MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            help(c,l,cons(x,y),z) -> if(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
            if(x,false(),z,c,l) -> help(s(c),l,x,z)
            if(x,true(),z,c,l) -> z
            length(cons(x,y)) -> s(length(y))
            length(nil()) -> 0()
            rev(x) -> if(x,eq(0(),length(x)),nil(),0(),length(x))
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1} / {0/0,cons/2,eq/2,false/0,nil/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,ge,help,if,length,rev} and constructors {0,cons,eq
            ,false,nil,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          append#(cons(x,y),z) -> c_1(append#(y,z))
          append#(nil(),y) -> c_2()
          ge#(x,0()) -> c_3()
          ge#(0(),s(y)) -> c_4()
          ge#(s(x),s(y)) -> c_5(ge#(x,y))
          help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                       ,append#(y,cons(x,nil()))
                                       ,ge#(c,l))
          if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
          if#(x,true(),z,c,l) -> c_8()
          length#(cons(x,y)) -> c_9(length#(y))
          length#(nil()) -> c_10()
          rev#(x) -> c_11(if#(x,eq(0(),length(x)),nil(),0(),length(x)),length#(x),length#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,y),z) -> c_1(append#(y,z))
            append#(nil(),y) -> c_2()
            ge#(x,0()) -> c_3()
            ge#(0(),s(y)) -> c_4()
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            if#(x,true(),z,c,l) -> c_8()
            length#(cons(x,y)) -> c_9(length#(y))
            length#(nil()) -> c_10()
            rev#(x) -> c_11(if#(x,eq(0(),length(x)),nil(),0(),length(x)),length#(x),length#(x))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            help(c,l,cons(x,y),z) -> if(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
            if(x,false(),z,c,l) -> help(s(c),l,x,z)
            if(x,true(),z,c,l) -> z
            length(cons(x,y)) -> s(length(y))
            length(nil()) -> 0()
            rev(x) -> if(x,eq(0(),length(x)),nil(),0(),length(x))
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          append(cons(x,y),z) -> cons(x,append(y,z))
          append(nil(),y) -> y
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          length(cons(x,y)) -> s(length(y))
          length(nil()) -> 0()
          append#(cons(x,y),z) -> c_1(append#(y,z))
          append#(nil(),y) -> c_2()
          ge#(x,0()) -> c_3()
          ge#(0(),s(y)) -> c_4()
          ge#(s(x),s(y)) -> c_5(ge#(x,y))
          help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                       ,append#(y,cons(x,nil()))
                                       ,ge#(c,l))
          if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
          if#(x,true(),z,c,l) -> c_8()
          length#(cons(x,y)) -> c_9(length#(y))
          length#(nil()) -> c_10()
          rev#(x) -> c_11(if#(x,eq(0(),length(x)),nil(),0(),length(x)),length#(x),length#(x))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,y),z) -> c_1(append#(y,z))
            append#(nil(),y) -> c_2()
            ge#(x,0()) -> c_3()
            ge#(0(),s(y)) -> c_4()
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            if#(x,true(),z,c,l) -> c_8()
            length#(cons(x,y)) -> c_9(length#(y))
            length#(nil()) -> c_10()
            rev#(x) -> c_11(if#(x,eq(0(),length(x)),nil(),0(),length(x)),length#(x),length#(x))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            length(cons(x,y)) -> s(length(y))
            length(nil()) -> 0()
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,4,8,10}
        by application of
          Pre({2,3,4,8,10}) = {1,5,6,9,11}.
        Here rules are labelled as follows:
          1: append#(cons(x,y),z) -> c_1(append#(y,z))
          2: append#(nil(),y) -> 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: help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                          ,append#(y,cons(x,nil()))
                                          ,ge#(c,l))
          7: if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
          8: if#(x,true(),z,c,l) -> c_8()
          9: length#(cons(x,y)) -> c_9(length#(y))
          10: length#(nil()) -> c_10()
          11: rev#(x) -> c_11(if#(x,eq(0(),length(x)),nil(),0(),length(x)),length#(x),length#(x))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,y),z) -> c_1(append#(y,z))
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
            rev#(x) -> c_11(if#(x,eq(0(),length(x)),nil(),0(),length(x)),length#(x),length#(x))
        - Weak DPs:
            append#(nil(),y) -> c_2()
            ge#(x,0()) -> c_3()
            ge#(0(),s(y)) -> c_4()
            if#(x,true(),z,c,l) -> c_8()
            length#(nil()) -> c_10()
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            length(cons(x,y)) -> s(length(y))
            length(nil()) -> 0()
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:append#(cons(x,y),z) -> c_1(append#(y,z))
             -->_1 append#(nil(),y) -> c_2():7
             -->_1 append#(cons(x,y),z) -> c_1(append#(y,z)):1
          
