*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        activate(X) -> X
        activate(n__d(X)) -> d(X)
        activate(n__f(X)) -> f(X)
        c(X) -> d(activate(X))
        d(X) -> n__d(X)
        f(X) -> n__f(X)
        f(f(X)) -> c(n__f(g(n__f(X))))
        h(X) -> c(n__d(X))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {activate/1,c/1,d/1,f/1,h/1} / {g/1,n__d/1,n__f/1}
      Obligation:
        Innermost
        basic terms: {activate,c,d,f,h}/{g,n__d,n__f}
    Applied Processor:
      InnermostRuleRemoval
    Proof:
      Arguments of following rules are not normal-forms.
        f(f(X)) -> c(n__f(g(n__f(X))))
      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__d(X)) -> d(X)
        activate(n__f(X)) -> f(X)
        c(X) -> d(activate(X))
        d(X) -> n__d(X)
        f(X) -> n__f(X)
        h(X) -> c(n__d(X))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {activate/1,c/1,d/1,f/1,h/1} / {g/1,n__d/1,n__f/1}
      Obligation:
        Innermost
        basic terms: {activate,c,d,f,h}/{g,n__d,n__f}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        activate#(X) -> c_1()
        activate#(n__d(X)) -> c_2(d#(X))
        activate#(n__f(X)) -> c_3(f#(X))
        c#(X) -> c_4(d#(activate(X)),activate#(X))
        d#(X) -> c_5()
        f#(X) -> c_6()
        h#(X) -> c_7(c#(n__d(X)))
      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__d(X)) -> c_2(d#(X))
        activate#(n__f(X)) -> c_3(f#(X))
        c#(X) -> c_4(d#(activate(X)),activate#(X))
        d#(X) -> c_5()
        f#(X) -> c_6()
        h#(X) -> c_7(c#(n__d(X)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        activate(X) -> X
        activate(n__d(X)) -> d(X)
        activate(n__f(X)) -> f(X)
        c(X) -> d(activate(X))
        d(X) -> n__d(X)
        f(X) -> n__f(X)
        h(X) -> c(n__d(X))
      Signature:
        {activate/1,c/1,d/1,f/1,h/1,activate#/1,c#/1,d#/1,f#/1,h#/1} / {g/1,n__d/1,n__f/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1}
      Obligation:
        Innermost
        basic terms: {activate#,c#,d#,f#,h#}/{g,n__d,n__f}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        activate(X) -> X
        activate(n__d(X)) -> d(X)
        activate(n__f(X)) -> f(X)
        d(X) -> n__d(X)
        f(X) -> n__f(X)
        activate#(X) -> c_1()
        activate#(n__d(X)) -> c_2(d#(X))
        activate#(n__f(X)) -> c_3(f#(X))
        c#(X) -> c_4(d#(activate(X)),activate#(X))
        d#(X) -> c_5()
        f#(X) -> c_6()
        h#(X) -> c_7(c#(n__d(X)))
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        activate#(X) -> c_1()
        activate#(n__d(X)) -> c_2(d#(X))
        activate#(n__f(X)) -> c_3(f#(X))
        c#(X) -> c_4(d#(activate(X)),activate#(X))
        d#(X) -> c_5()
        f#(X) -> c_6()
        h#(X) -> c_7(c#(n__d(X)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        activate(X) -> X
        activate(n__d(X)) -> d(X)
        activate(n__f(X)) -> f(X)
        d(X) -> n__d(X)
        f(X) -> n__f(X)
      Signature:
        {activate/1,c/1,d/1,f/1,h/1,activate#/1,c#/1,d#/1,f#/1,h#/1} / {g/1,n__d/1,n__f/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1}
      Obligation:
        Innermost
        basic terms: {activate#,c#,d#,f#,h#}/{g,n__d,n__f}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:activate#(X) -> c_1()
           
        
        2:S:activate#(n__d(X)) -> c_2(d#(X))
           -->_1 d#(X) -> c_5():5
        
        3:S:activate#(n__f(X)) -> c_3(f#(X))
           -->_1 f#(X) -> c_6():6
        
        4:S:c#(X) -> c_4(d#(activate(X)),activate#(X))
           -->_1 d#(X) -> c_5():5
           -->_2 activate#(n__f(X)) -> c_3(f#(X)):3
           -->_2 activate#(n__d(X)) -> c_2(d#(X)):2
           -->_2 activate#(X) -> c_1():1
        
        5:S:d#(X) -> c_5()
           
        
        6:S:f#(X) -> c_6()
           
        
        7:S:h#(X) -> c_7(c#(n__d(X)))
           -->_1 c#(X) -> c_4(d#(activate(X)),activate#(X)):4
        
      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:
        activate(X) -> X
        activate(n__d(X)) -> d(X)
        activate(n__f(X)) -> f(X)
        d(X) -> n__d(X)
        f(X) -> n__f(X)
      Signature:
        {activate/1,c/1,d/1,f/1,h/1,activate#/1,c#/1,d#/1,f#/1,h#/1} / {g/1,n__d/1,n__f/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1}
      Obligation:
        Innermost
        basic terms: {activate#,c#,d#,f#,h#}/{g,n__d,n__f}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).