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

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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:
             runtime complexity wrt. defined symbols {f} and constructors {*,+,0,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:
            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 runtime complexity wrt. defined symbols {f} and constructors {*,+,0,s}
    + Applied Processor:
        Bounds {initialAutomaton = perSymbol, enrichment = match}
    + Details:
        The problem is match-bounded by 2.
        The enriched problem is compatible with follwoing automaton.
          *_0(1,1) -> 1
          *_0(1,2) -> 1
          *_0(1,3) -> 1
          *_0(1,5) -> 1
          *_0(2,1) -> 1
          *_0(2,2) -> 1
          *_0(2,3) -> 1
          *_0(2,5) -> 1
          *_0(3,1) -> 1
          *_0(3,2) -> 1
          *_0(3,3) -> 1
          *_0(3,5) -> 1
          *_0(5,1) -> 1
          *_0(5,2) -> 1
          *_0(5,3) -> 1
          *_0(5,5) -> 1
          *_1(6,7) -> 4
          *_1(7,7) -> 4
          *_1(7,7) -> 7
          *_1(8,11) -> 4
          *_1(8,11) -> 7
          +_0(1,1) -> 2
          +_0(1,2) -> 2
          +_0(1,3) -> 2
          +_0(1,5) -> 2
          +_0(2,1) -> 2
          +_0(2,2) -> 2
          +_0(2,3) -> 2
          +_0(2,5) -> 2
          +_0(3,1) -> 2
          +_0(3,2) -> 2
          +_0(3,3) -> 2
          +_0(3,5) -> 2
          +_0(5,1) -> 2
          +_0(5,2) -> 2
          +_0(5,3) -> 2
          +_0(5,5) -> 2
          +_1(8,7) -> 4
          +_1(8,7) -> 7
          0_0() -> 3
          0_1() -> 10
          0_2() -> 12
          f_0(1) -> 4
          f_0(2) -> 4
          f_0(3) -> 4
          f_0(5) -> 4
          f_1(1) -> 6
          f_1(1) -> 7
          f_1(2) -> 7
          f_1(3) -> 7
          f_1(5) -> 7
          f_1(10) -> 11
          s_0(1) -> 5
          s_0(2) -> 5
          s_0(3) -> 5
          s_0(5) -> 5
          s_1(9) -> 4
          s_1(9) -> 7
          s_1(9) -> 8
          s_1(10) -> 4
          s_1(10) -> 7
          s_1(10) -> 9
          s_2(12) -> 11
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {f} 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))