*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a__c() -> a__f(g(c()))
        a__c() -> c()
        a__f(X) -> f(X)
        a__f(g(X)) -> g(X)
        mark(c()) -> a__c()
        mark(f(X)) -> a__f(X)
        mark(g(X)) -> g(X)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__c/0,a__f/1,mark/1} / {c/0,f/1,g/1}
      Obligation:
        Innermost
        basic terms: {a__c,a__f,mark}/{c,f,g}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        a__c#() -> c_1(a__f#(g(c())))
        a__c#() -> c_2()
        a__f#(X) -> c_3()
        a__f#(g(X)) -> c_4()
        mark#(c()) -> c_5(a__c#())
        mark#(f(X)) -> c_6(a__f#(X))
        mark#(g(X)) -> c_7()
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        a__c#() -> c_1(a__f#(g(c())))
        a__c#() -> c_2()
        a__f#(X) -> c_3()
        a__f#(g(X)) -> c_4()
        mark#(c()) -> c_5(a__c#())
        mark#(f(X)) -> c_6(a__f#(X))
        mark#(g(X)) -> c_7()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a__c() -> a__f(g(c()))
        a__c() -> c()
        a__f(X) -> f(X)
        a__f(g(X)) -> g(X)
        mark(c()) -> a__c()
        mark(f(X)) -> a__f(X)
        mark(g(X)) -> g(X)
      Signature:
        {a__c/0,a__f/1,mark/1,a__c#/0,a__f#/1,mark#/1} / {c/0,f/1,g/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0}
      Obligation:
        Innermost
        basic terms: {a__c#,a__f#,mark#}/{c,f,g}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        a__c#() -> c_1(a__f#(g(c())))
        a__c#() -> c_2()
        a__f#(X) -> c_3()
        a__f#(g(X)) -> c_4()
        mark#(c()) -> c_5(a__c#())
        mark#(f(X)) -> c_6(a__f#(X))
        mark#(g(X)) -> c_7()
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        a__c#() -> c_1(a__f#(g(c())))
        a__c#() -> c_2()
        a__f#(X) -> c_3()
        a__f#(g(X)) -> c_4()
        mark#(c()) -> c_5(a__c#())
        mark#(f(X)) -> c_6(a__f#(X))
        mark#(g(X)) -> c_7()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__c/0,a__f/1,mark/1,a__c#/0,a__f#/1,mark#/1} / {c/0,f/1,g/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0}
      Obligation:
        Innermost
        basic terms: {a__c#,a__f#,mark#}/{c,f,g}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:a__c#() -> c_1(a__f#(g(c())))
           -->_1 a__f#(g(X)) -> c_4():4
           -->_1 a__f#(X) -> c_3():3
        
        2:S:a__c#() -> c_2()
           
        
        3:S:a__f#(X) -> c_3()
           
        
        4:S:a__f#(g(X)) -> c_4()
           
        
        5:S:mark#(c()) -> c_5(a__c#())
           -->_1 a__c#() -> c_2():2
           -->_1 a__c#() -> c_1(a__f#(g(c()))):1
        
        6:S:mark#(f(X)) -> c_6(a__f#(X))
           -->_1 a__f#(g(X)) -> c_4():4
           -->_1 a__f#(X) -> c_3():3
        
        7:S:mark#(g(X)) -> c_7()
           
        
      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:
        {a__c/0,a__f/1,mark/1,a__c#/0,a__f#/1,mark#/1} / {c/0,f/1,g/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0}
      Obligation:
        Innermost
        basic terms: {a__c#,a__f#,mark#}/{c,f,g}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).