MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
            sum(cons(0(),xs)) -> sum(xs)
            sum(cons(s(x),xs)) -> s(sum(cons(x,xs)))
            sum(nil()) -> 0()
            times(x,y) -> sum(generate(x,y))
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge,gen,generate,if,sum,times} and constructors {0,cons
            ,false,nil,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          ge#(x,0()) -> c_1()
          ge#(0(),s(y)) -> c_2()
          ge#(s(x),s(y)) -> c_3(ge#(x,y))
          gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
          generate#(x,y) -> c_5(gen#(x,y,0()))
          if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
          if#(true(),x,y,z) -> c_7()
          sum#(cons(0(),xs)) -> c_8(sum#(xs))
          sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
          sum#(nil()) -> c_10()
          times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(x,0()) -> c_1()
            ge#(0(),s(y)) -> c_2()
            ge#(s(x),s(y)) -> c_3(ge#(x,y))
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
            if#(true(),x,y,z) -> c_7()
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            sum#(nil()) -> c_10()
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
            sum(cons(0(),xs)) -> sum(xs)
            sum(cons(s(x),xs)) -> s(sum(cons(x,xs)))
            sum(nil()) -> 0()
            times(x,y) -> sum(generate(x,y))
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,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)
          gen(x,y,z) -> if(ge(z,x),x,y,z)
          generate(x,y) -> gen(x,y,0())
          if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
          if(true(),x,y,z) -> nil()
          ge#(x,0()) -> c_1()
          ge#(0(),s(y)) -> c_2()
          ge#(s(x),s(y)) -> c_3(ge#(x,y))
          gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
          generate#(x,y) -> c_5(gen#(x,y,0()))
          if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
          if#(true(),x,y,z) -> c_7()
          sum#(cons(0(),xs)) -> c_8(sum#(xs))
          sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
          sum#(nil()) -> c_10()
          times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(x,0()) -> c_1()
            ge#(0(),s(y)) -> c_2()
            ge#(s(x),s(y)) -> c_3(ge#(x,y))
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
            if#(true(),x,y,z) -> c_7()
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            sum#(nil()) -> c_10()
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,7,10}
        by application of
          Pre({1,2,7,10}) = {3,4,8,11}.
        Here rules are labelled as follows:
          1: ge#(x,0()) -> c_1()
          2: ge#(0(),s(y)) -> c_2()
          3: ge#(s(x),s(y)) -> c_3(ge#(x,y))
          4: gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
          5: generate#(x,y) -> c_5(gen#(x,y,0()))
          6: if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
          7: if#(true(),x,y,z) -> c_7()
          8: sum#(cons(0(),xs)) -> c_8(sum#(xs))
          9: sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
          10: sum#(nil()) -> c_10()
          11: times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_3(ge#(x,y))
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak DPs:
            ge#(x,0()) -> c_1()
            ge#(0(),s(y)) -> c_2()
            if#(true(),x,y,z) -> c_7()
            sum#(nil()) -> c_10()
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:ge#(s(x),s(y)) -> c_3(ge#(x,y))
             -->_1 ge#(0(),s(y)) -> c_2():9
             -->_1 ge#(x,0()) -> c_1():8
             -->_1 ge#(s(x),s(y)) -> c_3(ge#(x,y)):1
          
          2:S:gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
             -->_1 if#(false(),x,y,z) -> c_6(gen#(x,y,s(z))):4
             -->_1 if#(true(),x,y,z) -> c_7():10
             -->_2 ge#(0(),s(y)) -> c_2():9
             -->_2 ge#(x,0()) -> c_1():8
             -->_2 ge#(s(x),s(y)) -> c_3(ge#(x,y)):1
          
          3:S:generate#(x,y) -> c_5(gen#(x,y,0()))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x)):2
          
          4:S:if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x)):2
          
          5:S:sum#(cons(0(),xs)) -> c_8(sum#(xs))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):6
             -->_1 sum#(nil()) -> c_10():11
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):5
          
          6:S:sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):6
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):5
          
          7:S:times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
             -->_1 sum#(nil()) -> c_10():11
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):6
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):5
             -->_2 generate#(x,y) -> c_5(gen#(x,y,0())):3
          
