                 
                  =  0()                 
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            fac(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1} / {*/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {*,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))