/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(?,O(1))
* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            p(f(f(x))) -> q(f(g(x)))
            p(g(g(x))) -> q(g(f(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1} / {f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {p,q} and constructors {f,g}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            p(f(f(x))) -> q(f(g(x)))
            p(g(g(x))) -> q(g(f(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1} / {f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {p,q} and constructors {f,g}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          p#(f(f(x))) -> c_1(q#(f(g(x))))
          p#(g(g(x))) -> c_2(q#(g(f(x))))
          q#(f(f(x))) -> c_3(p#(f(g(x))))
          q#(g(g(x))) -> c_4(p#(g(f(x))))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            p#(f(f(x))) -> c_1(q#(f(g(x))))
            p#(g(g(x))) -> c_2(q#(g(f(x))))
            q#(f(f(x))) -> c_3(p#(f(g(x))))
            q#(g(g(x))) -> c_4(p#(g(f(x))))
        - Weak TRS:
            p(f(f(x))) -> q(f(g(x)))
            p(g(g(x))) -> q(g(f(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1,p#/1,q#/1} / {f/1,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {p#,q#} and constructors {f,g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,4}
        by application of
          Pre({1,2,3,4}) = {}.
        Here rules are labelled as follows:
          1: p#(f(f(x))) -> c_1(q#(f(g(x))))
          2: p#(g(g(x))) -> c_2(q#(g(f(x))))
          3: q#(f(f(x))) -> c_3(p#(f(g(x))))
          4: q#(g(g(x))) -> c_4(p#(g(f(x))))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            p#(f(f(x))) -> c_1(q#(f(g(x))))
            p#(g(g(x))) -> c_2(q#(g(f(x))))
            q#(f(f(x))) -> c_3(p#(f(g(x))))
            q#(g(g(x))) -> c_4(p#(g(f(x))))
        - Weak TRS:
            p(f(f(x))) -> q(f(g(x)))
            p(g(g(x))) -> q(g(f(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1,p#/1,q#/1} / {f/1,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {p#,q#} and constructors {f,g}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:p#(f(f(x))) -> c_1(q#(f(g(x))))
             
          
          2:W:p#(g(g(x))) -> c_2(q#(g(f(x))))
             
          
          3:W:q#(f(f(x))) -> c_3(p#(f(g(x))))
             
          
          4:W:q#(g(g(x))) -> c_4(p#(g(f(x))))
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: q#(g(g(x))) -> c_4(p#(g(f(x))))
          3: q#(f(f(x))) -> c_3(p#(f(g(x))))
          2: p#(g(g(x))) -> c_2(q#(g(f(x))))
          1: p#(f(f(x))) -> c_1(q#(f(g(x))))
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            p(f(f(x))) -> q(f(g(x)))
            p(g(g(x))) -> q(g(f(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1,p#/1,q#/1} / {f/1,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {p#,q#} and constructors {f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))