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bf_stack.v
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bf_stack.v
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Require Import bf bf_equivalence bf_semantics.
Require Import Lists.Streams.
Require Import Setoid.
Require Import Arith.
Fixpoint add_n (n : nat) :=
match n with
| 0 => END
| S n => + (add_n n)
end.
Lemma about_add_n (n : nat) :
forall ls x rs stdin stdout,
iter (add_n n, state[ls, x, rs, stdin, stdout])
(END, state[ls, n+x, rs, stdin, stdout]).
Proof.
induction n.
intros.
bf_step.
intros.
bf_step.
simpl.
rewrite plus_n_Sm.
apply IHn.
Qed.
Definition push (n: nat) := > add_n n.
Theorem about_push (n: nat) :
forall ls x stdin stdout,
iter (push n, state[ls, x, zeroes, stdin, stdout])
(END, state[Cons x ls, n, zeroes, stdin, stdout]).
Proof.
unfold push.
intros.
bf_step.
simpl stepRight.
rewrite (plus_n_O n) at 2.
apply about_add_n.
Qed.
Definition add := [- < + >END]<END.
Theorem about_add :
forall ls y x stdin stdout,
iter (add, state[Cons x ls, y, zeroes, stdin, stdout])
(END, state[ls, y+x, zeroes, stdin, stdout]).
Proof.
unfold add.
intros ls y.
induction y.
intros.
repeat bf_step.
intros.
do 5 bf_step.
rewrite (plus_n_Sm).
apply IHy.
Qed.
Definition sub := [- < - >END]<END.
Theorem about_sub :
forall ls x y stdin stdout,
iter (sub, state[Cons y ls, x, zeroes, stdin, stdout])
(END, state[ls, y - x, zeroes, stdin, stdout]).
Proof.
unfold sub.
intros ls x.
induction x.
intros.
repeat bf_step.
rewrite <- minus_n_O.
bf_step.
intros.
do 4 bf_step.
replace (decrement state[ls, y, Cons x zeroes, stdin, stdout])
with state[ls, y - 1, Cons x zeroes, stdin, stdout]
by (destruct y;
[ reflexivity
| simpl; rewrite <- minus_n_O; reflexivity ]).
bf_step.
replace (y - S x)
with ((y - 1) - x)
by (destruct y; [ reflexivity
| simpl; rewrite <- minus_n_O; reflexivity ]).
apply IHx.
Qed.
Definition dup := [> + > + < < -END] (* Add top of stack to the two next cells *)
> > (* Move to the result *)
[< < + > > -END]<END (* Copy result to top of stack *)
.
Lemma about_dup_lemma1 :
forall ls x y stdin stdout c,
iter ([> + > + < < -END]c, state[ls, x, Cons y (Cons y zeroes), stdin, stdout])
(c, state[ls, 0, Cons (x+y) (Cons (x+y) zeroes), stdin, stdout]).
Proof.
intros ls x.
induction x.
intros.
repeat bf_step.
intros.
do 8 bf_step.
simpl.
rewrite plus_n_Sm.
apply IHx.
Qed.
Lemma about_dup_lemma2 :
forall ls x y z stdin stdout c,
iter
([< < + > > - END]c,
state[Cons y (Cons z ls), x, zeroes, stdin, stdout])
(c, state[Cons y (Cons (x+z) ls), 0, zeroes, stdin, stdout]).
Proof.
intros ls x.
induction x.
intros.
repeat bf_step.
intros.
do 7 bf_step.
simpl.
rewrite plus_n_Sm.
apply IHx.
Qed.
Theorem about_dup :
forall ls x stdin stdout,
iter (dup, state[ls, x, zeroes, stdin, stdout])
(END, state[Cons x ls, x, zeroes, stdin, stdout]).
Proof.
unfold dup.
intros.
apply (iter_trans _ (> > [< < + > > -END]<END,
state[ls, 0, Cons x (Cons x zeroes), stdin, stdout])).
rewrite plus_n_O at 2.
rewrite plus_n_O at 3.
apply (iter_trans _ ([> + > + < < - END] > > [< < + > > -END]<END,
state[ls, x, Cons 0 (Cons 0 zeroes), stdin, stdout])).
bf_step.
apply about_dup_lemma1.
do 2 bf_step.
rewrite (plus_n_O x) at 4.
simpl.
apply (iter_trans _ (< END,
state[Cons x (Cons (x+0) ls), 0, zeroes, stdin, stdout])).
apply about_dup_lemma2.
repeat bf_step.
simpl.
rewrite <- plus_n_O.
bf_step.