          2:S:ge#(s(x),s(y)) -> c_5(ge#(x,y))
             -->_1 ge#(0(),s(y)) -> c_4():9
             -->_1 ge#(x,0()) -> c_3():8
             -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):2
          
          3:S:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                           ,append#(y,cons(x,nil()))
                                           ,ge#(c,l))
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):4
             -->_1 if#(x,true(),z,c,l) -> c_8():10
             -->_3 ge#(0(),s(y)) -> c_4():9
             -->_3 ge#(x,0()) -> c_3():8
             -->_2 append#(nil(),y) -> c_2():7
             -->_3 ge#(s(x),s(y)) -> c_5(ge#(x,y)):2
             -->_2 append#(cons(x,y),z) -> c_1(append#(y,z)):1
          
          4:S:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                                ,append#(y,cons(x,nil()))
                                                ,ge#(c,l)):3
          
          5:S:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(nil()) -> c_10():11
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):5
          
          6:S:rev#(x) -> c_11(if#(x,eq(0(),length(x)),nil(),0(),length(x)),length#(x),length#(x))
             -->_3 length#(nil()) -> c_10():11
             -->_2 length#(nil()) -> c_10():11
             -->_3 length#(cons(x,y)) -> c_9(length#(y)):5
             -->_2 length#(cons(x,y)) -> c_9(length#(y)):5
          
          7:W:append#(nil(),y) -> c_2()
             
          
          8:W:ge#(x,0()) -> c_3()
             
          
          9:W:ge#(0(),s(y)) -> c_4()
             
          
          10:W:if#(x,true(),z,c,l) -> c_8()
             
          
          11:W:length#(nil()) -> c_10()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          11: length#(nil()) -> c_10()
          10: if#(x,true(),z,c,l) -> c_8()
          8: ge#(x,0()) -> c_3()
          9: ge#(0(),s(y)) -> c_4()
          7: append#(nil(),y) -> c_2()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,y),z) -> c_1(append#(y,z))
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
            rev#(x) -> c_11(if#(x,eq(0(),length(x)),nil(),0(),length(x)),length#(x),length#(x))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            length(cons(x,y)) -> s(length(y))
            length(nil()) -> 0()
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:append#(cons(x,y),z) -> c_1(append#(y,z))
             -->_1 append#(cons(x,y),z) -> c_1(append#(y,z)):1
          
          2:S:ge#(s(x),s(y)) -> c_5(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):2
          
          3:S:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                           ,append#(y,cons(x,nil()))
                                           ,ge#(c,l))
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):4
             -->_3 ge#(s(x),s(y)) -> c_5(ge#(x,y)):2
             -->_2 append#(cons(x,y),z) -> c_1(append#(y,z)):1
          
          4:S:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                                ,append#(y,cons(x,nil()))
                                                ,ge#(c,l)):3
          
