MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            a() -> b()
            a() -> c()
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifProd(false(),xs,x) -> prodIter(tail(xs),times(x,head(xs)))
            ifProd(true(),xs,x) -> x
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            prod(xs) -> prodIter(xs,s(0()))
            prodIter(xs,x) -> ifProd(isempty(xs),xs,x)
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,ge,head,ifProd,ifTimes,isempty,plus,prod,prodIter,tail
            ,times,timesIter} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a#() -> c_1()
          a#() -> c_2()
          ge#(x,0()) -> c_3()
          ge#(0(),s(y)) -> c_4()
          ge#(s(x),s(y)) -> c_5(ge#(x,y))
          head#(cons(x,xs)) -> c_6()
          head#(nil()) -> c_7()
          ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),tail#(xs),times#(x,head(xs)),head#(xs))
          ifProd#(true(),xs,x) -> c_9()
          ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
          ifTimes#(true(),x,y,z,u) -> c_11()
          isempty#(cons(x,xs)) -> c_12()
          isempty#(nil()) -> c_13()
          plus#(0(),y) -> c_14()
          plus#(s(x),y) -> c_15(plus#(x,y))
          prod#(xs) -> c_16(prodIter#(xs,s(0())))
          prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs))
          tail#(cons(x,xs)) -> c_18()
          tail#(nil()) -> c_19()
          times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
          timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            a#() -> c_1()
            a#() -> c_2()
            ge#(x,0()) -> c_3()
            ge#(0(),s(y)) -> c_4()
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            head#(cons(x,xs)) -> c_6()
            head#(nil()) -> c_7()
            ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),tail#(xs),times#(x,head(xs)),head#(xs))
            ifProd#(true(),xs,x) -> c_9()
            ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
            ifTimes#(true(),x,y,z,u) -> c_11()
            isempty#(cons(x,xs)) -> c_12()
            isempty#(nil()) -> c_13()
            plus#(0(),y) -> c_14()
            plus#(s(x),y) -> c_15(plus#(x,y))
            prod#(xs) -> c_16(prodIter#(xs,s(0())))
            prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs))
            tail#(cons(x,xs)) -> c_18()
            tail#(nil()) -> c_19()
            times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
            timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        - Weak TRS:
            a() -> b()
            a() -> c()
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifProd(false(),xs,x) -> prodIter(tail(xs),times(x,head(xs)))
            ifProd(true(),xs,x) -> x
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            prod(xs) -> prodIter(xs,s(0()))
            prodIter(xs,x) -> ifProd(isempty(xs),xs,x)
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
            ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/4,c_9/0
            ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/2,c_18/0,c_19/0,c_20/1,c_21/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
            ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          head(cons(x,xs)) -> x
          head(nil()) -> error()
          ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
          ifTimes(true(),x,y,z,u) -> z
          isempty(cons(x,xs)) -> false()
          isempty(nil()) -> true()
          plus(0(),y) -> y
          plus(s(x),y) -> s(plus(x,y))
          tail(cons(x,xs)) -> xs
          tail(nil()) -> nil()
          times(x,y) -> timesIter(x,y,0(),0())
          timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
          a#() -> c_1()
          a#() -> c_2()
          ge#(x,0()) -> c_3()
          ge#(0(),s(y)) -> c_4()
          ge#(s(x),s(y)) -> c_5(ge#(x,y))
          head#(cons(x,xs)) -> c_6()
          head#(nil()) -> c_7()
          ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),tail#(xs),times#(x,head(xs)),head#(xs))
          ifProd#(true(),xs,x) -> c_9()
          ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
          ifTimes#(true(),x,y,z,u) -> c_11()
          isempty#(cons(x,xs)) -> c_12()
          isempty#(nil()) -> c_13()
          plus#(0(),y) -> c_14()
          plus#(s(x),y) -> c_15(plus#(x,y))
