/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /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:
            a__f(X) -> f(X)
            a__f(f(a())) -> a__f(g(f(a())))
            mark(a()) -> a()
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
        - Signature:
            {a__f/1,mark/1} / {a/0,f/1,g/1}
        - Obligation:
             runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> a__f(g(f(a())))
            mark(a()) -> a()
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
        - Signature:
            {a__f/1,mark/1} / {a/0,f/1,g/1}
        - Obligation:
             runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> a__f(g(f(a())))
            mark(a()) -> a()
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
        - Signature:
            {a__f/1,mark/1} / {a/0,f/1,g/1}
        - Obligation:
             runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          mark(x){x -> f(x)} =
            mark(f(x)) ->^+ a__f(mark(x))
              = C[mark(x) = mark(x){}]

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> a__f(g(f(a())))
            mark(a()) -> a()
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
        - Signature:
            {a__f/1,mark/1} / {a/0,f/1,g/1}
        - Obligation:
             runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
** Step 1.b:2: Bounds.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> a__f(g(f(a())))
            mark(a()) -> a()
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
        - Signature:
            {a__f/1,mark/1} / {a/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 3.
        The enriched problem is compatible with follwoing automaton.
          a_0() -> 2
          a_1() -> 1
          a_1() -> 3
          a_1() -> 5
          a_2() -> 8
          a__f_0(2) -> 1
          a__f_1(3) -> 1
          a__f_1(3) -> 3
          a__f_2(6) -> 1
          a__f_2(6) -> 3
          f_0(2) -> 2
          f_1(2) -> 1
          f_1(5) -> 4
          f_2(3) -> 1
          f_2(3) -> 3
          f_2(8) -> 7
          f_3(6) -> 1
          f_3(6) -> 3
          g_0(2) -> 2
          g_1(2) -> 1
          g_1(2) -> 3
          g_1(4) -> 3
          g_2(7) -> 6
          mark_0(2) -> 1
          mark_1(2) -> 3
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> a__f(g(f(a())))
            mark(a()) -> a()
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
        - Signature:
            {a__f/1,mark/1} / {a/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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