*** 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:
        Innermost
        basic terms: {f,g}/{0,1,c,d}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
        f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
        g#(c(x)) -> c_3()
        g#(c(0())) -> c_4(g#(d(1())))
        g#(c(1())) -> c_5(g#(d(0())))
        g#(d(x)) -> c_6()
      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#(x))
        f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
        g#(c(x)) -> c_3()
        g#(c(0())) -> c_4(g#(d(1())))
        g#(c(1())) -> c_5(g#(d(0())))
        g#(d(x)) -> c_6()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak 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
      Signature:
        {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,c_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0}
      Obligation:
        Innermost
        basic terms: {f#,g#}/{0,1,c,d}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        g#(c(x)) -> c_3()
        g#(c(0())) -> c_4(g#(d(1())))
        g#(c(1())) -> c_5(g#(d(0())))
        g#(d(x)) -> c_6()
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(c(x)) -> c_3()
        g#(c(0())) -> c_4(g#(d(1())))
        g#(c(1())) -> c_5(g#(d(0())))
        g#(d(x)) -> c_6()
      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_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0}
      Obligation:
        Innermost
        basic terms: {f#,g#}/{0,1,c,d}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:g#(c(x)) -> c_3()
           
        
        2:S:g#(c(0())) -> c_4(g#(d(1())))
           -->_1 g#(d(x)) -> c_6():4
        
        3:S:g#(c(1())) -> c_5(g#(d(0())))
           -->_1 g#(d(x)) -> c_6():4
        
        4:S:g#(d(x)) -> c_6()
           
        
      The dependency graph contains no loops, we remove all dependency pairs.
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      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_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0}
      Obligation:
        Innermost
        basic terms: {f#,g#}/{0,1,c,d}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).