*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a__f(X) -> f(X)
        a__f(f(a())) -> a__f(g(f(a())))
        mark(a()) -> a()
        mark(f(X)) -> a__f(X)
        mark(g(X)) -> g(mark(X))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__f/1,mark/1} / {a/0,f/1,g/1}
      Obligation:
        Full
        basic terms: {a__f,mark}/{a,f,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(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a__f(X) -> f(X)
        a__f(f(a())) -> a__f(g(f(a())))
        mark(a()) -> a()
        mark(f(X)) -> a__f(X)
        mark(g(X)) -> g(mark(X))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__f/1,mark/1} / {a/0,f/1,g/1}
      Obligation:
        Innermost
        basic terms: {a__f,mark}/{a,f,g}
    Applied Processor:
      Bounds {initialAutomaton = perSymbol, enrichment = match}
    Proof:
      The problem is match-bounded by 2.
      The enriched problem is compatible with follwoing automaton.
        a_0() -> 1
        a_1() -> 5
        a_1() -> 8
        a_1() -> 9
        a__f_0(1) -> 2
        a__f_0(3) -> 2
        a__f_0(4) -> 2
        a__f_1(1) -> 5
        a__f_1(1) -> 9
        a__f_1(3) -> 5
        a__f_1(3) -> 9
        a__f_1(4) -> 5
        a__f_1(4) -> 9
        a__f_1(6) -> 2
        a__f_1(6) -> 5
        a__f_1(6) -> 9
        f_0(1) -> 3
        f_0(3) -> 3
        f_0(4) -> 3
        f_1(1) -> 2
        f_1(3) -> 2
        f_1(4) -> 2
        f_1(8) -> 7
        f_2(1) -> 5
        f_2(1) -> 9
        f_2(3) -> 5
        f_2(3) -> 9
        f_2(4) -> 5
        f_2(4) -> 9
        f_2(6) -> 2
        f_2(6) -> 5
        f_2(6) -> 9
        g_0(1) -> 4
        g_0(3) -> 4
        g_0(4) -> 4
        g_1(7) -> 6
        g_1(9) -> 5
        g_1(9) -> 9
        mark_0(1) -> 5
        mark_0(3) -> 5
        mark_0(4) -> 5
        mark_1(1) -> 9
        mark_1(3) -> 9
        mark_1(4) -> 9
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a__f(X) -> f(X)
        a__f(f(a())) -> a__f(g(f(a())))
        mark(a()) -> a()
        mark(f(X)) -> a__f(X)
        mark(g(X)) -> g(mark(X))
      Signature:
        {a__f/1,mark/1} / {a/0,f/1,g/1}
      Obligation:
        Innermost
        basic terms: {a__f,mark}/{a,f,g}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).