*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(0()) -> 0()
        f(s(x)) -> s(s(f(p(s(x)))))
        p(s(x)) -> x
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/1,p/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {f,p}/{0,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(0()) -> 0()
        f(s(x)) -> s(s(f(p(s(x)))))
        p(s(x)) -> x
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/1,p/1} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {f,p}/{0,s}
    Applied Processor:
      Bounds {initialAutomaton = minimal, enrichment = match}
    Proof:
      The problem is match-bounded by 1.
      The enriched problem is compatible with follwoing automaton.
        0_0() -> 1
        0_0() -> 2
        0_0() -> 5
        0_1() -> 1
        0_1() -> 4
        f_0(2) -> 1
        f_1(5) -> 4
        p_0(2) -> 1
        p_1(6) -> 5
        s_0(2) -> 1
        s_0(2) -> 2
        s_0(2) -> 5
        s_1(2) -> 6
        s_1(3) -> 1
        s_1(3) -> 4
        s_1(4) -> 3
        2 -> 1
        2 -> 5
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(0()) -> 0()
        f(s(x)) -> s(s(f(p(s(x)))))
        p(s(x)) -> x
      Signature:
        {f/1,p/1} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {f,p}/{0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).