/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__tail(X) -> tail(X)
            a__tail(cons(X,XS)) -> mark(XS)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(tail(X)) -> a__tail(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0}
        - Obligation:
             runtime complexity wrt. defined symbols {a__tail,a__zeros,mark} and constructors {0,cons,tail,zeros}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__tail(X) -> tail(X)
            a__tail(cons(X,XS)) -> mark(XS)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(tail(X)) -> a__tail(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0}
        - Obligation:
             runtime complexity wrt. defined symbols {a__tail,a__zeros,mark} and constructors {0,cons,tail,zeros}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__tail(X) -> tail(X)
            a__tail(cons(X,XS)) -> mark(XS)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(tail(X)) -> a__tail(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0}
        - Obligation:
             runtime complexity wrt. defined symbols {a__tail,a__zeros,mark} and constructors {0,cons,tail,zeros}
    + 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__tail(X) -> tail(X)
            a__tail(cons(X,XS)) -> mark(XS)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(tail(X)) -> a__tail(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0}
        - Obligation:
             runtime complexity wrt. defined symbols {a__tail,a__zeros,mark} and constructors {0,cons,tail,zeros}
    + 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__tail(X) -> tail(X)
            a__tail(cons(X,XS)) -> mark(XS)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(tail(X)) -> a__tail(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__tail,a__zeros,mark} and constructors {0,cons,tail
            ,zeros}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 4.
        The enriched problem is compatible with follwoing automaton.
          0_0() -> 2
          0_1() -> 1
          0_1() -> 3
          0_2() -> 5
          0_3() -> 7
          0_4() -> 9
          a__tail_0(2) -> 1
          a__tail_1(1) -> 1
          a__zeros_0() -> 1
          a__zeros_1() -> 1
          a__zeros_2() -> 1
          a__zeros_3() -> 1
          cons_0(2,2) -> 2
          cons_1(1,2) -> 1
          cons_1(3,4) -> 1
          cons_2(5,6) -> 1
          cons_3(7,8) -> 1
          cons_4(9,10) -> 1
          mark_0(2) -> 1
          mark_1(2) -> 1
          mark_2(2) -> 1
          mark_2(4) -> 1
          mark_2(6) -> 1
          mark_2(8) -> 1
          mark_2(10) -> 1
          tail_0(2) -> 2
          tail_1(2) -> 1
          tail_2(1) -> 1
          zeros_0() -> 2
          zeros_1() -> 1
          zeros_1() -> 4
          zeros_2() -> 1
          zeros_2() -> 6
          zeros_3() -> 1
          zeros_3() -> 8
          zeros_4() -> 1
          zeros_4() -> 10
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__tail(X) -> tail(X)
            a__tail(cons(X,XS)) -> mark(XS)
            a__zeros() -> cons(0(),zeros())
            a__zeros() -> zeros()
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(tail(X)) -> a__tail(mark(X))
            mark(zeros()) -> a__zeros()
        - Signature:
            {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__tail,a__zeros,mark} and constructors {0,cons,tail
            ,zeros}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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