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

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(c(X,s(Y))) -> f(c(s(X),Y))
            g(c(s(X),Y)) -> f(c(X,s(Y)))
        - Signature:
            {f/1,g/1} / {c/2,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {c,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(c(X,s(Y))) -> f(c(s(X),Y))
            g(c(s(X),Y)) -> f(c(X,s(Y)))
        - Signature:
            {f/1,g/1} / {c/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {c,s}
    + Applied Processor:
        Bounds {initialAutomaton = perSymbol, enrichment = match}
    + Details:
        The problem is match-bounded by 2.
        The enriched problem is compatible with follwoing automaton.
          c_0(1,1) -> 1
          c_0(1,4) -> 1
          c_0(4,1) -> 1
          c_0(4,4) -> 1
          c_1(1,6) -> 7
          c_1(4,6) -> 7
          c_1(6,1) -> 5
          c_1(6,4) -> 5
          c_1(10,1) -> 7
          c_1(10,4) -> 7
          c_2(9,1) -> 8
          c_2(9,4) -> 8
          c_2(9,6) -> 8
          f_0(1) -> 2
          f_0(4) -> 2
          f_1(5) -> 2
          f_1(7) -> 3
          f_2(8) -> 3
          g_0(1) -> 3
          g_0(4) -> 3
          s_0(1) -> 4
          s_0(4) -> 4
          s_1(1) -> 6
          s_1(4) -> 6
          s_1(6) -> 6
          s_1(9) -> 10
          s_1(10) -> 10
          s_2(1) -> 9
          s_2(4) -> 9
          s_2(9) -> 9
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(c(X,s(Y))) -> f(c(s(X),Y))
            g(c(s(X),Y)) -> f(c(X,s(Y)))
        - Signature:
            {f/1,g/1} / {c/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {c,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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