*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        activate(X) -> X
        activate(n__f(X)) -> f(X)
        f(X) -> n__f(X)
        f(f(a())) -> c(n__f(g(f(a()))))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {activate/1,f/1} / {a/0,c/1,g/1,n__f/1}
      Obligation:
        Innermost
        basic terms: {activate,f}/{a,c,g,n__f}
    Applied Processor:
      InnermostRuleRemoval
    Proof:
      Arguments of following rules are not normal-forms.
        f(f(a())) -> c(n__f(g(f(a()))))
      All above mentioned rules can be savely removed.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        activate(X) -> X
        activate(n__f(X)) -> f(X)
        f(X) -> n__f(X)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {activate/1,f/1} / {a/0,c/1,g/1,n__f/1}
      Obligation:
        Innermost
        basic terms: {activate,f}/{a,c,g,n__f}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        activate#(X) -> c_1()
        activate#(n__f(X)) -> c_2(f#(X))
        f#(X) -> c_3()
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        activate#(X) -> c_1()
        activate#(n__f(X)) -> c_2(f#(X))
        f#(X) -> c_3()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        activate(X) -> X
        activate(n__f(X)) -> f(X)
        f(X) -> n__f(X)
      Signature:
        {activate/1,f/1,activate#/1,f#/1} / {a/0,c/1,g/1,n__f/1,c_1/0,c_2/1,c_3/0}
      Obligation:
        Innermost
        basic terms: {activate#,f#}/{a,c,g,n__f}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        activate#(X) -> c_1()
        activate#(n__f(X)) -> c_2(f#(X))
        f#(X) -> c_3()
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        activate#(X) -> c_1()
        activate#(n__f(X)) -> c_2(f#(X))
        f#(X) -> c_3()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {activate/1,f/1,activate#/1,f#/1} / {a/0,c/1,g/1,n__f/1,c_1/0,c_2/1,c_3/0}
      Obligation:
        Innermost
        basic terms: {activate#,f#}/{a,c,g,n__f}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:activate#(X) -> c_1()
           
        
        2:S:activate#(n__f(X)) -> c_2(f#(X))
           -->_1 f#(X) -> c_3():3
        
        3:S:f#(X) -> c_3()
           
        
      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:
        
      Signature:
        {activate/1,f/1,activate#/1,f#/1} / {a/0,c/1,g/1,n__f/1,c_1/0,c_2/1,c_3/0}
      Obligation:
        Innermost
        basic terms: {activate#,f#}/{a,c,g,n__f}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).