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

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,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(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,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
          cons_0(2,2) -> 2
          cons_1(2,2) -> 4
          f_0(2) -> 1
          f_1(2) -> 1
          g_0(2) -> 1
          g_1(2) -> 1
          g_1(2) -> 3
          g_1(4) -> 3
          h_0(2) -> 1
          h_1(3) -> 1
          s_0(2) -> 2
          s_1(2) -> 1
          s_1(2) -> 3
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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