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)))
  EDG 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)) -> remove#(x,y)
    purge#(.(x,y)) -> purge#(remove(x,y)) -> purge#(.(x,y)) -> purge#(remove(x,y))
   SCC Processor:
    #sccs: 2
    #rules: 2
    #arcs: 4/9
    DPs:
     purge#(.(x,y)) -> purge#(remove(x,y))
    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:
      [purge#](x0) = x0,
      
      [if](x0, x1, x2) = 0,
      
      [=](x0, x1) = 0,
      
      [remove](x0, x1) = 0,
      
      [.](x0, x1) = 1,
      
      [purge](x0) = 1,
      
      [nil] = 0
     orientation:
      purge#(.(x,y)) = 1 >= 0 = purge#(remove(x,y))
      
      purge(nil()) = 1 >= 0 = nil()
      
      purge(.(x,y)) = 1 >= 1 = .(x,purge(remove(x,y)))
      
      remove(x,nil()) = 0 >= 0 = nil()
      
      remove(x,.(y,z)) = 0 >= 0 = 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
    
    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)))
    Subterm Criterion Processor:
     simple projection:
      pi(remove#) = 1
     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