          prod#(xs) -> c_16(prodIter#(xs,s(0())))
          prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs))
          tail#(cons(x,xs)) -> c_18()
          tail#(nil()) -> c_19()
          times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
          timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            a#() -> c_1()
            a#() -> c_2()
            ge#(x,0()) -> c_3()
            ge#(0(),s(y)) -> c_4()
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            head#(cons(x,xs)) -> c_6()
            head#(nil()) -> c_7()
            ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),tail#(xs),times#(x,head(xs)),head#(xs))
            ifProd#(true(),xs,x) -> c_9()
            ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
            ifTimes#(true(),x,y,z,u) -> c_11()
            isempty#(cons(x,xs)) -> c_12()
            isempty#(nil()) -> c_13()
            plus#(0(),y) -> c_14()
            plus#(s(x),y) -> c_15(plus#(x,y))
            prod#(xs) -> c_16(prodIter#(xs,s(0())))
            prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs))
            tail#(cons(x,xs)) -> c_18()
            tail#(nil()) -> c_19()
            times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
            timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
            ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/4,c_9/0
            ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/2,c_18/0,c_19/0,c_20/1,c_21/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
            ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,4,6,7,9,11,12,13,14,18,19}
        by application of
          Pre({1,2,3,4,6,7,9,11,12,13,14,18,19}) = {5,8,10,15,17,21}.
        Here rules are labelled as follows:
          1: a#() -> c_1()
          2: a#() -> c_2()
          3: ge#(x,0()) -> c_3()
          4: ge#(0(),s(y)) -> c_4()
          5: ge#(s(x),s(y)) -> c_5(ge#(x,y))
          6: head#(cons(x,xs)) -> c_6()
          7: head#(nil()) -> c_7()
          8: ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs)))
                                         ,tail#(xs)
                                         ,times#(x,head(xs))
                                         ,head#(xs))
          9: ifProd#(true(),xs,x) -> c_9()
          10: ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
          11: ifTimes#(true(),x,y,z,u) -> c_11()
          12: isempty#(cons(x,xs)) -> c_12()
          13: isempty#(nil()) -> c_13()
          14: plus#(0(),y) -> c_14()
          15: plus#(s(x),y) -> c_15(plus#(x,y))
          16: prod#(xs) -> c_16(prodIter#(xs,s(0())))
          17: prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs))
          18: tail#(cons(x,xs)) -> c_18()
          19: tail#(nil()) -> c_19()
          20: times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
          21: timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),tail#(xs),times#(x,head(xs)),head#(xs))
            ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
            plus#(s(x),y) -> c_15(plus#(x,y))
            prod#(xs) -> c_16(prodIter#(xs,s(0())))
            prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs))
            times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
            timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        - Weak DPs:
            a#() -> c_1()
            a#() -> c_2()
            ge#(x,0()) -> c_3()
            ge#(0(),s(y)) -> c_4()
            head#(cons(x,xs)) -> c_6()
            head#(nil()) -> c_7()
            ifProd#(true(),xs,x) -> c_9()
            ifTimes#(true(),x,y,z,u) -> c_11()
            isempty#(cons(x,xs)) -> c_12()
            isempty#(nil()) -> c_13()
            plus#(0(),y) -> c_14()
            tail#(cons(x,xs)) -> c_18()
            tail#(nil()) -> c_19()
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
            ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/4,c_9/0
            ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/2,c_18/0,c_19/0,c_20/1,c_21/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
            ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:ge#(s(x),s(y)) -> c_5(ge#(x,y))
             -->_1 ge#(0(),s(y)) -> c_4():12
             -->_1 ge#(x,0()) -> c_3():11
             -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
          
