*** 1 Progress [(O(1),O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        fac(s(x)) -> *(fac(p(s(x))),s(x))
        p(s(0())) -> 0()
        p(s(s(x))) -> s(p(s(x)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {fac/1,p/1} / {*/2,0/0,s/1}
      Obligation:
        Innermost
        basic terms: {fac,p}/{*,0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      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:
          {}
        TcT has computed the following interpretation:
            p(*) = [1] x1 + [0]
            p(0) = [0]         
          p(fac) = [1] x1 + [7]
            p(p) = [4]         
            p(s) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        p(s(0())) = [4]
                  > [0]
                  = 0()
        
        
        Following rules are (at-least) weakly oriented:
         fac(s(x)) =  [1] x + [8]         
                   >= [11]                
                   =  *(fac(p(s(x))),s(x))
        
        p(s(s(x))) =  [4]                 
                   >= [5]                 
                   =  s(p(s(x)))          
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        fac(s(x)) -> *(fac(p(s(x))),s(x))
        p(s(s(x))) -> s(p(s(x)))
      Weak DP Rules:
        
      Weak TRS Rules:
        p(s(0())) -> 0()
      Signature:
        {fac/1,p/1} / {*/2,0/0,s/1}
      Obligation:
        Innermost
        basic terms: {fac,p}/{*,0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      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:
          {}
        TcT has computed the following interpretation:
            p(*) = [1] x1 + [0]
            p(0) = [0]         
          p(fac) = [1] x1 + [0]
            p(p) = [0]         
            p(s) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        fac(s(x)) = [1] x + [1]         
                  > [0]                 
                  = *(fac(p(s(x))),s(x))
        
        
        Following rules are (at-least) weakly oriented:
         p(s(0())) =  [0]       
                   >= [0]       
                   =  0()       
        
        p(s(s(x))) =  [0]       
                   >= [1]       
                   =  s(p(s(x)))
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        p(s(s(x))) -> s(p(s(x)))
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {fac,p}/{*,0,s}
    Applied Processor:
      NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
    Proof:
      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 0] x1 + [0]
                 [0 0 0]      [0]
          p(0) = [2]             
                 [1]             
                 [1]             
        p(fac) = [1 0 0]      [0]
                 [0 0 3] x1 + [1]
                 [1 0 0]      [1]
          p(p) = [1 0 0]      [0]
                 [1 0 0] x1 + [1]
                 [0 1 0]      [1]
          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]
                   [1 0 4] x + [5]
                   [0 0 1]     [3]
                 > [1 0 4]     [2]
                   [0 0 1] x + [1]
                   [0 0 1]     [3]
                 = s(p(s(x)))     
      
      
      Following rules are (at-least) weakly oriented:
      fac(s(x)) =  [1 0 2]     [0]     
                   [0 0 3] x + [7]     
                   [1 0 2]     [1]     
                >= [1 0 2]     [0]     
                   [0 0 0] x + [0]     
                   [0 0 0]     [0]     
                =  *(fac(p(s(x))),s(x))
      
      p(s(0())) =  [4]                 
                   [5]                 
                   [2]                 
                >= [2]                 
                   [1]                 
                   [1]                 
                =  0()                 
      
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {fac,p}/{*,0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).