*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        g(f(x),y) -> f(h(x,y))
        h(x,y) -> g(x,f(y))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {g/2,h/2} / {f/1}
      Obligation:
        Innermost
        basic terms: {g,h}/{f}
    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) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
          p(f) = [1] x1 + [5]          
          p(g) = [4] x1 + [1] x2 + [0] 
          p(h) = [4] x1 + [1] x2 + [14]
        
        Following rules are strictly oriented:
        g(f(x),y) = [4] x + [1] y + [20]
                  > [4] x + [1] y + [19]
                  = f(h(x,y))           
        
           h(x,y) = [4] x + [1] y + [14]
                  > [4] x + [1] y + [5] 
                  = g(x,f(y))           
        
        
        Following rules are (at-least) weakly oriented:
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        g(f(x),y) -> f(h(x,y))
        h(x,y) -> g(x,f(y))
      Signature:
        {g/2,h/2} / {f/1}
      Obligation:
        Innermost
        basic terms: {g,h}/{f}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).