/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,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,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,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,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,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x,y){y -> g(y,z)} =
            f(x,g(y,z)) ->^+ g(f(x,y),z)
              = C[f(x,y) = f(x,y){}]

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,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,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 1.
        The enriched problem is compatible with follwoing automaton.
          0_0() -> 2
          0_1() -> 1
          0_1() -> 4
          f_0(2,2) -> 1
          f_1(2,2) -> 3
          g_0(2,2) -> 2
          g_1(2,2) -> 1
          g_1(3,2) -> 1
          g_1(3,2) -> 3
          nil_0() -> 2
          nil_1() -> 1
          nil_1() -> 3
          norm_0(2) -> 1
          norm_1(2) -> 4
          rem_0(2,2) -> 1
          rem_1(2,2) -> 1
          s_0(2) -> 2
          s_1(4) -> 1
          s_1(4) -> 4
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,g(y,z)) -> g(f(x,y),z)
            f(x,nil()) -> g(nil(),x)
            norm(g(x,y)) -> s(norm(x))
            norm(nil()) -> 0()
            rem(g(x,y),0()) -> g(x,y)
            rem(g(x,y),s(z)) -> rem(x,z)
            rem(nil(),y) -> nil()
        - Signature:
            {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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