          8:W:ge#(x,0()) -> c_1()
             
          
          9:W:ge#(0(),s(y)) -> c_2()
             
          
          10:W:if#(true(),x,y,z) -> c_7()
             
          
          11:W:sum#(nil()) -> c_10()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          11: sum#(nil()) -> c_10()
          10: if#(true(),x,y,z) -> c_7()
          8: ge#(x,0()) -> c_1()
          9: ge#(0(),s(y)) -> c_2()
* Step 5: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_3(ge#(x,y))
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,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_3(ge#(x,y))
          - Weak DPs:
              gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
              generate#(x,y) -> c_5(gen#(x,y,0()))
              if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
              sum#(cons(0(),xs)) -> c_8(sum#(xs))
              sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
              times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
          - Weak TRS:
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
              gen(x,y,z) -> if(ge(z,x),x,y,z)
              generate(x,y) -> gen(x,y,0())
              if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
              if(true(),x,y,z) -> nil()
          - Signature:
              {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
              ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
              ,cons,false,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
              generate#(x,y) -> c_5(gen#(x,y,0()))
              if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
              sum#(cons(0(),xs)) -> c_8(sum#(xs))
              sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
              times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
          - Weak DPs:
              ge#(s(x),s(y)) -> c_3(ge#(x,y))
          - Weak TRS:
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
              gen(x,y,z) -> if(ge(z,x),x,y,z)
              generate(x,y) -> gen(x,y,0())
              if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
              if(true(),x,y,z) -> nil()
          - Signature:
              {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
              ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
              ,cons,false,nil,s,true}
** Step 5.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_3(ge#(x,y))
        - Weak DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:ge#(s(x),s(y)) -> c_3(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_3(ge#(x,y)):1
          
          2:W:gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
             -->_2 ge#(s(x),s(y)) -> c_3(ge#(x,y)):1
             -->_1 if#(false(),x,y,z) -> c_6(gen#(x,y,s(z))):4
          
          3:W:generate#(x,y) -> c_5(gen#(x,y,0()))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x)):2
          
          4:W:if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x)):2
          
          5:W:sum#(cons(0(),xs)) -> c_8(sum#(xs))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):6
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):5
          
          6:W:sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):5
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):6
          
          7:W:times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
             -->_2 generate#(x,y) -> c_5(gen#(x,y,0())):3
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):5
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: sum#(cons(0(),xs)) -> c_8(sum#(xs))
          6: sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
** Step 5.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_3(ge#(x,y))
        - Weak DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:ge#(s(x),s(y)) -> c_3(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_3(ge#(x,y)):1
          
          2:W:gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
             -->_2 ge#(s(x),s(y)) -> c_3(ge#(x,y)):1
             -->_1 if#(false(),x,y,z) -> c_6(gen#(x,y,s(z))):4
          
          3:W:generate#(x,y) -> c_5(gen#(x,y,0()))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x)):2
          
          4:W:if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x)):2
          
          7:W:times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
             -->_2 generate#(x,y) -> c_5(gen#(x,y,0())):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          times#(x,y) -> c_11(generate#(x,y))
** Step 5.a:3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_3(ge#(x,y))
        - Weak DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
            times#(x,y) -> c_11(generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,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)
          ge#(s(x),s(y)) -> c_3(ge#(x,y))
          gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
          generate#(x,y) -> c_5(gen#(x,y,0()))
          if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
          times#(x,y) -> c_11(generate#(x,y))
** Step 5.a:4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          ge#(s(x),s(y)) -> c_3(ge#(x,y))
      - Weak DPs:
          gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
          generate#(x,y) -> c_5(gen#(x,y,0()))
          if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
          times#(x,y) -> c_11(generate#(x,y))
      - Weak TRS:
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
      - Signature:
          {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
          ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
          ,cons,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 5.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak DPs:
            ge#(s(x),s(y)) -> c_3(ge#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
             -->_2 ge#(s(x),s(y)) -> c_3(ge#(x,y)):7
             -->_1 if#(false(),x,y,z) -> c_6(gen#(x,y,s(z))):3
          
          2:S:generate#(x,y) -> c_5(gen#(x,y,0()))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x)):1
          
