*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(s(x),y,y) -> f(y,x,s(x))
        g(x,y) -> x
        g(x,y) -> y
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,g/2} / {s/1}
      Obligation:
        Full
        basic terms: {f,g}/{s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
        g#(x,y) -> c_2(x)
        g#(x,y) -> c_3(y)
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
        g#(x,y) -> c_2(x)
        g#(x,y) -> c_3(y)
      Strict TRS Rules:
        f(s(x),y,y) -> f(y,x,s(x))
        g(x,y) -> x
        g(x,y) -> y
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,g/2,f#/3,g#/2} / {s/1,c_1/1,c_2/1,c_3/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
        g#(x,y) -> c_2(x)
        g#(x,y) -> c_3(y)
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
        g#(x,y) -> c_2(x)
        g#(x,y) -> c_3(y)
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,g/2,f#/3,g#/2} / {s/1,c_1/1,c_2/1,c_3/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{s}
    Applied Processor:
      NaturalMI {miDimension = 1, miDegree = 0, 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 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) = [0]                  
          p(s) = [0]                  
         p(f#) = [6]                  
         p(g#) = [2] x1 + [1] x2 + [0]
        p(c_1) = [5]                  
        p(c_2) = [2] x1 + [0]         
        p(c_3) = [1] x1 + [0]         
      
      Following rules are strictly oriented:
      f#(s(x),y,y) = [6]              
                   > [5]              
                   = c_1(f#(y,x,s(x)))
      
      
      Following rules are (at-least) weakly oriented:
      g#(x,y) =  [2] x + [1] y + [0]
              >= [2] x + [0]        
              =  c_2(x)             
      
      g#(x,y) =  [2] x + [1] y + [0]
              >= [1] y + [0]        
              =  c_3(y)             
      
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(x,y) -> c_2(x)
        g#(x,y) -> c_3(y)
      Strict TRS Rules:
        
      Weak DP Rules:
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
      Weak TRS Rules:
        
      Signature:
        {f/3,g/2,f#/3,g#/2} / {s/1,c_1/1,c_2/1,c_3/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{s}
    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) = [8] x2 + [0]         
            p(g) = [1] x1 + [1] x2 + [0]
            p(s) = [0]                  
           p(f#) = [4] x1 + [8] x3 + [0]
           p(g#) = [9] x1 + [4]         
          p(c_1) = [2] x1 + [0]         
          p(c_2) = [2]                  
          p(c_3) = [2]                  
        
        Following rules are strictly oriented:
        g#(x,y) = [9] x + [4]
                > [2]        
                = c_2(x)     
        
        g#(x,y) = [9] x + [4]
                > [2]        
                = c_3(y)     
        
        
        Following rules are (at-least) weakly oriented:
        f#(s(x),y,y) =  [8] y + [0]      
                     >= [8] y + [0]      
                     =  c_1(f#(y,x,s(x)))
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
        g#(x,y) -> c_2(x)
        g#(x,y) -> c_3(y)
      Weak TRS Rules:
        
      Signature:
        {f/3,g/2,f#/3,g#/2} / {s/1,c_1/1,c_2/1,c_3/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).