/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "0") :: [] -(0)-> "A"(1)
          F (TrsFun "S") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "S") :: ["A"(1)] -(1)-> "A"(1)
          F (TrsFun "f") :: ["A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "g") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "main") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          f(x1,0()) -> g(x1,0())
          f(y,S(x)) -> f(S(y),x)
          g(0(),x2) -> x2
          g(S(x),y) -> g(x,S(y))
          main(x1,x2) -> g(x1,x2)
        2. Weak:
          
        

*** Step 1.a:1.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x,y){y -> S(y)} =
            f(x,S(y)) ->^+ f(S(x),y)
              = C[f(S(x),y) = f(x,y){x -> S(x)}]

** Step 1.b:1: Bounds.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 2.
        The enriched problem is compatible with follwoing automaton.
          0_0() -> 1
          0_0() -> 2
          0_1() -> 1
          0_1() -> 3
          S_0(2) -> 1
          S_0(2) -> 2
          S_1(2) -> 1
          S_1(2) -> 4
          S_1(3) -> 1
          S_1(3) -> 3
          S_1(4) -> 1
          S_1(4) -> 4
          S_1(5) -> 1
          S_1(5) -> 3
          S_2(3) -> 1
          S_2(3) -> 5
          S_2(5) -> 1
          S_2(5) -> 5
          f_0(2,2) -> 1
          f_1(4,2) -> 1
          g_0(2,2) -> 1
          g_1(2,3) -> 1
          g_1(2,4) -> 1
          g_1(4,3) -> 1
          g_2(2,5) -> 1
          g_2(4,5) -> 1
          2 -> 1
          3 -> 1
          4 -> 1
          5 -> 1
** Step 1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))