*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(a(),a()) -> f(a(),b())
        f(a(),b()) -> f(s(a()),c())
        f(c(),c()) -> f(a(),a())
        f(s(X),c()) -> f(X,c())
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2} / {a/0,b/0,c/0,s/1}
      Obligation:
        Full
        basic terms: {f}/{a,b,c,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:
        f(a(),a()) -> f(a(),b())
        f(a(),b()) -> f(s(a()),c())
        f(c(),c()) -> f(a(),a())
        f(s(X),c()) -> f(X,c())
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2} / {a/0,b/0,c/0,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{a,b,c,s}
    Applied Processor:
      Bounds {initialAutomaton = minimal, enrichment = match}
    Proof:
      The problem is match-bounded by 4.
      The enriched problem is compatible with follwoing automaton.
        a_0() -> 2
        a_1() -> 3
        a_2() -> 7
        a_3() -> 10
        b_0() -> 2
        b_1() -> 4
        b_2() -> 9
        c_0() -> 2
        c_1() -> 4
        c_2() -> 6
        c_3() -> 8
        c_4() -> 11
        f_0(2,2) -> 1
        f_1(2,4) -> 1
        f_1(3,3) -> 1
        f_1(3,4) -> 1
        f_2(3,6) -> 1
        f_2(5,6) -> 1
        f_2(7,9) -> 1
        f_3(7,8) -> 1
        f_4(10,11) -> 1
        s_0(2) -> 2
        s_1(3) -> 3
        s_2(7) -> 5
        s_3(10) -> 7
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(a(),a()) -> f(a(),b())
        f(a(),b()) -> f(s(a()),c())
        f(c(),c()) -> f(a(),a())
        f(s(X),c()) -> f(X,c())
      Signature:
        {f/2} / {a/0,b/0,c/0,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{a,b,c,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).