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

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(x,s(y)) -> g(f(x,y),0())
            g(0(),f(x,x)) -> x
            g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
            g(s(x),y) -> g(f(x,y),0())
        - Signature:
            {g/2} / {0/0,f/2,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {g} and constructors {0,f,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:
            g(x,s(y)) -> g(f(x,y),0())
            g(0(),f(x,x)) -> x
            g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
            g(s(x),y) -> g(f(x,y),0())
        - Signature:
            {g/2} / {0/0,f/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,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_0() -> 1
          0_0() -> 3
          0_1() -> 6
          0_2() -> 11
          0_2() -> 12
          f_0(1,1) -> 2
          f_0(1,1) -> 3
          f_0(1,2) -> 2
          f_0(1,2) -> 3
          f_0(1,4) -> 2
          f_0(1,4) -> 3
          f_0(2,1) -> 2
          f_0(2,1) -> 3
          f_0(2,2) -> 2
          f_0(2,2) -> 3
          f_0(2,4) -> 2
          f_0(2,4) -> 3
          f_0(4,1) -> 2
          f_0(4,1) -> 3
          f_0(4,2) -> 2
          f_0(4,2) -> 3
          f_0(4,4) -> 2
          f_0(4,4) -> 3
          f_1(1,1) -> 5
          f_1(1,2) -> 5
          f_1(1,4) -> 5
          f_1(1,6) -> 1
          f_1(1,6) -> 3
          f_1(1,12) -> 13
          f_1(2,1) -> 5
          f_1(2,2) -> 5
          f_1(2,4) -> 5
          f_1(2,6) -> 1
          f_1(2,6) -> 3
          f_1(2,12) -> 13
          f_1(4,1) -> 5
          f_1(4,2) -> 5
          f_1(4,4) -> 5
          f_1(4,6) -> 1
          f_1(4,6) -> 3
          f_1(4,12) -> 13
          f_1(7,8) -> 3
          f_1(8,8) -> 3
          f_1(8,8) -> 8
          f_1(8,8) -> 10
          f_2(9,10) -> 3
          f_2(10,10) -> 3
          f_2(10,14) -> 7
          f_2(10,14) -> 8
          f_2(10,14) -> 9
          f_2(10,14) -> 10
          g_0(1,1) -> 3
          g_0(1,2) -> 3
          g_0(1,4) -> 3
          g_0(2,1) -> 3
          g_0(2,2) -> 3
          g_0(2,4) -> 3
          g_0(4,1) -> 3
          g_0(4,2) -> 3
          g_0(4,4) -> 3
          g_1(1,6) -> 7
          g_1(1,6) -> 8
          g_1(2,6) -> 8
          g_1(4,6) -> 8
          g_1(5,6) -> 3
          g_1(6,6) -> 8
          g_1(13,6) -> 10
          g_2(1,11) -> 9
          g_2(1,12) -> 10
          g_2(2,12) -> 10
          g_2(4,12) -> 10
          g_2(6,12) -> 14
          g_2(12,12) -> 14
          s_0(1) -> 3
          s_0(1) -> 4
          s_0(2) -> 3
          s_0(2) -> 4
          s_0(4) -> 3
          s_0(4) -> 4
          1 -> 3
          2 -> 3
          4 -> 3
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            g(x,s(y)) -> g(f(x,y),0())
            g(0(),f(x,x)) -> x
            g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
            g(s(x),y) -> g(f(x,y),0())
        - Signature:
            {g/2} / {0/0,f/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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