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Require Import Parametricity.
Require Import List.
Import ListNotations.
Definition bind_option {A B} (f : A -> option B) (x : option A) :
option B :=
match x with
| Some x => f x
| None => None
end.
Notation "'do' X <- A 'in' B" := (bind_option (fun X => B) A)
(at level 200, X ident, A at level 100, B at level 200).
Definition bind_option2 {A B C} (f : A -> B -> option C)
(x : option (A * B)) : option C :=
do yz <- x in let (y, z) := yz : A * B in f y z.
Notation "'do' X , Y <- A 'in' B" := (bind_option2 (fun X Y => B) A)
(at level 200, X ident, Y ident, A at level 100, B at level 200).
Require Import List.
Record Queue := {
t :> Type;
empty : t;
push : nat -> t -> t;
pop : t -> option (nat * t)
}.
Definition program (Q : Queue) (n : nat) : option nat :=
(* q := 0::1::2::...::n *)
let q : Q :=
nat_rect (fun _ => Q) Q.(empty) Q.(push) (S n)
in
let q : option Q := nat_rect (fun _ => option Q) (Some q)
(fun _ (q : option Q) =>
do q <- q in
do x, q <- Q.(pop) q in
do y, q <- Q.(pop) q in
Some (Q.(push) (x + y) q)) n
in
do q <- q in
option_map fst (Q.(pop) q).
Definition ListQueue := {|
t := list nat;
empty := nil;
push := @cons nat;
pop := fun l =>
match rev l with
| nil => None
| hd :: tl => Some (hd, rev tl) end
|}.
Definition DListQueue := {|
t := list nat * list nat;
empty := (nil, nil);
push x l :=
let (back, front) := l in
(cons x back,front);
pop := fun l =>
let (back, front) := l in
match front with
| [] =>
match rev back with
| [] => None
| hd :: tl => Some (hd, (nil, tl))
end
| hd :: tl => Some (hd, (back, tl))
end
|}.
Parametricity Recursive nat.
Print nat_R.
Lemma nat_R_equal :
forall x y, nat_R x y -> x = y.
intros x y H; induction H; subst; trivial.
Defined.
Lemma equal_nat_R :
forall x y, x = y -> nat_R x y.
intros x y H; subst.
induction y; constructor; trivial.
Defined.
Parametricity Recursive option.
Lemma option_nat_R_equal :
forall x y, option_R nat nat nat_R x y -> x = y.
intros x1 x2 H; destruct H as [x1 x2 x_R | ].
rewrite (nat_R_equal _ _ x_R); reflexivity.
reflexivity.
Defined.
Lemma equal_option_nat_R :
forall x y, x = y -> option_R nat nat nat_R x y.
intros x y H; subst.
destruct y; constructor; apply equal_nat_R; reflexivity.
Defined.
Parametricity Recursive prod.
Parametricity Recursive Queue.
Print Queue_R.
Check Queue_R.
Notation Bisimilar := Queue_R.
Print Queue_R.
Definition R (l1 : list nat) (l2 : list nat * list nat) :=
let (back, front) := l2 in
l1 = back ++ rev front.
Lemma rev_app :
forall A (l1 l2 : list A),
rev (l1 ++ l2) = rev l2 ++ rev l1.
induction l1.
intro; symmetry; apply app_nil_r.
intro; simpl; rewrite IHl1; rewrite app_ass.
reflexivity.
Defined.
Lemma rev_list_rect A :
forall P:list A-> Type,
P [] ->
(forall (a:A) (l:list A), P (rev l) -> P (rev (a :: l))) ->
forall l:list A, P (rev l).
Proof.
induction l; auto.
Defined.
Theorem rev_rect A :
forall P:list A -> Type,
P [] ->
(forall (x:A) (l:list A), P l -> P (l ++ [x])) ->
forall l:list A, P l.
Proof.
intros.
generalize (rev_involutive l).
intros E; rewrite <- E.
apply (rev_list_rect _ P).
auto.
simpl.
intros.
apply (X0 a (rev l0)).
auto.
Defined.
Lemma bisim_list_dlist : Bisimilar ListQueue DListQueue.
apply (Queue_R_Build_Queue_R _ _ R).
* reflexivity.
* intros n1 n2 n_R.
pose (nat_R_equal _ _ n_R) as H.
destruct H. clear n_R.
intros l [back front].
unfold R.
simpl.
intro; subst.
simpl.
reflexivity.
* intros l [back front].
generalize l. clear l.
unfold R; fold R.
pattern back.
apply rev_rect.
intros l H; subst.
rewrite rev_app.
simpl.
rewrite app_nil_r.
rewrite rev_involutive.
destruct front.
constructor.
repeat constructor.
apply equal_nat_R; reflexivity.
clear back; intros hd back IHR l H.
subst.
rewrite rev_app.
rewrite rev_involutive.
rewrite rev_app.
simpl.
destruct front.
simpl.
repeat constructor.
apply equal_nat_R; reflexivity.
simpl.
repeat constructor.
apply equal_nat_R; reflexivity.
unfold R.
rewrite rev_app.
simpl.
rewrite rev_involutive.
reflexivity.
Defined.
Print program.
Check program.
Parametricity Recursive program.
Check program_R.
Lemma program_independent :
forall n,
program ListQueue n = program DListQueue n.
intro n.
apply option_nat_R_equal.
apply program_R.
apply bisim_list_dlist.
apply equal_nat_R.
reflexivity.
Defined.
Print program.
Print program_R.
|
