*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        b(X) -> a(X)
        f(a(g(X))) -> b(X)
        f(f(X)) -> f(a(b(f(X))))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {b/1,f/1} / {a/1,g/1}
      Obligation:
        Innermost
        basic terms: {b,f}/{a,g}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        b#(X) -> c_1()
        f#(a(g(X))) -> c_2(b#(X))
        f#(f(X)) -> c_3(f#(a(b(f(X)))),b#(f(X)),f#(X))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        b#(X) -> c_1()
        f#(a(g(X))) -> c_2(b#(X))
        f#(f(X)) -> c_3(f#(a(b(f(X)))),b#(f(X)),f#(X))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        b(X) -> a(X)
        f(a(g(X))) -> b(X)
        f(f(X)) -> f(a(b(f(X))))
      Signature:
        {b/1,f/1,b#/1,f#/1} / {a/1,g/1,c_1/0,c_2/1,c_3/3}
      Obligation:
        Innermost
        basic terms: {b#,f#}/{a,g}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        b#(X) -> c_1()
        f#(a(g(X))) -> c_2(b#(X))
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        b#(X) -> c_1()
        f#(a(g(X))) -> c_2(b#(X))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {b/1,f/1,b#/1,f#/1} / {a/1,g/1,c_1/0,c_2/1,c_3/3}
      Obligation:
        Innermost
        basic terms: {b#,f#}/{a,g}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:b#(X) -> c_1()
           
        
        2:S:f#(a(g(X))) -> c_2(b#(X))
           -->_1 b#(X) -> c_1():1
        
      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:
        {b/1,f/1,b#/1,f#/1} / {a/1,g/1,c_1/0,c_2/1,c_3/3}
      Obligation:
        Innermost
        basic terms: {b#,f#}/{a,g}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).