          3:S:if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x)):1
          
          4:S:sum#(cons(0(),xs)) -> c_8(sum#(xs))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):5
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):4
          
          5:S:sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):5
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):4
          
          6:S:times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):5
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):4
             -->_2 generate#(x,y) -> c_5(gen#(x,y,0())):2
          
          7:W:ge#(s(x),s(y)) -> c_3(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_3(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_3(ge#(x,y))
** Step 5.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x))
             -->_1 if#(false(),x,y,z) -> c_6(gen#(x,y,s(z))):3
          
          2:S:generate#(x,y) -> c_5(gen#(x,y,0()))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x)):1
          
          3:S:if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z),ge#(z,x)):1
          
          4:S:sum#(cons(0(),xs)) -> c_8(sum#(xs))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):5
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):4
          
          5:S:sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):5
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):4
          
          6:S:times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):5
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):4
             -->_2 generate#(x,y) -> c_5(gen#(x,y,0())):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
** Step 5.b:3: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,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:
              gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
              if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
          - Weak DPs:
              generate#(x,y) -> c_5(gen#(x,y,0()))
              sum#(cons(0(),xs)) -> c_8(sum#(xs))
              sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
              times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
          - Weak TRS:
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
              gen(x,y,z) -> if(ge(z,x),x,y,z)
              generate(x,y) -> gen(x,y,0())
              if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
              if(true(),x,y,z) -> nil()
          - Signature:
              {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
              ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
              ,cons,false,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              generate#(x,y) -> c_5(gen#(x,y,0()))
              sum#(cons(0(),xs)) -> c_8(sum#(xs))
              sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
              times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
          - Weak DPs:
              gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
              if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
          - Weak TRS:
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
              gen(x,y,z) -> if(ge(z,x),x,y,z)
              generate(x,y) -> gen(x,y,0())
              if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
              if(true(),x,y,z) -> nil()
          - Signature:
              {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
              ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
              ,cons,false,nil,s,true}
*** Step 5.b:3.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
        - Weak DPs:
            generate#(x,y) -> c_5(gen#(x,y,0()))
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
             -->_1 if#(false(),x,y,z) -> c_6(gen#(x,y,s(z))):3
          
          2:W:generate#(x,y) -> c_5(gen#(x,y,0()))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z)):1
          
          3:S:if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z)):1
          
          4:W:sum#(cons(0(),xs)) -> c_8(sum#(xs))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):5
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):4
          
          5:W:sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):4
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):5
          
          6:W:times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
             -->_2 generate#(x,y) -> c_5(gen#(x,y,0())):2
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):4
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: sum#(cons(0(),xs)) -> c_8(sum#(xs))
          5: sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
*** Step 5.b:3.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
        - Weak DPs:
            generate#(x,y) -> c_5(gen#(x,y,0()))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
             -->_1 if#(false(),x,y,z) -> c_6(gen#(x,y,s(z))):3
          
          2:W:generate#(x,y) -> c_5(gen#(x,y,0()))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z)):1
          
          3:S:if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z)):1
          
          6:W:times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
             -->_2 generate#(x,y) -> c_5(gen#(x,y,0())):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          times#(x,y) -> c_11(generate#(x,y))
*** Step 5.b:3.a:3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
        - Weak DPs:
            generate#(x,y) -> c_5(gen#(x,y,0()))
            times#(x,y) -> c_11(generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,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)
          gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
          generate#(x,y) -> c_5(gen#(x,y,0()))
          if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
          times#(x,y) -> c_11(generate#(x,y))
*** Step 5.b:3.a:4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
          if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
      - Weak DPs:
          generate#(x,y) -> c_5(gen#(x,y,0()))
          times#(x,y) -> c_11(generate#(x,y))
      - Weak TRS:
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
      - Signature:
          {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
          ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
          ,cons,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 5.b:3.b:1: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            generate#(x,y) -> c_5(gen#(x,y,0()))
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {4}.
        Here rules are labelled as follows:
          1: generate#(x,y) -> c_5(gen#(x,y,0()))
          2: sum#(cons(0(),xs)) -> c_8(sum#(xs))
          3: sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
          4: times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
          5: gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
          6: if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
*** Step 5.b:3.b:2: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak DPs:
            gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
            generate#(x,y) -> c_5(gen#(x,y,0()))
            if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sum#(cons(0(),xs)) -> c_8(sum#(xs))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):2
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):1
          