          5:S:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):5
          
          6:S:rev#(x) -> c_11(if#(x,eq(0(),length(x)),nil(),0(),length(x)),length#(x),length#(x))
             -->_3 length#(cons(x,y)) -> c_9(length#(y)):5
             -->_2 length#(cons(x,y)) -> c_9(length#(y)):5
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          rev#(x) -> c_11(length#(x),length#(x))
* Step 6: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,y),z) -> c_1(append#(y,z))
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
            rev#(x) -> c_11(length#(x),length#(x))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            length(cons(x,y)) -> s(length(y))
            length(nil()) -> 0()
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          append(cons(x,y),z) -> cons(x,append(y,z))
          append(nil(),y) -> y
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          append#(cons(x,y),z) -> c_1(append#(y,z))
          ge#(s(x),s(y)) -> c_5(ge#(x,y))
          help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                       ,append#(y,cons(x,nil()))
                                       ,ge#(c,l))
          if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
          length#(cons(x,y)) -> c_9(length#(y))
          rev#(x) -> c_11(length#(x),length#(x))
* Step 7: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,y),z) -> c_1(append#(y,z))
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
            rev#(x) -> c_11(length#(x),length#(x))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:append#(cons(x,y),z) -> c_1(append#(y,z))
           -->_1 append#(cons(x,y),z) -> c_1(append#(y,z)):1
        
        2:S:ge#(s(x),s(y)) -> c_5(ge#(x,y))
           -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):2
        
        3:S:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
           -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):4
           -->_3 ge#(s(x),s(y)) -> c_5(ge#(x,y)):2
           -->_2 append#(cons(x,y),z) -> c_1(append#(y,z)):1
        
        4:S:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
           -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                              ,append#(y,cons(x,nil()))
                                              ,ge#(c,l)):3
        
        5:S:length#(cons(x,y)) -> c_9(length#(y))
           -->_1 length#(cons(x,y)) -> c_9(length#(y)):5
        
        6:S:rev#(x) -> c_11(length#(x),length#(x))
           -->_2 length#(cons(x,y)) -> c_9(length#(y)):5
           -->_1 length#(cons(x,y)) -> c_9(length#(y)):5
        
        
        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).
        
        [(6,rev#(x) -> c_11(length#(x),length#(x)))]
* Step 8: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,y),z) -> c_1(append#(y,z))
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,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:
              append#(cons(x,y),z) -> c_1(append#(y,z))
          - Weak DPs:
              ge#(s(x),s(y)) -> c_5(ge#(x,y))
              help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                           ,append#(y,cons(x,nil()))
                                           ,ge#(c,l))
              if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
              length#(cons(x,y)) -> c_9(length#(y))
          - Weak TRS:
              append(cons(x,y),z) -> cons(x,append(y,z))
              append(nil(),y) -> y
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
          - Signature:
              {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2
              ,eq/2,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
              ,cons,eq,false,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              ge#(s(x),s(y)) -> c_5(ge#(x,y))
              help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                           ,append#(y,cons(x,nil()))
                                           ,ge#(c,l))
              if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
              length#(cons(x,y)) -> c_9(length#(y))
          - Weak DPs:
              append#(cons(x,y),z) -> c_1(append#(y,z))
          - Weak TRS:
              append(cons(x,y),z) -> cons(x,append(y,z))
              append(nil(),y) -> y
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
          - Signature:
              {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2
              ,eq/2,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
              ,cons,eq,false,nil,s,true}
** Step 8.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,y),z) -> c_1(append#(y,z))
        - Weak DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:append#(cons(x,y),z) -> c_1(append#(y,z))
             -->_1 append#(cons(x,y),z) -> c_1(append#(y,z)):1
          
          2:W:ge#(s(x),s(y)) -> c_5(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):2
          
          3:W:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                           ,append#(y,cons(x,nil()))
                                           ,ge#(c,l))
             -->_2 append#(cons(x,y),z) -> c_1(append#(y,z)):1
             -->_3 ge#(s(x),s(y)) -> c_5(ge#(x,y)):2
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):4
          
          4:W:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                                ,append#(y,cons(x,nil()))
                                                ,ge#(c,l)):3
          
          5:W:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: length#(cons(x,y)) -> c_9(length#(y))
          2: ge#(s(x),s(y)) -> c_5(ge#(x,y))
** Step 8.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,y),z) -> c_1(append#(y,z))
        - Weak DPs:
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:append#(cons(x,y),z) -> c_1(append#(y,z))
             -->_1 append#(cons(x,y),z) -> c_1(append#(y,z)):1
          
