import data.list
open nat list

context
  parameter {A : Type}
  parameter (p : A → Prop)
  parameter [H : decidable_pred p]
  include H

  definition filter : list A → list A,
  filter nil      := nil,
  filter (a :: l) := if p a then a :: filter l else filter l

  theorem filter_nil : filter nil = nil :=
  rfl

  theorem filter_cons (a : A) (l : list A) : filter (a :: l) = if p a then a :: filter l else filter l :=
  rfl

  open eq.ops

  theorem filter_cons_of_pos {a : A} (l : list A) (h : p a) : filter (a :: l) = a :: filter l :=
  if_pos h ▸ filter_cons a l

  theorem filter_cons_of_neg {a : A} (l : list A) (h : ¬ p a) : filter (a :: l) = filter l :=
  if_neg h ▸ filter_cons a l
end

check @filter
check @filter_cons_of_pos
check @filter_cons_of_neg