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)))
  LPO Processor:
   argument filtering:
    pi(nil) = []
    pi(purge) = [0]
    pi(.) = [1]
    pi(remove) = 1
    pi(=) = 1
    pi(if) = 2
    pi(purge#) = 0
    pi(remove#) = 1
   precedence:
    purge > remove# ~ purge# ~ if ~ = ~ remove ~ . ~ 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