YES

Problem:
 purge(nil()) -> nil()
 purge(.(x,y)) -> .(x,purge(remove(x,y)))
 remove(x,nil()) -> nil()
 remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))

Proof:
 DP Processor:
  DPs:
   purge#(.(x,y)) -> remove#(x,y)
   purge#(.(x,y)) -> purge#(remove(x,y))
   remove#(x,.(y,z)) -> remove#(x,z)
  TRS:
   purge(nil()) -> nil()
   purge(.(x,y)) -> .(x,purge(remove(x,y)))
   remove(x,nil()) -> nil()
   remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
  TDG Processor:
   DPs:
    purge#(.(x,y)) -> remove#(x,y)
    purge#(.(x,y)) -> purge#(remove(x,y))
    remove#(x,.(y,z)) -> remove#(x,z)
   TRS:
    purge(nil()) -> nil()
    purge(.(x,y)) -> .(x,purge(remove(x,y)))
    remove(x,nil()) -> nil()
    remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
   graph:
    remove#(x,.(y,z)) -> remove#(x,z) ->
    remove#(x,.(y,z)) -> remove#(x,z)
    purge#(.(x,y)) -> remove#(x,y) ->
    remove#(x,.(y,z)) -> remove#(x,z)
    purge#(.(x,y)) -> purge#(remove(x,y)) ->
    purge#(.(x,y)) -> purge#(remove(x,y))
    purge#(.(x,y)) -> purge#(remove(x,y)) -> purge#(.(x,y)) -> remove#(x,y)
   CDG Processor:
    DPs:
     purge#(.(x,y)) -> remove#(x,y)
     purge#(.(x,y)) -> purge#(remove(x,y))
     remove#(x,.(y,z)) -> remove#(x,z)
    TRS:
     purge(nil()) -> nil()
     purge(.(x,y)) -> .(x,purge(remove(x,y)))
     remove(x,nil()) -> nil()
     remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
    graph:
     remove#(x,.(y,z)) -> remove#(x,z) ->
     remove#(x,.(y,z)) -> remove#(x,z)
     purge#(.(x,y)) -> remove#(x,y) -> remove#(x,.(y,z)) -> remove#(x,z)
    SCC Processor:
     #sccs: 1
     #rules: 1
     #arcs: 2/9
     DPs:
      remove#(x,.(y,z)) -> remove#(x,z)
     TRS:
      purge(nil()) -> nil()
      purge(.(x,y)) -> .(x,purge(remove(x,y)))
      remove(x,nil()) -> nil()
      remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
     KBO Processor:
      argument filtering:
       pi(nil) = []
       pi(purge) = [0]
       pi(.) = [1]
       pi(remove) = []
       pi(=) = 1
       pi(if) = 1
       pi(remove#) = 1
      weight function:
       w0 = 1
       w(remove#) = w(if) = w(=) = w(remove) = w(.) = w(nil) = 1
       w(purge) = 0
      precedence:
       if ~ = > remove# ~ remove ~ purge > . ~ nil
      problem:
       DPs:
        
       TRS:
        purge(nil()) -> nil()
        purge(.(x,y)) -> .(x,purge(remove(x,y)))
        remove(x,nil()) -> nil()
        remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
      Qed