          3:W:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                           ,append#(y,cons(x,nil()))
                                           ,ge#(c,l))
             -->_2 append#(cons(x,y),z) -> c_1(append#(y,z)):1
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):4
          
          4:W:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                                ,append#(y,cons(x,nil()))
                                                ,ge#(c,l)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),append#(y,cons(x,nil())))
** Step 8.a:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          append#(cons(x,y),z) -> c_1(append#(y,z))
      - Weak DPs:
          help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),append#(y,cons(x,nil())))
          if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
      - Weak TRS:
          append(cons(x,y),z) -> cons(x,append(y,z))
          append(nil(),y) -> y
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
      - Signature:
          {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
          ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
          ,cons,eq,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 8.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak DPs:
            append#(cons(x,y),z) -> c_1(append#(y,z))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,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:S:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                           ,append#(y,cons(x,nil()))
                                           ,ge#(c,l))
             -->_2 append#(cons(x,y),z) -> c_1(append#(y,z)):5
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):3
             -->_3 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
          
          3:S:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                                ,append#(y,cons(x,nil()))
                                                ,ge#(c,l)):2
          
          4:S:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):4
          
          5:W:append#(cons(x,y),z) -> c_1(append#(y,z))
             -->_1 append#(cons(x,y),z) -> c_1(append#(y,z)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: append#(cons(x,y),z) -> c_1(append#(y,z))
** Step 8.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                         ,append#(y,cons(x,nil()))
                                         ,ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,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:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                           ,append#(y,cons(x,nil()))
                                           ,ge#(c,l))
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):3
             -->_3 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
          
          3:S:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)
                                                ,append#(y,cons(x,nil()))
                                                ,ge#(c,l)):2
          
          4:S:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):4
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
** Step 8.b:3: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,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:
              help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
              if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
              length#(cons(x,y)) -> c_9(length#(y))
          - Weak TRS:
              append(cons(x,y),z) -> cons(x,append(y,z))
              append(nil(),y) -> y
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
          - Signature:
              {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2
              ,eq/2,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
              ,cons,eq,false,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
              if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
              length#(cons(x,y)) -> c_9(length#(y))
          - Weak DPs:
              ge#(s(x),s(y)) -> c_5(ge#(x,y))
          - Weak TRS:
              append(cons(x,y),z) -> cons(x,append(y,z))
              append(nil(),y) -> y
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
          - Signature:
              {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2
              ,eq/2,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
              ,cons,eq,false,nil,s,true}
*** Step 8.b:3.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
        - Weak DPs:
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,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:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
             -->_2 ge#(s(x),s(y)) -> c_5(ge#(x,y)):1
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):3
          
          3:W:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l)):2
          
          4:W:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: length#(cons(x,y)) -> c_9(length#(y))
*** Step 8.b:3.a:2: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          ge#(s(x),s(y)) -> c_5(ge#(x,y))
      - Weak DPs:
          help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
          if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
      - Weak TRS:
          append(cons(x,y),z) -> cons(x,append(y,z))
          append(nil(),y) -> y
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
      - Signature:
          {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
          ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
          ,cons,eq,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 8.b:3.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak DPs:
            ge#(s(x),s(y)) -> c_5(ge#(x,y))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
             -->_2 ge#(s(x),s(y)) -> c_5(ge#(x,y)):4
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):2
          
          2:S:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l)):1
          
          3:S:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):3
          
          4:W:ge#(s(x),s(y)) -> c_5(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_5(ge#(x,y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: ge#(s(x),s(y)) -> c_5(ge#(x,y))
*** Step 8.b:3.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l))
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):2
          