          2:S:ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs)))
                                          ,tail#(xs)
                                          ,times#(x,head(xs))
                                          ,head#(xs))
             -->_3 times#(x,y) -> c_20(timesIter#(x,y,0(),0())):7
             -->_1 prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs)):6
             -->_2 tail#(nil()) -> c_19():21
             -->_2 tail#(cons(x,xs)) -> c_18():20
             -->_4 head#(nil()) -> c_7():14
             -->_4 head#(cons(x,xs)) -> c_6():13
          
          3:S:ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):8
             -->_2 plus#(s(x),y) -> c_15(plus#(x,y)):4
             -->_2 plus#(0(),y) -> c_14():19
          
          4:S:plus#(s(x),y) -> c_15(plus#(x,y))
             -->_1 plus#(0(),y) -> c_14():19
             -->_1 plus#(s(x),y) -> c_15(plus#(x,y)):4
          
          5:S:prod#(xs) -> c_16(prodIter#(xs,s(0())))
             -->_1 prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs)):6
          
          6:S:prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs))
             -->_2 isempty#(nil()) -> c_13():18
             -->_2 isempty#(cons(x,xs)) -> c_12():17
             -->_1 ifProd#(true(),xs,x) -> c_9():15
             -->_1 ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs)))
                                               ,tail#(xs)
                                               ,times#(x,head(xs))
                                               ,head#(xs)):2
          
          7:S:times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):8
          
          8:S:timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
             -->_1 ifTimes#(true(),x,y,z,u) -> c_11():16
             -->_2 ge#(0(),s(y)) -> c_4():12
             -->_2 ge#(x,0()) -> c_3():11
             -->_1 ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z)):3
             -->_2 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
          
          9:W:a#() -> c_1()
             
          
          10:W:a#() -> c_2()
             
          
          11:W:ge#(x,0()) -> c_3()
             
          
          12:W:ge#(0(),s(y)) -> c_4()
             
          
          13:W:head#(cons(x,xs)) -> c_6()
             
          
          14:W:head#(nil()) -> c_7()
             
          
          15:W:ifProd#(true(),xs,x) -> c_9()
             
          
          16:W:ifTimes#(true(),x,y,z,u) -> c_11()
             
          
          17:W:isempty#(cons(x,xs)) -> c_12()
             
          
          18:W:isempty#(nil()) -> c_13()
             
          
          19:W:plus#(0(),y) -> c_14()
             
          
          20:W:tail#(cons(x,xs)) -> c_18()
             
          
          21:W:tail#(nil()) -> c_19()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          10: a#() -> c_2()
          9: a#() -> c_1()
          13: head#(cons(x,xs)) -> c_6()
          14: head#(nil()) -> c_7()
          20: tail#(cons(x,xs)) -> c_18()
          21: tail#(nil()) -> c_19()
          15: ifProd#(true(),xs,x) -> c_9()
          17: isempty#(cons(x,xs)) -> c_12()
          18: isempty#(nil()) -> c_13()
          19: plus#(0(),y) -> c_14()
          16: ifTimes#(true(),x,y,z,u) -> c_11()
          11: ge#(x,0()) -> c_3()
          12: ge#(0(),s(y)) -> c_4()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),tail#(xs),times#(x,head(xs)),head#(xs))
            ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
            plus#(s(x),y) -> c_15(plus#(x,y))
            prod#(xs) -> c_16(prodIter#(xs,s(0())))
            prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs))
            times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
            timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
            ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/4,c_9/0
            ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/2,c_18/0,c_19/0,c_20/1,c_21/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
            ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:ge#(s(x),s(y)) -> c_5(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
          
          2:S:ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs)))
                                          ,tail#(xs)
                                          ,times#(x,head(xs))
                                          ,head#(xs))
             -->_3 times#(x,y) -> c_20(timesIter#(x,y,0(),0())):7
             -->_1 prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs)):6
          
          3:S:ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):8
             -->_2 plus#(s(x),y) -> c_15(plus#(x,y)):4
          
          4:S:plus#(s(x),y) -> c_15(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_15(plus#(x,y)):4
          
          5:S:prod#(xs) -> c_16(prodIter#(xs,s(0())))
             -->_1 prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs)):6
          
          6:S:prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x),isempty#(xs))
             -->_1 ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs)))
                                               ,tail#(xs)
                                               ,times#(x,head(xs))
                                               ,head#(xs)):2
          
          7:S:times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):8
          
          8:S:timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
             -->_1 ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z)):3
             -->_2 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
          prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
* Step 6: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
            ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
            plus#(s(x),y) -> c_15(plus#(x,y))
            prod#(xs) -> c_16(prodIter#(xs,s(0())))
            prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
            times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
            timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
            ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0
            ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
            ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:ge#(s(x),s(y)) -> c_5(ge#(x,y))
           -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
        
