*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a() -> n__a()
        activate(X) -> X
        activate(n__a()) -> a()
        activate(n__f(X)) -> f(X)
        activate(n__g(X)) -> g(X)
        f(X) -> n__f(X)
        f(n__f(n__a())) -> f(n__g(f(n__a())))
        g(X) -> n__g(X)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1}
      Obligation:
        Full
        basic terms: {a,activate,f,g}/{n__a,n__f,n__g}
    Applied Processor:
      ToInnermost
    Proof:
      switch to innermost, as the system is overlay and right linear and does not contain weak rules
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a() -> n__a()
        activate(X) -> X
        activate(n__a()) -> a()
        activate(n__f(X)) -> f(X)
        activate(n__g(X)) -> g(X)
        f(X) -> n__f(X)
        f(n__f(n__a())) -> f(n__g(f(n__a())))
        g(X) -> n__g(X)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1}
      Obligation:
        Innermost
        basic terms: {a,activate,f,g}/{n__a,n__f,n__g}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        a#() -> c_1()
        activate#(X) -> c_2()
        activate#(n__a()) -> c_3(a#())
        activate#(n__f(X)) -> c_4(f#(X))
        activate#(n__g(X)) -> c_5(g#(X))
        f#(X) -> c_6()
        f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a()))
        g#(X) -> c_8()
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        a#() -> c_1()
        activate#(X) -> c_2()
        activate#(n__a()) -> c_3(a#())
        activate#(n__f(X)) -> c_4(f#(X))
        activate#(n__g(X)) -> c_5(g#(X))
        f#(X) -> c_6()
        f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a()))
        g#(X) -> c_8()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a() -> n__a()
        activate(X) -> X
        activate(n__a()) -> a()
        activate(n__f(X)) -> f(X)
        activate(n__g(X)) -> g(X)
        f(X) -> n__f(X)
        f(n__f(n__a())) -> f(n__g(f(n__a())))
        g(X) -> n__g(X)
      Signature:
        {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/2,c_8/0}
      Obligation:
        Innermost
        basic terms: {a#,activate#,f#,g#}/{n__a,n__f,n__g}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        f(X) -> n__f(X)
        a#() -> c_1()
        activate#(X) -> c_2()
        activate#(n__a()) -> c_3(a#())
        activate#(n__f(X)) -> c_4(f#(X))
        activate#(n__g(X)) -> c_5(g#(X))
        f#(X) -> c_6()
        f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a()))
        g#(X) -> c_8()
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        a#() -> c_1()
        activate#(X) -> c_2()
        activate#(n__a()) -> c_3(a#())
        activate#(n__f(X)) -> c_4(f#(X))
        activate#(n__g(X)) -> c_5(g#(X))
        f#(X) -> c_6()
        f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a()))
        g#(X) -> c_8()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(X) -> n__f(X)
      Signature:
        {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/2,c_8/0}
      Obligation:
        Innermost
        basic terms: {a#,activate#,f#,g#}/{n__a,n__f,n__g}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:a#() -> c_1()
           
        
        2:S:activate#(X) -> c_2()
           
        
        3:S:activate#(n__a()) -> c_3(a#())
           -->_1 a#() -> c_1():1
        
        4:S:activate#(n__f(X)) -> c_4(f#(X))
           -->_1 f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a())):7
           -->_1 f#(X) -> c_6():6
        
        5:S:activate#(n__g(X)) -> c_5(g#(X))
           -->_1 g#(X) -> c_8():8
        
        6:S:f#(X) -> c_6()
           
        
        7:S:f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a()))
           -->_2 f#(X) -> c_6():6
           -->_1 f#(X) -> c_6():6
        
        8:S:g#(X) -> c_8()
           
        
      The dependency graph contains no loops, we remove all dependency pairs.
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(X) -> n__f(X)
      Signature:
        {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/2,c_8/0}
      Obligation:
        Innermost
        basic terms: {a#,activate#,f#,g#}/{n__a,n__f,n__g}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).