/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(Omega(n^1),O(n^3))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^3))
    + Considered Problem:
        - Strict TRS:
            fac(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1} / {*/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {*,0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            fac(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1} / {*/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {*,0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            fac(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1} / {*/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {*,0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          p(s(x)){x -> s(x)} =
            p(s(s(x))) ->^+ s(p(s(x)))
              = C[p(s(x)) = p(s(x)){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            fac(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1} / {*/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p} and constructors {*,0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
          p#(s(0())) -> c_2()
          p#(s(s(x))) -> c_3(p#(s(x)))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
            p#(s(0())) -> c_2()
            p#(s(s(x))) -> c_3(p#(s(x)))
        - Weak TRS:
            fac(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {*/2,0/0,s/1,c_1/2,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {*,0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1,3}.
        Here rules are labelled as follows:
          1: fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
          2: p#(s(0())) -> c_2()
          3: p#(s(s(x))) -> c_3(p#(s(x)))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
            p#(s(s(x))) -> c_3(p#(s(x)))
        - Weak DPs:
            p#(s(0())) -> c_2()
        - Weak TRS:
            fac(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {*/2,0/0,s/1,c_1/2,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {*,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
             -->_2 p#(s(s(x))) -> c_3(p#(s(x))):2
             -->_2 p#(s(0())) -> c_2():3
             -->_1 fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x))):1
          
          2:S:p#(s(s(x))) -> c_3(p#(s(x)))
             -->_1 p#(s(0())) -> c_2():3
             -->_1 p#(s(s(x))) -> c_3(p#(s(x))):2
          
          3:W:p#(s(0())) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: p#(s(0())) -> c_2()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
            p#(s(s(x))) -> c_3(p#(s(x)))
        - Weak TRS:
            fac(s(x)) -> *(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {*/2,0/0,s/1,c_1/2,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {*,0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          p(s(0())) -> 0()
          p(s(s(x))) -> s(p(s(x)))
          fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
          p#(s(s(x))) -> c_3(p#(s(x)))
** Step 1.b:5: DecomposeDG.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
            p#(s(s(x))) -> c_3(p#(s(x)))
        - Weak TRS:
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {*/2,0/0,s/1,c_1/2,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {*,0,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
        and a lower component
          p#(s(s(x))) -> c_3(p#(s(x)))
        Further, following extension rules are added to the lower component.
          fac#(s(x)) -> fac#(p(s(x)))
          fac#(s(x)) -> p#(s(x))
*** Step 1.b:5.a:1: SimplifyRHS.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
        - Weak TRS:
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {*/2,0/0,s/1,c_1/2,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {*,0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:fac#(s(x)) -> c_1(fac#(p(s(x))),p#(s(x)))
             -->_1 fac#(s(x)) -> c_1(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_1(fac#(p(s(x))))
*** Step 1.b:5.a:2: Ara.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_1(fac#(p(s(x))))
        - Weak TRS:
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {*/2,0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {*,0,s}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(13, 13)
          F (TrsFun "0") :: [] -(0)-> "A"(15, 15)
          F (TrsFun "p") :: ["A"(0, 13)] -(1)-> "A"(13, 13)
          F (TrsFun "s") :: ["A"(13, 13)] -(13)-> "A"(13, 13)
          F (TrsFun "s") :: ["A"(13, 13)] -(0)-> "A"(0, 13)
          F (DpFun "fac") :: ["A"(13, 13)] -(7)-> "A"(0, 14)
          F (ComFun 1) :: ["A"(0, 14)] -(0)-> "A"(14, 14)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
          "F (ComFun 1)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0)
          "F (ComFun 1)_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1)
          "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0)
          "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1)
          "F (TrsFun \"s\")_A" :: ["A"(0, 0)] -(1)-> "A"(1, 0)
          "F (TrsFun \"s\")_A" :: ["A"(1, 1)] -(0)-> "A"(0, 1)
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          fac#(s(x)) -> c_1(fac#(p(s(x))))
        2. Weak:
          
        

*** Step 1.b:5.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            p#(s(s(x))) -> c_3(p#(s(x)))
        - Weak DPs:
            fac#(s(x)) -> fac#(p(s(x)))
            fac#(s(x)) -> p#(s(x))
        - Weak TRS:
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {*/2,0/0,s/1,c_1/2,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {*,0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {p,fac#,p#}
        TcT has computed the following interpretation:
             p(*) = [2]          
             p(0) = [15]         
           p(fac) = [1] x1 + [1] 
             p(p) = [1] x1 + [0] 
             p(s) = [1] x1 + [4] 
          p(fac#) = [3] x1 + [12]
            p(p#) = [2] x1 + [8] 
           p(c_1) = [4] x2 + [2] 
           p(c_2) = [1]          
           p(c_3) = [1] x1 + [6] 
        
        Following rules are strictly oriented:
        p#(s(s(x))) = [2] x + [24] 
                    > [2] x + [22] 
                    = c_3(p#(s(x)))
        
        
        Following rules are (at-least) weakly oriented:
        fac#(s(x)) =  [3] x + [24] 
                   >= [3] x + [24] 
                   =  fac#(p(s(x)))
        
        fac#(s(x)) =  [3] x + [24] 
                   >= [2] x + [16] 
                   =  p#(s(x))     
        
         p(s(0())) =  [19]         
                   >= [15]         
                   =  0()          
        
        p(s(s(x))) =  [1] x + [8]  
                   >= [1] x + [8]  
                   =  s(p(s(x)))   
        
*** Step 1.b:5.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            fac#(s(x)) -> fac#(p(s(x)))
            fac#(s(x)) -> p#(s(x))
            p#(s(s(x))) -> c_3(p#(s(x)))
        - Weak TRS:
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {fac/1,p/1,fac#/1,p#/1} / {*/2,0/0,s/1,c_1/2,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {*,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^3))