        2:S:ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
           -->_2 times#(x,y) -> c_20(timesIter#(x,y,0(),0())):7
           -->_1 prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x)):6
        
        3:S:ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
           -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):8
           -->_2 plus#(s(x),y) -> c_15(plus#(x,y)):4
        
        4:S:plus#(s(x),y) -> c_15(plus#(x,y))
           -->_1 plus#(s(x),y) -> c_15(plus#(x,y)):4
        
        5:S:prod#(xs) -> c_16(prodIter#(xs,s(0())))
           -->_1 prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x)):6
        
        6:S:prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
           -->_1 ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs))):2
        
        7:S:times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
           -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):8
        
        8:S:timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
           -->_1 ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z)):3
           -->_2 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(5,prod#(xs) -> c_16(prodIter#(xs,s(0()))))]
* Step 7: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
            ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
            plus#(s(x),y) -> c_15(plus#(x,y))
            prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
            times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
            timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
            ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0
            ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
            ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              ge#(s(x),s(y)) -> c_5(ge#(x,y))
          - Weak DPs:
              ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
              ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
              plus#(s(x),y) -> c_15(plus#(x,y))
              prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
              times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
              timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
          - Weak TRS:
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
              head(cons(x,xs)) -> x
              head(nil()) -> error()
              ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
              ifTimes(true(),x,y,z,u) -> z
              isempty(cons(x,xs)) -> false()
              isempty(nil()) -> true()
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
              tail(cons(x,xs)) -> xs
              tail(nil()) -> nil()
              times(x,y) -> timesIter(x,y,0(),0())
              timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
          - Signature:
              {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0
              ,ge#/2,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2
              ,timesIter#/4} / {0/0,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0
              ,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
              ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
              ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
              plus#(s(x),y) -> c_15(plus#(x,y))
              prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
              times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
              timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
          - Weak DPs:
              ge#(s(x),s(y)) -> c_5(ge#(x,y))
          - Weak TRS:
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
              head(cons(x,xs)) -> x
              head(nil()) -> error()
              ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
              ifTimes(true(),x,y,z,u) -> z
              isempty(cons(x,xs)) -> false()
              isempty(nil()) -> true()
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
              tail(cons(x,xs)) -> xs
              tail(nil()) -> nil()
              times(x,y) -> timesIter(x,y,0(),0())
              timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
          - Signature:
              {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0
              ,ge#/2,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2
              ,timesIter#/4} / {0/0,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0
              ,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
              ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
** Step 7.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
        - Weak DPs:
            ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
            ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
            plus#(s(x),y) -> c_15(plus#(x,y))
            prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
            times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
            timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
            ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0
            ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
            ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:ge#(s(x),s(y)) -> c_5(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
          
          2:W:ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
             -->_2 times#(x,y) -> c_20(timesIter#(x,y,0(),0())):7
             -->_1 prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x)):6
          
          3:W:ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):8
             -->_2 plus#(s(x),y) -> c_15(plus#(x,y)):4
          
          4:W:plus#(s(x),y) -> c_15(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_15(plus#(x,y)):4
          
          6:W:prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
             -->_1 ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs))):2
          
          7:W:times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):8
          
          8:W:timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
             -->_2 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
             -->_1 ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: plus#(s(x),y) -> c_15(plus#(x,y))
** Step 7.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
        - Weak DPs:
            ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
            ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
            prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
            times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
            timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
            ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0
            ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
            ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:ge#(s(x),s(y)) -> c_5(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
          
          2:W:ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
             -->_2 times#(x,y) -> c_20(timesIter#(x,y,0(),0())):7
             -->_1 prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x)):6
          
          3:W:ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):8
          
          6:W:prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
             -->_1 ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs))):2
          
