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

** Step 1.b:1: DependencyPairs.  MAYBE
    + Considered Problem:
        - Strict TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,+,fact,p} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          *#(0(),y) -> c_1()
          *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
          +#(x,0()) -> c_3()
          +#(x,s(y)) -> c_4(+#(x,y))
          fact#(0()) -> c_5()
          fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x)))
          p#(s(x)) -> c_7()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  MAYBE
    + Considered Problem:
        - Strict DPs:
            *#(0(),y) -> c_1()
            *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
            +#(x,0()) -> c_3()
            +#(x,s(y)) -> c_4(+#(x,y))
            fact#(0()) -> c_5()
            fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x)))
            p#(s(x)) -> c_7()
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/3,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,5,7}
        by application of
          Pre({1,3,5,7}) = {2,4,6}.
        Here rules are labelled as follows:
          1: *#(0(),y) -> c_1()
          2: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
          3: +#(x,0()) -> c_3()
          4: +#(x,s(y)) -> c_4(+#(x,y))
          5: fact#(0()) -> c_5()
          6: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x)))
          7: p#(s(x)) -> c_7()
** Step 1.b:3: RemoveWeakSuffixes.  MAYBE
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
            +#(x,s(y)) -> c_4(+#(x,y))
            fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x)))
        - Weak DPs:
            *#(0(),y) -> c_1()
            +#(x,0()) -> c_3()
            fact#(0()) -> c_5()
            p#(s(x)) -> c_7()
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/3,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:*#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
             -->_1 +#(x,s(y)) -> c_4(+#(x,y)):2
             -->_1 +#(x,0()) -> c_3():5
             -->_2 *#(0(),y) -> c_1():4
             -->_2 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1
          
          2:S:+#(x,s(y)) -> c_4(+#(x,y))
             -->_1 +#(x,0()) -> c_3():5
             -->_1 +#(x,s(y)) -> c_4(+#(x,y)):2
          
          3:S:fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x)))
             -->_3 p#(s(x)) -> c_7():7
             -->_2 fact#(0()) -> c_5():6
             -->_2 fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))):3
             -->_1 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1
          
          4:W:*#(0(),y) -> c_1()
             
          
          5:W:+#(x,0()) -> c_3()
             
          
          6:W:fact#(0()) -> c_5()
             
          
          7:W:p#(s(x)) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: fact#(0()) -> c_5()
          7: p#(s(x)) -> c_7()
          4: *#(0(),y) -> c_1()
          5: +#(x,0()) -> c_3()
** Step 1.b:4: SimplifyRHS.  MAYBE
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
            +#(x,s(y)) -> c_4(+#(x,y))
            fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x)))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/3,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:*#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
             -->_1 +#(x,s(y)) -> c_4(+#(x,y)):2
             -->_2 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1
          
          2:S:+#(x,s(y)) -> c_4(+#(x,y))
             -->_1 +#(x,s(y)) -> c_4(+#(x,y)):2
          
          3:S:fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x)))
             -->_2 fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))):3
             -->_1 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))))
** Step 1.b:5: DecomposeDG.  MAYBE
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
            +#(x,s(y)) -> c_4(+#(x,y))
            fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,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
          fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))))
        and a lower component
          *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
          +#(x,s(y)) -> c_4(+#(x,y))
        Further, following extension rules are added to the lower component.
          fact#(s(x)) -> *#(s(x),fact(p(s(x))))
          fact#(s(x)) -> fact#(p(s(x)))
*** Step 1.b:5.a:1: SimplifyRHS.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))))
             -->_2 fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fact#(s(x)) -> c_6(fact#(p(s(x))))
*** Step 1.b:5.a:2: UsableRules.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            fact#(s(x)) -> c_6(fact#(p(s(x))))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          p(s(x)) -> x
          fact#(s(x)) -> c_6(fact#(p(s(x))))
*** Step 1.b:5.a:3: Ara.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            fact#(s(x)) -> c_6(fact#(p(s(x))))
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,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 "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 "fact") :: ["A"(4, 2)] -(0)-> "A"(0, 0)
          F (ComFun 6) :: ["A"(0, 0)] -(0)-> "A"(0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          fact#(s(x)) -> c_6(fact#(p(s(x))))
        2. Weak:
          
        

