*** 1 Progress [(O(1),O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(x,c(y)) -> f(x,s(f(y,y)))
        f(s(x),s(y)) -> f(x,s(c(s(y))))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2} / {c/1,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{c,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(f) = {2},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
          p(c) = [1] x1 + [14]
          p(f) = [1] x2 + [0] 
          p(s) = [1] x1 + [2] 
        
        Following rules are strictly oriented:
        f(x,c(y)) = [1] y + [14]  
                  > [1] y + [2]   
                  = f(x,s(f(y,y)))
        
        
        Following rules are (at-least) weakly oriented:
        f(s(x),s(y)) =  [1] y + [2]    
                     >= [1] y + [18]   
                     =  f(x,s(c(s(y))))
        
      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:
        f(s(x),s(y)) -> f(x,s(c(s(y))))
      Weak DP Rules:
        
      Weak TRS Rules:
        f(x,c(y)) -> f(x,s(f(y,y)))
      Signature:
        {f/2} / {c/1,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{c,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(f) = {2},
        uargs(s) = {1}
      
      Following symbols are considered usable:
        {f}
      TcT has computed the following interpretation:
        p(c) = [1 1 0]      [0]             
               [0 0 3] x1 + [3]             
               [0 0 0]      [0]             
        p(f) = [0 0 3]      [1 1 0]      [1]
               [0 0 0] x1 + [0 0 0] x2 + [0]
               [1 0 0]      [0 2 0]      [1]
        p(s) = [1 0 0]      [0]             
               [0 0 0] x1 + [2]             
               [0 0 1]      [1]             
      
      Following rules are strictly oriented:
      f(s(x),s(y)) = [0 0 3]     [1 0 0]     [6]
                     [0 0 0] x + [0 0 0] y + [0]
                     [1 0 0]     [0 0 0]     [5]
                   > [0 0 3]     [1 0 0]     [5]
                     [0 0 0] x + [0 0 0] y + [0]
                     [1 0 0]     [0 0 0]     [5]
                   = f(x,s(c(s(y))))            
      
      
      Following rules are (at-least) weakly oriented:
      f(x,c(y)) =  [0 0 3]     [1 1 3]     [4]
                   [0 0 0] x + [0 0 0] y + [0]
                   [1 0 0]     [0 0 6]     [7]
                >= [0 0 3]     [1 1 3]     [4]
                   [0 0 0] x + [0 0 0] y + [0]
                   [1 0 0]     [0 0 0]     [5]
                =  f(x,s(f(y,y)))             
      
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(x,c(y)) -> f(x,s(f(y,y)))
        f(s(x),s(y)) -> f(x,s(c(s(y))))
      Signature:
        {f/2} / {c/1,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{c,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).