/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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(0()) -> cons(0(),f(s(0())))
            a__f(s(0())) -> a__f(a__p(s(0())))
            a__p(X) -> p(X)
            a__p(s(0())) -> 0()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(f(X)) -> a__f(mark(X))
            mark(p(X)) -> a__p(mark(X))
            mark(s(X)) -> s(mark(X))
        - Signature:
            {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s}
    + 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(0()) -> cons(0(),f(s(0())))
            a__f(s(0())) -> a__f(a__p(s(0())))
            a__p(X) -> p(X)
            a__p(s(0())) -> 0()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(f(X)) -> a__f(mark(X))
            mark(p(X)) -> a__p(mark(X))
            mark(s(X)) -> s(mark(X))
        - Signature:
            {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s}
    + 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(0()) -> cons(0(),f(s(0())))
            a__f(s(0())) -> a__f(a__p(s(0())))
            a__p(X) -> p(X)
            a__p(s(0())) -> 0()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(f(X)) -> a__f(mark(X))
            mark(p(X)) -> a__p(mark(X))
            mark(s(X)) -> s(mark(X))
        - Signature:
            {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          mark(x){x -> cons(x,y)} =
            mark(cons(x,y)) ->^+ cons(mark(x),y)
              = 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(0()) -> cons(0(),f(s(0())))
            a__f(s(0())) -> a__f(a__p(s(0())))
            a__p(X) -> p(X)
            a__p(s(0())) -> 0()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(f(X)) -> a__f(mark(X))
            mark(p(X)) -> a__p(mark(X))
            mark(s(X)) -> s(mark(X))
        - Signature:
            {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s}
    + 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(0()) -> cons(0(),f(s(0())))
            a__f(s(0())) -> a__f(a__p(s(0())))
            a__p(X) -> p(X)
            a__p(s(0())) -> 0()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(f(X)) -> a__f(mark(X))
            mark(p(X)) -> a__p(mark(X))
            mark(s(X)) -> s(mark(X))
        - Signature:
            {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 3.
        The enriched problem is compatible with follwoing automaton.
          0_0() -> 2
          0_1() -> 1
          0_1() -> 3
          0_1() -> 6
          0_1() -> 8
          0_2() -> 1
          0_2() -> 7
          0_2() -> 8
          0_3() -> 11
          a__f_0(2) -> 1
          a__f_1(7) -> 1
          a__f_1(8) -> 1
          a__f_1(8) -> 8
          a__f_2(11) -> 1
          a__f_2(11) -> 8
          a__p_0(2) -> 1
          a__p_1(5) -> 7
          a__p_1(8) -> 1
          a__p_1(8) -> 8
          a__p_2(10) -> 11
          cons_0(2,2) -> 2
          cons_1(3,4) -> 1
          cons_1(8,2) -> 1
          cons_1(8,2) -> 8
          cons_2(7,9) -> 1
          cons_2(7,9) -> 8
          cons_3(11,12) -> 1
          cons_3(11,12) -> 8
          f_0(2) -> 2
          f_1(2) -> 1
          f_1(5) -> 4
          f_2(7) -> 1
          f_2(8) -> 1
          f_2(8) -> 8
          f_2(10) -> 9
          f_3(11) -> 1
          f_3(11) -> 8
          f_3(13) -> 12
          mark_0(2) -> 1
          mark_1(2) -> 8
          p_0(2) -> 2
          p_1(2) -> 1
          p_2(5) -> 7
          p_2(8) -> 1
          p_2(8) -> 8
          p_3(10) -> 11
          s_0(2) -> 2
          s_1(6) -> 5
          s_1(8) -> 1
          s_1(8) -> 8
          s_2(7) -> 10
          s_3(11) -> 13
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__f(X) -> f(X)
            a__f(0()) -> cons(0(),f(s(0())))
            a__f(s(0())) -> a__f(a__p(s(0())))
            a__p(X) -> p(X)
            a__p(s(0())) -> 0()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(f(X)) -> a__f(mark(X))
            mark(p(X)) -> a__p(mark(X))
            mark(s(X)) -> s(mark(X))
        - Signature:
            {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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