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

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(x,y) -> h(x,y)
            h(f(x),y) -> f(g(x,y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
             runtime complexity wrt. defined symbols {g,h} and constructors {f}
    + 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,y) -> h(x,y)
            h(f(x),y) -> f(g(x,y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g,h} and constructors {f}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 2.
        The enriched problem is compatible with follwoing automaton.
          f_0(2) -> 2
          f_1(3) -> 1
          f_1(3) -> 3
          g_0(2,2) -> 1
          g_1(2,2) -> 3
          h_0(2,2) -> 1
          h_1(2,2) -> 1
          h_2(2,2) -> 3
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            g(x,y) -> h(x,y)
            h(f(x),y) -> f(g(x,y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g,h} and constructors {f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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