MAYBE

Problem:
 double(0()) -> 0()
 double(s(x)) -> s(s(double(x)))
 del(x,nil()) -> nil()
 del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
 if(true(),x,y,xs) -> xs
 if(false(),x,y,xs) -> cons(y,del(x,xs))
 eq(0(),0()) -> true()
 eq(0(),s(y)) -> false()
 eq(s(x),0()) -> false()
 eq(s(x),s(y)) -> eq(x,y)
 first(nil()) -> 0()
 first(cons(x,xs)) -> x
 doublelist(nil()) -> nil()
 doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))

Proof:
 DP Processor:
  DPs:
   double#(s(x)) -> double#(x)
   del#(x,cons(y,xs)) -> eq#(x,y)
   del#(x,cons(y,xs)) -> if#(eq(x,y),x,y,xs)
   if#(false(),x,y,xs) -> del#(x,xs)
   eq#(s(x),s(y)) -> eq#(x,y)
   doublelist#(cons(x,xs)) -> first#(cons(x,xs))
   doublelist#(cons(x,xs)) -> del#(first(cons(x,xs)),cons(x,xs))
   doublelist#(cons(x,xs)) -> doublelist#(del(first(cons(x,xs)),cons(x,xs)))
   doublelist#(cons(x,xs)) -> double#(x)
  TRS:
   double(0()) -> 0()
   double(s(x)) -> s(s(double(x)))
   del(x,nil()) -> nil()
   del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
   if(true(),x,y,xs) -> xs
   if(false(),x,y,xs) -> cons(y,del(x,xs))
   eq(0(),0()) -> true()
   eq(0(),s(y)) -> false()
   eq(s(x),0()) -> false()
   eq(s(x),s(y)) -> eq(x,y)
   first(nil()) -> 0()
   first(cons(x,xs)) -> x
   doublelist(nil()) -> nil()
   doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
  TDG Processor:
   DPs:
    double#(s(x)) -> double#(x)
    del#(x,cons(y,xs)) -> eq#(x,y)
    del#(x,cons(y,xs)) -> if#(eq(x,y),x,y,xs)
    if#(false(),x,y,xs) -> del#(x,xs)
    eq#(s(x),s(y)) -> eq#(x,y)
    doublelist#(cons(x,xs)) -> first#(cons(x,xs))
    doublelist#(cons(x,xs)) -> del#(first(cons(x,xs)),cons(x,xs))
    doublelist#(cons(x,xs)) -> doublelist#(del(first(cons(x,xs)),cons(x,xs)))
    doublelist#(cons(x,xs)) -> double#(x)
   TRS:
    double(0()) -> 0()
    double(s(x)) -> s(s(double(x)))
    del(x,nil()) -> nil()
    del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
    if(true(),x,y,xs) -> xs
    if(false(),x,y,xs) -> cons(y,del(x,xs))
    eq(0(),0()) -> true()
    eq(0(),s(y)) -> false()
    eq(s(x),0()) -> false()
    eq(s(x),s(y)) -> eq(x,y)
    first(nil()) -> 0()
    first(cons(x,xs)) -> x
    doublelist(nil()) -> nil()
    doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
   graph:
    doublelist#(cons(x,xs)) -> doublelist#(del(first(cons(x,xs)),cons(x,xs))) ->
    doublelist#(cons(x,xs)) -> double#(x)
    doublelist#(cons(x,xs)) -> doublelist#(del(first(cons(x,xs)),cons(x,xs))) ->
    doublelist#(cons(x,xs)) -> doublelist#(del(first(cons(x,xs)),cons(x,xs)))
    doublelist#(cons(x,xs)) -> doublelist#(del(first(cons(x,xs)),cons(x,xs))) ->
    doublelist#(cons(x,xs)) -> del#(first(cons(x,xs)),cons(x,xs))
    doublelist#(cons(x,xs)) -> doublelist#(del(first(cons(x,xs)),cons(x,xs))) ->
    doublelist#(cons(x,xs)) -> first#(cons(x,xs))
    doublelist#(cons(x,xs)) -> del#(first(cons(x,xs)),cons(x,xs)) ->
    del#(x,cons(y,xs)) -> if#(eq(x,y),x,y,xs)
    doublelist#(cons(x,xs)) -> del#(first(cons(x,xs)),cons(x,xs)) ->
    del#(x,cons(y,xs)) -> eq#(x,y)
    doublelist#(cons(x,xs)) -> double#(x) ->
    double#(s(x)) -> double#(x)
    if#(false(),x,y,xs) -> del#(x,xs) ->
    del#(x,cons(y,xs)) -> if#(eq(x,y),x,y,xs)
    if#(false(),x,y,xs) -> del#(x,xs) -> del#(x,cons(y,xs)) -> eq#(x,y)
    eq#(s(x),s(y)) -> eq#(x,y) ->
    eq#(s(x),s(y)) -> eq#(x,y)
    del#(x,cons(y,xs)) -> if#(eq(x,y),x,y,xs) ->
    if#(false(),x,y,xs) -> del#(x,xs)
    del#(x,cons(y,xs)) -> eq#(x,y) -> eq#(s(x),s(y)) -> eq#(x,y)
    double#(s(x)) -> double#(x) -> double#(s(x)) -> double#(x)
   SCC Processor:
    #sccs: 4
    #rules: 5
    #arcs: 13/81
    DPs:
     doublelist#(cons(x,xs)) -> doublelist#(del(first(cons(x,xs)),cons(x,xs)))
    TRS:
     double(0()) -> 0()
     double(s(x)) -> s(s(double(x)))
     del(x,nil()) -> nil()
     del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
     if(true(),x,y,xs) -> xs
     if(false(),x,y,xs) -> cons(y,del(x,xs))
     eq(0(),0()) -> true()
     eq(0(),s(y)) -> false()
     eq(s(x),0()) -> false()
     eq(s(x),s(y)) -> eq(x,y)
     first(nil()) -> 0()
     first(cons(x,xs)) -> x
     doublelist(nil()) -> nil()
     doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
    Open
    
    DPs:
     double#(s(x)) -> double#(x)
    TRS:
     double(0()) -> 0()
     double(s(x)) -> s(s(double(x)))
     del(x,nil()) -> nil()
     del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
     if(true(),x,y,xs) -> xs
     if(false(),x,y,xs) -> cons(y,del(x,xs))
     eq(0(),0()) -> true()
     eq(0(),s(y)) -> false()
     eq(s(x),0()) -> false()
     eq(s(x),s(y)) -> eq(x,y)
     first(nil()) -> 0()
     first(cons(x,xs)) -> x
     doublelist(nil()) -> nil()
     doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
    Subterm Criterion Processor:
     simple projection:
      pi(double#) = 0
     problem:
      DPs:
       
      TRS:
       double(0()) -> 0()
       double(s(x)) -> s(s(double(x)))
       del(x,nil()) -> nil()
       del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
       if(true(),x,y,xs) -> xs
       if(false(),x,y,xs) -> cons(y,del(x,xs))
       eq(0(),0()) -> true()
       eq(0(),s(y)) -> false()
       eq(s(x),0()) -> false()
       eq(s(x),s(y)) -> eq(x,y)
       first(nil()) -> 0()
       first(cons(x,xs)) -> x
       doublelist(nil()) -> nil()
       doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
     Qed
    
    DPs:
     del#(x,cons(y,xs)) -> if#(eq(x,y),x,y,xs)
     if#(false(),x,y,xs) -> del#(x,xs)
    TRS:
     double(0()) -> 0()
     double(s(x)) -> s(s(double(x)))
     del(x,nil()) -> nil()
     del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
     if(true(),x,y,xs) -> xs
     if(false(),x,y,xs) -> cons(y,del(x,xs))
     eq(0(),0()) -> true()
     eq(0(),s(y)) -> false()
     eq(s(x),0()) -> false()
     eq(s(x),s(y)) -> eq(x,y)
     first(nil()) -> 0()
     first(cons(x,xs)) -> x
     doublelist(nil()) -> nil()
     doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
    Subterm Criterion Processor:
     simple projection:
      pi(del#) = 1
      pi(if#) = 3
     problem:
      DPs:
       if#(false(),x,y,xs) -> del#(x,xs)
      TRS:
       double(0()) -> 0()
       double(s(x)) -> s(s(double(x)))
       del(x,nil()) -> nil()
       del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
       if(true(),x,y,xs) -> xs
       if(false(),x,y,xs) -> cons(y,del(x,xs))
       eq(0(),0()) -> true()
       eq(0(),s(y)) -> false()
       eq(s(x),0()) -> false()
       eq(s(x),s(y)) -> eq(x,y)
       first(nil()) -> 0()
       first(cons(x,xs)) -> x
       doublelist(nil()) -> nil()
       doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/1
      
    
    DPs:
     eq#(s(x),s(y)) -> eq#(x,y)
    TRS:
     double(0()) -> 0()
     double(s(x)) -> s(s(double(x)))
     del(x,nil()) -> nil()
     del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
     if(true(),x,y,xs) -> xs
     if(false(),x,y,xs) -> cons(y,del(x,xs))
     eq(0(),0()) -> true()
     eq(0(),s(y)) -> false()
     eq(s(x),0()) -> false()
     eq(s(x),s(y)) -> eq(x,y)
     first(nil()) -> 0()
     first(cons(x,xs)) -> x
     doublelist(nil()) -> nil()
     doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
    Subterm Criterion Processor:
     simple projection:
      pi(eq#) = 1
     problem:
      DPs:
       
      TRS:
       double(0()) -> 0()
       double(s(x)) -> s(s(double(x)))
       del(x,nil()) -> nil()
       del(x,cons(y,xs)) -> if(eq(x,y),x,y,xs)
       if(true(),x,y,xs) -> xs
       if(false(),x,y,xs) -> cons(y,del(x,xs))
       eq(0(),0()) -> true()
       eq(0(),s(y)) -> false()
       eq(s(x),0()) -> false()
       eq(s(x),s(y)) -> eq(x,y)
       first(nil()) -> 0()
       first(cons(x,xs)) -> x
       doublelist(nil()) -> nil()
       doublelist(cons(x,xs)) -> cons(double(x),doublelist(del(first(cons(x,xs)),cons(x,xs))))
     Qed