Section Frequency.
Variable A : Type.
Variable eqA_dec : forall a b : A, {a = b} + {a <> b}.
(* Create a list of a given length with a given element  *)
Fixpoint id_list (a : A) (n : nat) {struct n} : list A :=
match n with
| O => nil
| S n1 => a :: id_list a n1
end.
(* An element in a frequency list *)
Fixpoint add_frequency_list (a : A) (l : list (A * nat)) {struct l} :
list (A * nat) :=
match l with
| nil => (a, 1) :: nil
| (b, n) :: l1 =>
match eqA_dec a b with
| left _ => (a, S n) :: l1
| right _ => (b, n) :: add_frequency_list a l1
end
end.

(* Adding an element gives the same list upto permutation if the frequency list is developed *)
Theorem add_frequency_list_perm :
forall (a : A) l,
permutation (a :: flat_map (fun p => id_list (fst p) (snd p)) l)
(flat_map (fun p => id_list (fst p) (snd p)) (add_frequency_list a l)).