*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(+(x,y)) -> *(f(x),f(y))
        f(+(x,s(0()))) -> +(s(s(0())),f(x))
        f(0()) -> s(0())
        f(s(0())) -> *(s(s(0())),f(0()))
        f(s(0())) -> s(s(0()))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/1} / {*/2,+/2,0/0,s/1}
      Obligation:
        Full
        basic terms: {f}/{*,+,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(+(x,y)) -> *(f(x),f(y))
        f(+(x,s(0()))) -> +(s(s(0())),f(x))
        f(0()) -> s(0())
        f(s(0())) -> *(s(s(0())),f(0()))
        f(s(0())) -> s(s(0()))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/1} / {*/2,+/2,0/0,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{*,+,0,s}
    Applied Processor:
      Bounds {initialAutomaton = minimal, enrichment = match}
    Proof:
      The problem is match-bounded by 2.
      The enriched problem is compatible with follwoing automaton.
        *_0(2,2) -> 2
        *_1(3,4) -> 1
        *_1(4,4) -> 3
        *_1(4,4) -> 4
        *_1(5,8) -> 1
        *_1(5,8) -> 3
        *_1(5,8) -> 4
        +_0(2,2) -> 2
        +_1(5,4) -> 1
        +_1(5,4) -> 3
        +_1(5,4) -> 4
        0_0() -> 2
        0_1() -> 7
        0_2() -> 9
        f_0(2) -> 1
        f_1(2) -> 3
        f_1(2) -> 4
        f_1(7) -> 8
        s_0(2) -> 2
        s_1(6) -> 1
        s_1(6) -> 3
        s_1(6) -> 4
        s_1(6) -> 5
        s_1(7) -> 1
        s_1(7) -> 3
        s_1(7) -> 4
        s_1(7) -> 6
        s_2(9) -> 8
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(+(x,y)) -> *(f(x),f(y))
        f(+(x,s(0()))) -> +(s(s(0())),f(x))
        f(0()) -> s(0())
        f(s(0())) -> *(s(s(0())),f(0()))
        f(s(0())) -> s(s(0()))
      Signature:
        {f/1} / {*/2,+/2,0/0,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{*,+,0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).