Qed.
Definition reset := [-END]END.
Theorem about_reset : forall ls curr rs stdin stdout,
iter (reset, state[ls, curr, rs, stdin, stdout])
(END, state[ls, 0, rs, stdin, stdout]).
Proof.
unfold reset.
intros.
induction curr.
repeat bf_step.
repeat bf_step.
exact IHcurr.
Qed.
Definition mult :=
(* [...;x1][x2][] Move x1 *)
<[> > + < < -END]> >
(* [...;0;x2][x1][] *)
[<
(* [...;i*x2][x2][x1-i] Add and copy x2 *)
[< + > > > + < < -END]
(* [...;(i+1)*x2][0][x1-i;x2] Move to x2 *)
> >
(* [...;(i+1)*x2;0;x1-i][x2][] Move back x2 *)
[< < + > > -END]<
(* [...;(i+1)*x2;x2][x1-i][] Decrement loop counter *)
-END
(* [...;(i+1)*x2;x2][x1-(i+1)][] *)
]<
(* [...;x1*x2][x2][] Cleanup *)
reset;
<
(* [...][x1*x2][] *)
END.
Example mult_example1 :
forall stdin stdout,
iter (mult, state[Cons 2 zeroes, 3, zeroes, stdin, stdout])
(END, state[zeroes, 6, zeroes, stdin, stdout]).
Proof.
unfold mult.
intros.
repeat bf_step.
Qed.
Example mult_example2 :
forall stdin stdout,
iter (mult, state[Cons 0 zeroes, 3, zeroes, stdin, stdout])
(END, state[zeroes, 0, zeroes, stdin, stdout]).
Proof.
unfold mult.
intros.
repeat bf_step.
Qed.
Example mult_example3 :
forall stdin stdout,
iter (mult, state[Cons 2 zeroes, 0, zeroes, stdin, stdout])
(END, state[zeroes, 0, zeroes, stdin, stdout]).
Proof.
unfold mult.
intros.
repeat bf_step.
Qed.
Theorem about_sequence :
forall c c' s s' s'',
iter (c, s)
(END, s') ->
iter (c', s')
(END, s'') ->
iter (c;c', s)
(END, s'').
Proof.
induction c;
try (
intros;
simpl sequence;
inversion H; subst;
inversion H1; subst;
try discriminate H5;
inversion H1; subst;
match goal with
| [ H1 : step_rel ?C (?C', ?S) |- ?P ] =>
apply (iter_step _ (c;c', S));
[constructor |
apply (IHc c' S s'); assumption]
end).
intros.
destruct s as [? []].
bf_step.
apply (IHc2 _ _ s').
inversion H; inversion H1; subst.
discriminate H7.
simpl in H1, H2.
assumption.
assumption.
bf_step.
admit.
intros.
simpl sequence.
inversion H; subst.
assert ((c', s) ≡ (c', s')) as Hequiv.
bf_reflexivity.
inversion H1; assumption.
apply (iter_injective' (c', s') (c', s) (END, s'')).
assumption.
apply EqBf_sym; assumption.
inversion H1.
Qed.
(* This proof is a total mess! *)
Theorem about_mult :
forall ls x1 x2 stdin stdout,
iter (mult, state[Cons x1 ls, x2, zeroes, stdin, stdout])
(END, state[ls, x1 * x2, zeroes, stdin, stdout]).
Proof.
unfold mult, reset.
intros.
repeat bf_step; simpl.
generalize dependent x1; generalize dependent x2.
assert
(forall x1 x2 k c,
iter ([> > + < < -END]c, state[ls, x1, Cons x2 (Cons k zeroes), stdin, stdout])
(c, state[ls, 0, Cons x2 (Cons (x1+k) zeroes), stdin, stdout]))
as Hmv2.
induction x1; intros.
repeat bf_step.
repeat bf_step; simpl.
rewrite plus_n_Sm.
apply IHx1.
intros.
apply (iter_trans _ (> > [<
[< + > > > + < < -END]> >
[< < + > > -END]
< -END]
< [-END]
<END,
state[ls, 0, Cons x2 (Cons (x1+0) zeroes), stdin, stdout])).