*** Step 1.b:5.b:1: DecomposeDG.  MAYBE
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
            +#(x,s(y)) -> c_4(+#(x,y))
        - Weak DPs:
            fact#(s(x)) -> *#(s(x),fact(p(s(x))))
            fact#(s(x)) -> fact#(p(s(x)))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,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
          *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
          fact#(s(x)) -> *#(s(x),fact(p(s(x))))
          fact#(s(x)) -> fact#(p(s(x)))
        and a lower component
          +#(x,s(y)) -> c_4(+#(x,y))
        Further, following extension rules are added to the lower component.
          *#(s(x),y) -> *#(x,y)
          *#(s(x),y) -> +#(*(x,y),y)
          fact#(s(x)) -> *#(s(x),fact(p(s(x))))
          fact#(s(x)) -> fact#(p(s(x)))
**** Step 1.b:5.b:1.a:1: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
        - Weak DPs:
            fact#(s(x)) -> *#(s(x),fact(p(s(x))))
            fact#(s(x)) -> fact#(p(s(x)))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:*#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y))
             -->_2 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1
          
          2:W:fact#(s(x)) -> *#(s(x),fact(p(s(x))))
             -->_1 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1
          
          3:W:fact#(s(x)) -> fact#(p(s(x)))
             -->_1 fact#(s(x)) -> fact#(p(s(x))):3
             -->_1 fact#(s(x)) -> *#(s(x),fact(p(s(x)))):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          *#(s(x),y) -> c_2(*#(x,y))
**** Step 1.b:5.b:1.a:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(*#(x,y))
        - Weak DPs:
            fact#(s(x)) -> *#(s(x),fact(p(s(x))))
            fact#(s(x)) -> fact#(p(s(x)))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,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_2) = {1}
        
        Following symbols are considered usable:
          {p,*#,+#,fact#,p#}
        TcT has computed the following interpretation:
              p(*) = [4] x1 + [4]
              p(+) = [5]         
              p(0) = [4]         
           p(fact) = [1]         
              p(p) = [1] x1 + [0]
              p(s) = [1] x1 + [4]
             p(*#) = [4] x1 + [0]
             p(+#) = [1]         
          p(fact#) = [4] x1 + [0]
             p(p#) = [1] x1 + [1]
            p(c_1) = [1]         
            p(c_2) = [1] x1 + [8]
            p(c_3) = [2]         
            p(c_4) = [4] x1 + [1]
            p(c_5) = [0]         
            p(c_6) = [1] x2 + [1]
            p(c_7) = [1]         
        
        Following rules are strictly oriented:
        *#(s(x),y) = [4] x + [16]
                   > [4] x + [8] 
                   = c_2(*#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        fact#(s(x)) =  [4] x + [16]          
                    >= [4] x + [16]          
                    =  *#(s(x),fact(p(s(x))))
        
        fact#(s(x)) =  [4] x + [16]          
                    >= [4] x + [16]          
                    =  fact#(p(s(x)))        
        
            p(s(x)) =  [1] x + [4]           
                    >= [1] x + [0]           
                    =  x                     
        
**** Step 1.b:5.b:1.a:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(s(x),y) -> c_2(*#(x,y))
            fact#(s(x)) -> *#(s(x),fact(p(s(x))))
            fact#(s(x)) -> fact#(p(s(x)))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(*(x,y),y)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fact(0()) -> s(0())
            fact(s(x)) -> *(s(x),fact(p(s(x))))
            p(s(x)) -> x
        - Signature:
            {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:5.b:1.b:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          +#(x,s(y)) -> c_4(+#(x,y))
      - Weak DPs:
          *#(s(x),y) -> *#(x,y)
          *#(s(x),y) -> +#(*(x,y),y)
          fact#(s(x)) -> *#(s(x),fact(p(s(x))))
          fact#(s(x)) -> fact#(p(s(x)))
      - Weak TRS:
          *(0(),y) -> 0()
          *(s(x),y) -> +(*(x,y),y)
          +(x,0()) -> x
          +(x,s(y)) -> s(+(x,y))
          fact(0()) -> s(0())
          fact(s(x)) -> *(s(x),fact(p(s(x))))
          p(s(x)) -> x
      - Signature:
          {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
WORST_CASE(Omega(n^1),?)