*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(f(x)) -> f(c(f(x)))
        f(f(x)) -> f(d(f(x)))
        g(c(x)) -> x
        g(c(0())) -> g(d(1()))
        g(c(1())) -> g(d(0()))
        g(d(x)) -> x
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/1,g/1} / {0/0,1/0,c/1,d/1}
      Obligation:
        Full
        basic terms: {f,g}/{0,1,c,d}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        f#(f(x)) -> c_1(f#(c(f(x))))
        f#(f(x)) -> c_2(f#(d(f(x))))
        g#(c(x)) -> c_3(x)
        g#(c(0())) -> c_4(g#(d(1())))
        g#(c(1())) -> c_5(g#(d(0())))
        g#(d(x)) -> c_6(x)
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(f(x)) -> c_1(f#(c(f(x))))
        f#(f(x)) -> c_2(f#(d(f(x))))
        g#(c(x)) -> c_3(x)
        g#(c(0())) -> c_4(g#(d(1())))
        g#(c(1())) -> c_5(g#(d(0())))
        g#(d(x)) -> c_6(x)
      Strict TRS Rules:
        f(f(x)) -> f(c(f(x)))
        f(f(x)) -> f(d(f(x)))
        g(c(x)) -> x
        g(c(0())) -> g(d(1()))
        g(c(1())) -> g(d(0()))
        g(d(x)) -> x
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{0,1,c,d}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        g#(c(x)) -> c_3(x)
        g#(c(0())) -> c_4(g#(d(1())))
        g#(c(1())) -> c_5(g#(d(0())))
        g#(d(x)) -> c_6(x)
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(c(x)) -> c_3(x)
        g#(c(0())) -> c_4(g#(d(1())))
        g#(c(1())) -> c_5(g#(d(0())))
        g#(d(x)) -> c_6(x)
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{0,1,c,d}
    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:
        uargs(c_4) = {1},
        uargs(c_5) = {1}
      
      Following symbols are considered usable:
        {}
      TcT has computed the following interpretation:
          p(0) = [0]          
          p(1) = [0]          
          p(c) = [3]          
          p(d) = [0]          
          p(f) = [1] x1 + [2] 
          p(g) = [8] x1 + [1] 
         p(f#) = [2]          
         p(g#) = [8] x1 + [1] 
        p(c_1) = [2] x1 + [0] 
        p(c_2) = [1] x1 + [1] 
        p(c_3) = [8]          
        p(c_4) = [2] x1 + [15]
        p(c_5) = [4] x1 + [15]
        p(c_6) = [0]          
      
      Following rules are strictly oriented:
        g#(c(x)) = [25]           
                 > [8]            
                 = c_3(x)         
      
      g#(c(0())) = [25]           
                 > [17]           
                 = c_4(g#(d(1())))
      
      g#(c(1())) = [25]           
                 > [19]           
                 = c_5(g#(d(0())))
      
        g#(d(x)) = [1]            
                 > [0]            
                 = c_6(x)         
      
      
      Following rules are (at-least) weakly oriented:
      
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        g#(c(x)) -> c_3(x)
        g#(c(0())) -> c_4(g#(d(1())))
        g#(c(1())) -> c_5(g#(d(0())))
        g#(d(x)) -> c_6(x)
      Weak TRS Rules:
        
      Signature:
        {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{0,1,c,d}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).