/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?,O(n^2))
* Step 1: Sum.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1} / {0/0,s/1,times/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1} / {0/0,s/1,times/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          fac#(0()) -> c_1()
          fac#(s(x)) -> c_2(fac#(p(s(x))),p#(s(x)))
          p#(s(x)) -> c_3()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: PredecessorEstimation.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            fac#(0()) -> c_1()
            fac#(s(x)) -> c_2(fac#(p(s(x))),p#(s(x)))
            p#(s(x)) -> c_3()
        - Weak TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/2,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3}
        by application of
          Pre({1,3}) = {2}.
        Here rules are labelled as follows:
          1: fac#(0()) -> c_1()
          2: fac#(s(x)) -> c_2(fac#(p(s(x))),p#(s(x)))
          3: p#(s(x)) -> c_3()
* Step 4: RemoveWeakSuffixes.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_2(fac#(p(s(x))),p#(s(x)))
        - Weak DPs:
            fac#(0()) -> c_1()
            p#(s(x)) -> c_3()
        - Weak TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/2,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fac#(s(x)) -> c_2(fac#(p(s(x))),p#(s(x)))
             -->_2 p#(s(x)) -> c_3():3
             -->_1 fac#(0()) -> c_1():2
             -->_1 fac#(s(x)) -> c_2(fac#(p(s(x))),p#(s(x))):1
          
          2:W:fac#(0()) -> c_1()
             
          
          3:W:p#(s(x)) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: fac#(0()) -> c_1()
          3: p#(s(x)) -> c_3()
* Step 5: SimplifyRHS.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_2(fac#(p(s(x))),p#(s(x)))
        - Weak TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/2,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:fac#(s(x)) -> c_2(fac#(p(s(x))),p#(s(x)))
             -->_1 fac#(s(x)) -> c_2(fac#(p(s(x))),p#(s(x))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fac#(s(x)) -> c_2(fac#(p(s(x))))
* Step 6: UsableRules.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_2(fac#(p(s(x))))
        - Weak TRS:
            fac(0()) -> s(0())
            fac(s(x)) -> times(s(x),fac(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          p(s(x)) -> x
          fac#(s(x)) -> c_2(fac#(p(s(x))))
* Step 7: Ara.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_2(fac#(p(s(x))))
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "p") :: ["A"(2, 2)] -(0)-> "A"(4, 2)
          F (TrsFun "s") :: ["A"(6, 2)] -(4)-> "A"(4, 2)
          F (TrsFun "s") :: ["A"(4, 2)] -(2)-> "A"(2, 2)
          F (TrsFun "s") :: ["A"(5, 2)] -(3)-> "A"(3, 2)
          F (DpFun "fac") :: ["A"(4, 2)] -(0)-> "A"(0, 0)
          F (ComFun 2) :: ["A"(0, 0)] -(0)-> "A"(0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          fac#(s(x)) -> c_2(fac#(p(s(x))))
        2. Weak:
          
        

WORST_CASE(?,O(n^2))