*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(x,x,y) -> x
        f(x,y,y) -> y
        f(x,y,g(y)) -> x
        f(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
        f(g(x),x,y) -> y
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3} / {g/1}
      Obligation:
        Full
        basic terms: {f}/{g}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        f#(x,x,y) -> c_1(x)
        f#(x,y,y) -> c_2(y)
        f#(x,y,g(y)) -> c_3(x)
        f#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)))
        f#(g(x),x,y) -> c_5(y)
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(x,x,y) -> c_1(x)
        f#(x,y,y) -> c_2(y)
        f#(x,y,g(y)) -> c_3(x)
        f#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)))
        f#(g(x),x,y) -> c_5(y)
      Strict TRS Rules:
        f(x,x,y) -> x
        f(x,y,y) -> y
        f(x,y,g(y)) -> x
        f(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
        f(g(x),x,y) -> y
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,f#/3} / {g/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1}
      Obligation:
        Full
        basic terms: {f#}/{g}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        f#(x,x,y) -> c_1(x)
        f#(x,y,y) -> c_2(y)
        f#(x,y,g(y)) -> c_3(x)
        f#(g(x),x,y) -> c_5(y)
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(x,x,y) -> c_1(x)
        f#(x,y,y) -> c_2(y)
        f#(x,y,g(y)) -> c_3(x)
        f#(g(x),x,y) -> c_5(y)
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,f#/3} / {g/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1}
      Obligation:
        Full
        basic terms: {f#}/{g}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following constant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          none
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
            p(f) = [0]                  
            p(g) = [10]                 
           p(f#) = [1] x2 + [2] x3 + [3]
          p(c_1) = [0]                  
          p(c_2) = [3] x1 + [1]         
          p(c_3) = [1]                  
          p(c_4) = [1]                  
          p(c_5) = [2] x1 + [2]         
        
        Following rules are strictly oriented:
           f#(x,x,y) = [1] x + [2] y + [3]
                     > [0]                
                     = c_1(x)             
        
           f#(x,y,y) = [3] y + [3]        
                     > [3] y + [1]        
                     = c_2(y)             
        
        f#(x,y,g(y)) = [1] y + [23]       
                     > [1]                
                     = c_3(x)             
        
        f#(g(x),x,y) = [1] x + [2] y + [3]
                     > [2] y + [2]        
                     = c_5(y)             
        
        
        Following rules are (at-least) weakly oriented:
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        f#(x,x,y) -> c_1(x)
        f#(x,y,y) -> c_2(y)
        f#(x,y,g(y)) -> c_3(x)
        f#(g(x),x,y) -> c_5(y)
      Weak TRS Rules:
        
      Signature:
        {f/3,f#/3} / {g/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1}
      Obligation:
        Full
        basic terms: {f#}/{g}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).