*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a__f(X) -> f(X)
        a__f(0()) -> cons(0(),f(s(0())))
        a__f(s(0())) -> a__f(a__p(s(0())))
        a__p(X) -> p(X)
        a__p(s(0())) -> 0()
        mark(0()) -> 0()
        mark(cons(X1,X2)) -> cons(mark(X1),X2)
        mark(f(X)) -> a__f(mark(X))
        mark(p(X)) -> a__p(mark(X))
        mark(s(X)) -> s(mark(X))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1}
      Obligation:
        Full
        basic terms: {a__f,a__p,mark}/{0,cons,f,p,s}
    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(0()) -> cons(0(),f(s(0())))
        a__f(s(0())) -> a__f(a__p(s(0())))
        a__p(X) -> p(X)
        a__p(s(0())) -> 0()
        mark(0()) -> 0()
        mark(cons(X1,X2)) -> cons(mark(X1),X2)
        mark(f(X)) -> a__f(mark(X))
        mark(p(X)) -> a__p(mark(X))
        mark(s(X)) -> s(mark(X))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1}
      Obligation:
        Innermost
        basic terms: {a__f,a__p,mark}/{0,cons,f,p,s}
    Applied Processor:
      Bounds {initialAutomaton = minimal, enrichment = match}
    Proof:
      The problem is match-bounded by 3.
      The enriched problem is compatible with follwoing automaton.
        0_0() -> 2
        0_1() -> 1
        0_1() -> 3
        0_1() -> 6
        0_1() -> 8
        0_2() -> 1
        0_2() -> 7
        0_2() -> 8
        0_3() -> 11
        a__f_0(2) -> 1
        a__f_1(7) -> 1
        a__f_1(8) -> 1
        a__f_1(8) -> 8
        a__f_2(11) -> 1
        a__f_2(11) -> 8
        a__p_0(2) -> 1
        a__p_1(5) -> 7
        a__p_1(8) -> 1
        a__p_1(8) -> 8
        a__p_2(10) -> 11
        cons_0(2,2) -> 2
        cons_1(3,4) -> 1
        cons_1(8,2) -> 1
        cons_1(8,2) -> 8
        cons_2(7,9) -> 1
        cons_2(7,9) -> 8
        cons_3(11,12) -> 1
        cons_3(11,12) -> 8
        f_0(2) -> 2
        f_1(2) -> 1
        f_1(5) -> 4
        f_2(7) -> 1
        f_2(8) -> 1
        f_2(8) -> 8
        f_2(10) -> 9
        f_3(11) -> 1
        f_3(11) -> 8
        f_3(13) -> 12
        mark_0(2) -> 1
        mark_1(2) -> 8
        p_0(2) -> 2
        p_1(2) -> 1
        p_2(5) -> 7
        p_2(8) -> 1
        p_2(8) -> 8
        p_3(10) -> 11
        s_0(2) -> 2
        s_1(6) -> 5
        s_1(8) -> 1
        s_1(8) -> 8
        s_2(7) -> 10
        s_3(11) -> 13
*** 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(0()) -> cons(0(),f(s(0())))
        a__f(s(0())) -> a__f(a__p(s(0())))
        a__p(X) -> p(X)
        a__p(s(0())) -> 0()
        mark(0()) -> 0()
        mark(cons(X1,X2)) -> cons(mark(X1),X2)
        mark(f(X)) -> a__f(mark(X))
        mark(p(X)) -> a__p(mark(X))
        mark(s(X)) -> s(mark(X))
      Signature:
        {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1}
      Obligation:
        Innermost
        basic terms: {a__f,a__p,mark}/{0,cons,f,p,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).