          7:W:times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):8
          
          8:W:timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
             -->_2 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
             -->_1 ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)))
** Step 7.a:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          ge#(s(x),s(y)) -> c_5(ge#(x,y))
      - Weak DPs:
          ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
          ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)))
          prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
          times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
          timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
      - Weak TRS:
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          head(cons(x,xs)) -> x
          head(nil()) -> error()
          ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
          ifTimes(true(),x,y,z,u) -> z
          isempty(cons(x,xs)) -> false()
          isempty(nil()) -> true()
          plus(0(),y) -> y
          plus(s(x),y) -> s(plus(x,y))
          tail(cons(x,xs)) -> xs
          tail(nil()) -> nil()
          times(x,y) -> timesIter(x,y,0(),0())
          timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
      - Signature:
          {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
          ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
          ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0
          ,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
          ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 7.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
            ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
            plus#(s(x),y) -> c_15(plus#(x,y))
            prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
            times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
            timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        - Weak DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
            ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0
            ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
            ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
             -->_2 times#(x,y) -> c_20(timesIter#(x,y,0(),0())):5
             -->_1 prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x)):4
          
          2:S:ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):6
             -->_2 plus#(s(x),y) -> c_15(plus#(x,y)):3
          
          3:S:plus#(s(x),y) -> c_15(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_15(plus#(x,y)):3
          
          4:S:prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
             -->_1 ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs))):1
          
          5:S:times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):6
          
          6:S:timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
             -->_2 ge#(s(x),s(y)) -> c_5(ge#(x,y)):7
             -->_1 ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z)):2
          
          7:W:ge#(s(x),s(y)) -> c_5(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):7
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: ge#(s(x),s(y)) -> c_5(ge#(x,y))
** Step 7.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
            ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
            plus#(s(x),y) -> c_15(plus#(x,y))
            prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
            times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
            timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            head(cons(x,xs)) -> x
            head(nil()) -> error()
            ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
            ifTimes(true(),x,y,z,u) -> z
            isempty(cons(x,xs)) -> false()
            isempty(nil()) -> true()
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
            times(x,y) -> timesIter(x,y,0(),0())
            timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
        - Signature:
            {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
            ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
            ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0
            ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
            ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
             -->_2 times#(x,y) -> c_20(timesIter#(x,y,0(),0())):5
             -->_1 prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x)):4
          
          2:S:ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):6
             -->_2 plus#(s(x),y) -> c_15(plus#(x,y)):3
          
          3:S:plus#(s(x),y) -> c_15(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_15(plus#(x,y)):3
          
          4:S:prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
             -->_1 ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs))):1
          
          5:S:times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
             -->_1 timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x)):6
          
          6:S:timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u),ge#(u,x))
             -->_1 ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u))
** Step 7.b:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          ifProd#(false(),xs,x) -> c_8(prodIter#(tail(xs),times(x,head(xs))),times#(x,head(xs)))
          ifTimes#(false(),x,y,z,u) -> c_10(timesIter#(x,y,plus(y,z),s(u)),plus#(y,z))
          plus#(s(x),y) -> c_15(plus#(x,y))
          prodIter#(xs,x) -> c_17(ifProd#(isempty(xs),xs,x))
          times#(x,y) -> c_20(timesIter#(x,y,0(),0()))
          timesIter#(x,y,z,u) -> c_21(ifTimes#(ge(u,x),x,y,z,u))
      - Weak TRS:
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          head(cons(x,xs)) -> x
          head(nil()) -> error()
          ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u))
          ifTimes(true(),x,y,z,u) -> z
          isempty(cons(x,xs)) -> false()
          isempty(nil()) -> true()
          plus(0(),y) -> y
          plus(s(x),y) -> s(plus(x,y))
          tail(cons(x,xs)) -> xs
          tail(nil()) -> nil()
          times(x,y) -> timesIter(x,y,0(),0())
          timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u)
      - Signature:
          {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4,a#/0,ge#/2
          ,head#/1,ifProd#/3,ifTimes#/5,isempty#/1,plus#/2,prod#/1,prodIter#/2,tail#/1,times#/2,timesIter#/4} / {0/0
          ,b/0,c/0,cons/2,error/0,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0
          ,c_10/2,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {a#,ge#,head#,ifProd#,ifTimes#,isempty#,plus#,prod#
          ,prodIter#,tail#,times#,timesIter#} and constructors {0,b,c,cons,error,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE