MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
            del(x,nil()) -> nil()
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
            doublelist(nil()) -> nil()
            eq(0(),0()) -> true()
            eq(0(),s(y)) -> false()
            eq(s(x),0()) -> false()
            eq(s(x),s(y)) -> eq(x,y)
            first(cons(x,xs)) -> x
            first(nil()) -> 0()
            if(false(),x,y,xs) -> cons(y,del(x,xs))
            if(true(),x,y,xs) -> xs
        - Signature:
            {del/2,double/1,doublelist/1,eq/2,first/1,if/4} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {del,double,doublelist,eq,first,if} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y))
          del#(x,nil()) -> c_2()
          double#(0()) -> c_3()
          double#(s(x)) -> c_4(double#(x))
          doublelist#(cons(x,xs)) -> c_5(double#(x)
                                        ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                        ,del#(first(cons(x,xs)),cons(x,xs))
                                        ,first#(cons(x,xs)))
          doublelist#(nil()) -> c_6()
          eq#(0(),0()) -> c_7()
          eq#(0(),s(y)) -> c_8()
          eq#(s(x),0()) -> c_9()
          eq#(s(x),s(y)) -> c_10(eq#(x,y))
          first#(cons(x,xs)) -> c_11()
          first#(nil()) -> c_12()
          if#(false(),x,y,xs) -> c_13(del#(x,xs))
          if#(true(),x,y,xs) -> c_14()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y))
            del#(x,nil()) -> c_2()
            double#(0()) -> c_3()
            double#(s(x)) -> c_4(double#(x))
            doublelist#(cons(x,xs)) -> c_5(double#(x)
                                          ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                          ,del#(first(cons(x,xs)),cons(x,xs))
                                          ,first#(cons(x,xs)))
            doublelist#(nil()) -> c_6()
            eq#(0(),0()) -> c_7()
            eq#(0(),s(y)) -> c_8()
            eq#(s(x),0()) -> c_9()
            eq#(s(x),s(y)) -> c_10(eq#(x,y))
            first#(cons(x,xs)) -> c_11()
            first#(nil()) -> c_12()
            if#(false(),x,y,xs) -> c_13(del#(x,xs))
            if#(true(),x,y,xs) -> c_14()
        - Weak TRS:
            del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
            del(x,nil()) -> nil()
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
            doublelist(nil()) -> nil()
            eq(0(),0()) -> true()
            eq(0(),s(y)) -> false()
            eq(s(x),0()) -> false()
            eq(s(x),s(y)) -> eq(x,y)
            first(cons(x,xs)) -> x
            first(nil()) -> 0()
            if(false(),x,y,xs) -> cons(y,del(x,xs))
            if(true(),x,y,xs) -> xs
        - Signature:
            {del/2,double/1,doublelist/1,eq/2,first/1,if/4,del#/2,double#/1,doublelist#/1,eq#/2,first#/1,if#/4} / {0/0
            ,cons/2,false/0,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/4,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0
            ,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {del#,double#,doublelist#,eq#,first#
            ,if#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
          del(x,nil()) -> nil()
          eq(0(),0()) -> true()
          eq(0(),s(y)) -> false()
          eq(s(x),0()) -> false()
          eq(s(x),s(y)) -> eq(x,y)
          first(cons(x,xs)) -> x
          if(false(),x,y,xs) -> cons(y,del(x,xs))
          if(true(),x,y,xs) -> xs
          del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y))
          del#(x,nil()) -> c_2()
          double#(0()) -> c_3()
          double#(s(x)) -> c_4(double#(x))
          doublelist#(cons(x,xs)) -> c_5(double#(x)
                                        ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                        ,del#(first(cons(x,xs)),cons(x,xs))
                                        ,first#(cons(x,xs)))
          doublelist#(nil()) -> c_6()
          eq#(0(),0()) -> c_7()
          eq#(0(),s(y)) -> c_8()
          eq#(s(x),0()) -> c_9()
          eq#(s(x),s(y)) -> c_10(eq#(x,y))
          first#(cons(x,xs)) -> c_11()
          first#(nil()) -> c_12()
          if#(false(),x,y,xs) -> c_13(del#(x,xs))
          if#(true(),x,y,xs) -> c_14()
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y))
            del#(x,nil()) -> c_2()
            double#(0()) -> c_3()
            double#(s(x)) -> c_4(double#(x))
            doublelist#(cons(x,xs)) -> c_5(double#(x)
                                          ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                          ,del#(first(cons(x,xs)),cons(x,xs))
                                          ,first#(cons(x,xs)))
            doublelist#(nil()) -> c_6()
            eq#(0(),0()) -> c_7()
            eq#(0(),s(y)) -> c_8()
            eq#(s(x),0()) -> c_9()
            eq#(s(x),s(y)) -> c_10(eq#(x,y))
            first#(cons(x,xs)) -> c_11()
            first#(nil()) -> c_12()
            if#(false(),x,y,xs) -> c_13(del#(x,xs))
            if#(true(),x,y,xs) -> c_14()
        - Weak TRS:
            del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
            del(x,nil()) -> nil()
            eq(0(),0()) -> true()
            eq(0(),s(y)) -> false()
            eq(s(x),0()) -> false()
            eq(s(x),s(y)) -> eq(x,y)
            first(cons(x,xs)) -> x
            if(false(),x,y,xs) -> cons(y,del(x,xs))
            if(true(),x,y,xs) -> xs
        - Signature:
            {del/2,double/1,doublelist/1,eq/2,first/1,if/4,del#/2,double#/1,doublelist#/1,eq#/2,first#/1,if#/4} / {0/0
            ,cons/2,false/0,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/4,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0
            ,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {del#,double#,doublelist#,eq#,first#
            ,if#} 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
          {2,3,6,7,8,9,11,12,14}
        by application of
          Pre({2,3,6,7,8,9,11,12,14}) = {1,4,5,10,13}.
        Here rules are labelled as follows:
          1: del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y))
          2: del#(x,nil()) -> c_2()
          3: double#(0()) -> c_3()
          4: double#(s(x)) -> c_4(double#(x))
          5: doublelist#(cons(x,xs)) -> c_5(double#(x)
                                           ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                           ,del#(first(cons(x,xs)),cons(x,xs))
                                           ,first#(cons(x,xs)))
          6: doublelist#(nil()) -> c_6()
          7: eq#(0(),0()) -> c_7()
          8: eq#(0(),s(y)) -> c_8()
          9: eq#(s(x),0()) -> c_9()
          10: eq#(s(x),s(y)) -> c_10(eq#(x,y))
          11: first#(cons(x,xs)) -> c_11()
          12: first#(nil()) -> c_12()
          13: if#(false(),x,y,xs) -> c_13(del#(x,xs))
          14: if#(true(),x,y,xs) -> c_14()
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y))
            double#(s(x)) -> c_4(double#(x))
            doublelist#(cons(x,xs)) -> c_5(double#(x)
                                          ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                          ,del#(first(cons(x,xs)),cons(x,xs))
                                          ,first#(cons(x,xs)))
            eq#(s(x),s(y)) -> c_10(eq#(x,y))
            if#(false(),x,y,xs) -> c_13(del#(x,xs))
        - Weak DPs:
            del#(x,nil()) -> c_2()
            double#(0()) -> c_3()
            doublelist#(nil()) -> c_6()
            eq#(0(),0()) -> c_7()
            eq#(0(),s(y)) -> c_8()
            eq#(s(x),0()) -> c_9()
            first#(cons(x,xs)) -> c_11()
            first#(nil()) -> c_12()
            if#(true(),x,y,xs) -> c_14()
        - Weak TRS:
            del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
            del(x,nil()) -> nil()
            eq(0(),0()) -> true()
            eq(0(),s(y)) -> false()
            eq(s(x),0()) -> false()
            eq(s(x),s(y)) -> eq(x,y)
            first(cons(x,xs)) -> x
            if(false(),x,y,xs) -> cons(y,del(x,xs))
            if(true(),x,y,xs) -> xs
        - Signature:
            {del/2,double/1,doublelist/1,eq/2,first/1,if/4,del#/2,double#/1,doublelist#/1,eq#/2,first#/1,if#/4} / {0/0
            ,cons/2,false/0,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/4,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0
            ,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {del#,double#,doublelist#,eq#,first#
            ,if#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y))
             -->_1 if#(false(),x,y,xs) -> c_13(del#(x,xs)):5
             -->_2 eq#(s(x),s(y)) -> c_10(eq#(x,y)):4
             -->_1 if#(true(),x,y,xs) -> c_14():14
             -->_2 eq#(s(x),0()) -> c_9():11
             -->_2 eq#(0(),s(y)) -> c_8():10
             -->_2 eq#(0(),0()) -> c_7():9
          
          2:S:double#(s(x)) -> c_4(double#(x))
             -->_1 double#(0()) -> c_3():7
             -->_1 double#(s(x)) -> c_4(double#(x)):2
          
          3:S:doublelist#(cons(x,xs)) -> c_5(double#(x)
                                            ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                            ,del#(first(cons(x,xs)),cons(x,xs))
                                            ,first#(cons(x,xs)))
             -->_4 first#(cons(x,xs)) -> c_11():12
             -->_2 doublelist#(nil()) -> c_6():8
             -->_1 double#(0()) -> c_3():7
             -->_2 doublelist#(cons(x,xs)) -> c_5(double#(x)
                                                 ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                                 ,del#(first(cons(x,xs)),cons(x,xs))
                                                 ,first#(cons(x,xs))):3
             -->_1 double#(s(x)) -> c_4(double#(x)):2
             -->_3 del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y)):1
          
          4:S:eq#(s(x),s(y)) -> c_10(eq#(x,y))
             -->_1 eq#(s(x),0()) -> c_9():11
             -->_1 eq#(0(),s(y)) -> c_8():10
             -->_1 eq#(0(),0()) -> c_7():9
             -->_1 eq#(s(x),s(y)) -> c_10(eq#(x,y)):4
          
          5:S:if#(false(),x,y,xs) -> c_13(del#(x,xs))
             -->_1 del#(x,nil()) -> c_2():6
             -->_1 del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y)):1
          
          6:W:del#(x,nil()) -> c_2()
             
          
          7:W:double#(0()) -> c_3()
             
          
          8:W:doublelist#(nil()) -> c_6()
             
          
          9:W:eq#(0(),0()) -> c_7()
             
          
          10:W:eq#(0(),s(y)) -> c_8()
             
          
          11:W:eq#(s(x),0()) -> c_9()
             
          
          12:W:first#(cons(x,xs)) -> c_11()
             
          
          13:W:first#(nil()) -> c_12()
             
          
          14:W:if#(true(),x,y,xs) -> c_14()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          13: first#(nil()) -> c_12()
          8: doublelist#(nil()) -> c_6()
          12: first#(cons(x,xs)) -> c_11()
          7: double#(0()) -> c_3()
          14: if#(true(),x,y,xs) -> c_14()
          9: eq#(0(),0()) -> c_7()
          10: eq#(0(),s(y)) -> c_8()
          11: eq#(s(x),0()) -> c_9()
          6: del#(x,nil()) -> c_2()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y))
            double#(s(x)) -> c_4(double#(x))
            doublelist#(cons(x,xs)) -> c_5(double#(x)
                                          ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                          ,del#(first(cons(x,xs)),cons(x,xs))
                                          ,first#(cons(x,xs)))
            eq#(s(x),s(y)) -> c_10(eq#(x,y))
            if#(false(),x,y,xs) -> c_13(del#(x,xs))
        - Weak TRS:
            del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
            del(x,nil()) -> nil()
            eq(0(),0()) -> true()
            eq(0(),s(y)) -> false()
            eq(s(x),0()) -> false()
            eq(s(x),s(y)) -> eq(x,y)
            first(cons(x,xs)) -> x
            if(false(),x,y,xs) -> cons(y,del(x,xs))
            if(true(),x,y,xs) -> xs
        - Signature:
            {del/2,double/1,doublelist/1,eq/2,first/1,if/4,del#/2,double#/1,doublelist#/1,eq#/2,first#/1,if#/4} / {0/0
            ,cons/2,false/0,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/4,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0
            ,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {del#,double#,doublelist#,eq#,first#
            ,if#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y))
             -->_1 if#(false(),x,y,xs) -> c_13(del#(x,xs)):5
             -->_2 eq#(s(x),s(y)) -> c_10(eq#(x,y)):4
          
          2:S:double#(s(x)) -> c_4(double#(x))
             -->_1 double#(s(x)) -> c_4(double#(x)):2
          
          3:S:doublelist#(cons(x,xs)) -> c_5(double#(x)
                                            ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                            ,del#(first(cons(x,xs)),cons(x,xs))
                                            ,first#(cons(x,xs)))
             -->_2 doublelist#(cons(x,xs)) -> c_5(double#(x)
                                                 ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                                 ,del#(first(cons(x,xs)),cons(x,xs))
                                                 ,first#(cons(x,xs))):3
             -->_1 double#(s(x)) -> c_4(double#(x)):2
             -->_3 del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y)):1
          
          4:S:eq#(s(x),s(y)) -> c_10(eq#(x,y))
             -->_1 eq#(s(x),s(y)) -> c_10(eq#(x,y)):4
          
          5:S:if#(false(),x,y,xs) -> c_13(del#(x,xs))
             -->_1 del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          doublelist#(cons(x,xs)) -> c_5(double#(x)
                                        ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                        ,del#(first(cons(x,xs)),cons(x,xs)))
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          del#(x,cons(y,xs)) -> c_1(if#(eq(x,y),x,y,xs),eq#(x,y))
          double#(s(x)) -> c_4(double#(x))
          doublelist#(cons(x,xs)) -> c_5(double#(x)
                                        ,doublelist#(del(first(cons(x,xs)),cons(x,xs)))
                                        ,del#(first(cons(x,xs)),cons(x,xs)))
          eq#(s(x),s(y)) -> c_10(eq#(x,y))
          if#(false(),x,y,xs) -> c_13(del#(x,xs))
      - Weak TRS:
          del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
          del(x,nil()) -> nil()
          eq(0(),0()) -> true()
          eq(0(),s(y)) -> false()
          eq(s(x),0()) -> false()
          eq(s(x),s(y)) -> eq(x,y)
          first(cons(x,xs)) -> x
          if(false(),x,y,xs) -> cons(y,del(x,xs))
          if(true(),x,y,xs) -> xs
      - Signature:
          {del/2,double/1,doublelist/1,eq/2,first/1,if/4,del#/2,double#/1,doublelist#/1,eq#/2,first#/1,if#/4} / {0/0
          ,cons/2,false/0,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/3,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0
          ,c_13/1,c_14/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {del#,double#,doublelist#,eq#,first#
          ,if#} and constructors {0,cons,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE