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)))
  Usable Rule Processor:
   DPs:
    purge#(.(x,y)) -> remove#(x,y)
    purge#(.(x,y)) -> purge#(remove(x,y))
    remove#(x,.(y,z)) -> remove#(x,z)
   TRS:
    f8(x,y) -> x
    f8(x,y) -> 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:
     f8(x,y) -> x
     f8(x,y) -> 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:
      f8(x,y) -> x
      f8(x,y) -> 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)
     Restore Modifier:
      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)))
      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)))
       Matrix Interpretation Processor:
        dimension: 1
        interpretation:
         [remove#](x0, x1) = x1 + 1,
         
         [if](x0, x1, x2) = x1 + 1,
         
         [=](x0, x1) = 0,
         
         [remove](x0, x1) = x1,
         
         [.](x0, x1) = x1 + 1,
         
         [purge](x0) = x0,
         
         [nil] = 0
        orientation:
         remove#(x,.(y,z)) = z + 2 >= z + 1 = remove#(x,z)
         
         purge(nil()) = 0 >= 0 = nil()
         
         purge(.(x,y)) = y + 1 >= y + 1 = .(x,purge(remove(x,y)))
         
         remove(x,nil()) = 0 >= 0 = nil()
         
         remove(x,.(y,z)) = z + 1 >= z + 1 = if(=(x,y),remove(x,z),.(y,remove(x,z)))
        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