          2:S:sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):2
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):1
          
          3:S:times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
             -->_2 generate#(x,y) -> c_5(gen#(x,y,0())):5
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):2
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):1
          
          4:W:gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
             -->_1 if#(false(),x,y,z) -> c_6(gen#(x,y,s(z))):6
          
          5:W:generate#(x,y) -> c_5(gen#(x,y,0()))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z)):4
          
          6:W:if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
             -->_1 gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: generate#(x,y) -> c_5(gen#(x,y,0()))
          4: gen#(x,y,z) -> c_4(if#(ge(z,x),x,y,z))
          6: if#(false(),x,y,z) -> c_6(gen#(x,y,s(z)))
*** Step 5.b:3.b:3: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sum#(cons(0(),xs)) -> c_8(sum#(xs))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):2
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):1
          
          2:S:sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):2
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):1
          
          3:S:times#(x,y) -> c_11(sum#(generate(x,y)),generate#(x,y))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):2
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          times#(x,y) -> c_11(sum#(generate(x,y)))
*** Step 5.b:3.b:4: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,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:
              sum#(cons(0(),xs)) -> c_8(sum#(xs))
              sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
          - Weak DPs:
              times#(x,y) -> c_11(sum#(generate(x,y)))
          - Weak TRS:
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
              gen(x,y,z) -> if(ge(z,x),x,y,z)
              generate(x,y) -> gen(x,y,0())
              if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
              if(true(),x,y,z) -> nil()
          - Signature:
              {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
              ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
              ,cons,false,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              times#(x,y) -> c_11(sum#(generate(x,y)))
          - Weak DPs:
              sum#(cons(0(),xs)) -> c_8(sum#(xs))
              sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
          - Weak TRS:
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
              gen(x,y,z) -> if(ge(z,x),x,y,z)
              generate(x,y) -> gen(x,y,0())
              if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
              if(true(),x,y,z) -> nil()
          - Signature:
              {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
              ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
              ,cons,false,nil,s,true}
**** Step 5.b:3.b:4.a:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          sum#(cons(0(),xs)) -> c_8(sum#(xs))
          sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
      - Weak DPs:
          times#(x,y) -> c_11(sum#(generate(x,y)))
      - Weak TRS:
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          gen(x,y,z) -> if(ge(z,x),x,y,z)
          generate(x,y) -> gen(x,y,0())
          if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
          if(true(),x,y,z) -> nil()
      - Signature:
          {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
          ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
          ,cons,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
**** Step 5.b:3.b:4.b:1: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            times#(x,y) -> c_11(sum#(generate(x,y)))
        - Weak DPs:
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: times#(x,y) -> c_11(sum#(generate(x,y)))
          2: sum#(cons(0(),xs)) -> c_8(sum#(xs))
          3: sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
**** Step 5.b:3.b:4.b:2: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(cons(0(),xs)) -> c_8(sum#(xs))
            sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
            times#(x,y) -> c_11(sum#(generate(x,y)))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sum#(cons(0(),xs)) -> c_8(sum#(xs))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):2
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):1
          
          2:W:sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):2
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):1
          
          3:W:times#(x,y) -> c_11(sum#(generate(x,y)))
             -->_1 sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs))):2
             -->_1 sum#(cons(0(),xs)) -> c_8(sum#(xs)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: times#(x,y) -> c_11(sum#(generate(x,y)))
          1: sum#(cons(0(),xs)) -> c_8(sum#(xs))
          2: sum#(cons(s(x),xs)) -> c_9(sum#(cons(x,xs)))
**** Step 5.b:3.b:4.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gen(x,y,z) -> if(ge(z,x),x,y,z)
            generate(x,y) -> gen(x,y,0())
            if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
            if(true(),x,y,z) -> nil()
        - Signature:
            {ge/2,gen/3,generate/2,if/4,sum/1,times/2,ge#/2,gen#/3,generate#/2,if#/4,sum#/1,times#/2} / {0/0,cons/2
            ,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ge#,gen#,generate#,if#,sum#,times#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

MAYBE