*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(x,y,s(z)) -> s(f(0(),1(),z))
        f(0(),1(),x) -> f(s(x),x,x)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3} / {0/0,1/0,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{0,1,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(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
          p(0) = [1]          
          p(1) = [1]          
          p(f) = [8] x3 + [14]
          p(s) = [1] x1 + [2] 
        
        Following rules are strictly oriented:
        f(x,y,s(z)) = [8] z + [30]   
                    > [8] z + [16]   
                    = s(f(0(),1(),z))
        
        
        Following rules are (at-least) weakly oriented:
        f(0(),1(),x) =  [8] x + [14]
                     >= [8] x + [14]
                     =  f(s(x),x,x) 
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(0(),1(),x) -> f(s(x),x,x)
      Weak DP Rules:
        
      Weak TRS Rules:
        f(x,y,s(z)) -> s(f(0(),1(),z))
      Signature:
        {f/3} / {0/0,1/0,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{0,1,s}
    Applied Processor:
      NaturalMI {miDimension = 2, miDegree = 1, 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 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
      The following argument positions are considered usable:
        uargs(s) = {1}
      
      Following symbols are considered usable:
        {f}
      TcT has computed the following interpretation:
        p(0) = [1]                      
               [2]                      
        p(1) = [0]                      
               [4]                      
        p(f) = [0 4] x1 + [5 4] x3 + [1]
               [0 0]      [0 0]      [8]
        p(s) = [1 2] x1 + [5]           
               [0 0]      [1]           
      
      Following rules are strictly oriented:
      f(0(),1(),x) = [5 4] x + [9]
                     [0 0]     [8]
                   > [5 4] x + [5]
                     [0 0]     [8]
                   = f(s(x),x,x)  
      
      
      Following rules are (at-least) weakly oriented:
      f(x,y,s(z)) =  [0 4] x + [5 10] z + [30]
                     [0 0]     [0  0]     [8] 
                  >= [5 4] z + [30]           
                     [0 0]     [1]            
                  =  s(f(0(),1(),z))          
      
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(x,y,s(z)) -> s(f(0(),1(),z))
        f(0(),1(),x) -> f(s(x),x,x)
      Signature:
        {f/3} / {0/0,1/0,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{0,1,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).