          2:S:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l),ge#(c,l)):1
          
          3:S:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l))
*** Step 8.b:3.b:3: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,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:
              help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l))
              if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
          - Weak DPs:
              length#(cons(x,y)) -> c_9(length#(y))
          - Weak TRS:
              append(cons(x,y),z) -> cons(x,append(y,z))
              append(nil(),y) -> y
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
          - Signature:
              {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2
              ,eq/2,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
              ,cons,eq,false,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              length#(cons(x,y)) -> c_9(length#(y))
          - Weak DPs:
              help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l))
              if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
          - Weak TRS:
              append(cons(x,y),z) -> cons(x,append(y,z))
              append(nil(),y) -> y
              ge(x,0()) -> true()
              ge(0(),s(y)) -> false()
              ge(s(x),s(y)) -> ge(x,y)
          - Signature:
              {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2
              ,eq/2,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
              ,cons,eq,false,nil,s,true}
**** Step 8.b:3.b:3.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
        - Weak DPs:
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l))
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):2
          
          2:S:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)):1
          
          3:W:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: length#(cons(x,y)) -> c_9(length#(y))
**** Step 8.b:3.b:3.a:2: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l))
          if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
      - Weak TRS:
          append(cons(x,y),z) -> cons(x,append(y,z))
          append(nil(),y) -> y
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
      - Signature:
          {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
          ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
          ,cons,eq,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
**** Step 8.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak DPs:
            help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l))
            if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):1
          
          2:W:help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l))
             -->_1 if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z)):3
          
          3:W:if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
             -->_1 help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: help#(c,l,cons(x,y),z) -> c_6(if#(append(y,cons(x,nil())),ge(c,l),cons(x,z),c,l))
          3: if#(x,false(),z,c,l) -> c_7(help#(s(c),l,x,z))
**** Step 8.b:3.b:3.b:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(cons(x,y)) -> c_9(length#(y))
        - Weak TRS:
            append(cons(x,y),z) -> cons(x,append(y,z))
            append(nil(),y) -> y
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          length#(cons(x,y)) -> c_9(length#(y))
**** Step 8.b:3.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(cons(x,y)) -> c_9(length#(y))
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: length#(cons(x,y)) -> c_9(length#(y))
          
        The strictly oriented rules are moved into the weak component.
***** Step 8.b:3.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(cons(x,y)) -> c_9(length#(y))
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {append#,ge#,help#,if#,length#,rev#}
        TcT has computed the following interpretation:
                p(0) = [0]                  
           p(append) = [0]                  
             p(cons) = [1] x1 + [1] x2 + [1]
               p(eq) = [0]                  
            p(false) = [0]                  
               p(ge) = [0]                  
             p(help) = [0]                  
               p(if) = [0]                  
           p(length) = [0]                  
              p(nil) = [0]                  
              p(rev) = [0]                  
                p(s) = [0]                  
             p(true) = [0]                  
          p(append#) = [0]                  
              p(ge#) = [0]                  
            p(help#) = [0]                  
              p(if#) = [0]                  
          p(length#) = [1] x1 + [5]         
             p(rev#) = [1]                  
              p(c_1) = [0]                  
              p(c_2) = [0]                  
              p(c_3) = [0]                  
              p(c_4) = [0]                  
              p(c_5) = [0]                  
              p(c_6) = [0]                  
              p(c_7) = [0]                  
              p(c_8) = [0]                  
              p(c_9) = [1] x1 + [0]         
             p(c_10) = [0]                  
             p(c_11) = [0]                  
        
        Following rules are strictly oriented:
        length#(cons(x,y)) = [1] x + [1] y + [6]
                           > [1] y + [5]        
                           = c_9(length#(y))    
        
        
        Following rules are (at-least) weakly oriented:
        
***** Step 8.b:3.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            length#(cons(x,y)) -> c_9(length#(y))
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 8.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            length#(cons(x,y)) -> c_9(length#(y))
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:length#(cons(x,y)) -> c_9(length#(y))
             -->_1 length#(cons(x,y)) -> c_9(length#(y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: length#(cons(x,y)) -> c_9(length#(y))
***** Step 8.b:3.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {append/2,ge/2,help/4,if/5,length/1,rev/1,append#/2,ge#/2,help#/4,if#/5,length#/1,rev#/1} / {0/0,cons/2,eq/2
            ,false/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,ge#,help#,if#,length#,rev#} and constructors {0
            ,cons,eq,false,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

MAYBE