apply (iter_trans _ ([> > + < < -END]
> > [<
[< + > > > + < < -END]> >
[< < + > > -END]
< -END]
< [-END]
<END,
state[ls, x1, Cons x2 (Cons 0 zeroes), stdin, stdout])).
bf_step.
apply Hmv2.
clear Hmv2.
rewrite <- plus_n_O.
do 2 bf_step; simpl.
generalize dependent x1; generalize dependent x2.
assert
(forall x1 x2 k c,
iter ([<
[< + > > > + < < -END]
> >
[< < + > > -END]
< -END]c,
state[Cons x2 (Cons k ls), x1, zeroes, stdin, stdout])
(c, state[Cons x2 (Cons (x1 * x2 + k) ls), 0, zeroes, stdin, stdout]))
as H.
induction x1.
intros.
repeat bf_step.
intros; bf_step.
assert
(forall x1 x2 k i c,
iter (< [< + > > > + < < -END]> >[< < + > > -END]< -c,
state[Cons x2 (Cons (i*x2+k) ls), S x1, zeroes, stdin, stdout])
(c,
state[Cons x2 (Cons (S i*x2+k) ls), x1, zeroes, stdin, stdout]))
as Hstep.
intros.
bf_step; simpl.
assert
(forall x1 x2 k k' c,
iter ([< + > > > + < < -END]c,
state[Cons (k) ls, x2, Cons x1 (Cons k' zeroes), stdin, stdout])
(c, state[Cons (x2+k) ls, 0, Cons x1 (Cons (x2+k') zeroes), stdin, stdout]))
as Hsomething.
intros x x'.
induction x'.
intros.
repeat bf_step.
intros.
repeat bf_step; simpl.
replace (S (x' + k')) with (x' + S k') by auto with arith.
replace (S (x' + k1)) with (x' + S k1) by auto with arith.
apply IHx'.
apply (iter_trans _
(> >[< < + > > -END]< - c0,
state[Cons (x3+(i*x3+k0)) ls, 0,
Cons (S x0) (Cons (x3+0) zeroes), stdin, stdout])).
apply (iter_trans _
([< + > > > + < < - END]> > [< < + > > - END]< - c0,
state[Cons (i * x3 + k0) ls, x3,
Cons (S x0) (Cons 0 zeroes), stdin, stdout])).
bf_step.
apply (Hsomething (S x0) x3 (i*x3+k0) 0).
clear Hsomething.
do 2 bf_step; simpl.
apply (iter_trans
_ (< - c0,
state[Cons (S x0) (Cons x3 (Cons (x3+(i*x3+k0)) ls)),
0, zeroes, stdin, stdout])).
rewrite plus_0_r.
assert
(forall k k',
iter
([< < + > > - END]< - c0,
state[Cons (S x0) (Cons k (Cons (k' + (i * k' + k0)) ls)),
x3, zeroes, stdin, stdout])
(< - c0,
state[Cons (S x0) (Cons (x3+k) (Cons (k' + (i * k' + k0)) ls)),
0, zeroes, stdin, stdout]))
as Hind.
induction x3.
intros.
repeat bf_step.
intros.
repeat bf_step; simpl.
replace (S (x3 + k1)) with (x3 + S k1) by auto with arith.
apply (IHx3 (S k1)).
rewrite <- (plus_0_r x3) at 4.
apply Hind.
repeat bf_step; simpl.
replace (x3 + (i * x3 + k0)) with (x3 + i * x3 + k0) by auto with arith.
repeat bf_step.
apply (iter_trans
_
([< [< + > > > + < < - END]> > [< < + > > - END]< - END]c,
state[Cons x2 (Cons (x2+k) ls), x1, zeroes, stdin, stdout])).
rewrite <- (plus_0_l k) at 1.
rewrite <- (mult_0_l x2) at 1.
rewrite <- (mult_1_l x2) at 4.
apply (Hstep x1 x2 k 0).
replace (x2 + x1*x2 + k) with (x1*x2+(x2+k)).
apply (IHx1 x2 (x2+k)).
rewrite (plus_comm x2 (x1*x2)).
rewrite plus_assoc.
reflexivity.
intros.
apply (iter_trans
_
(<[-END]<END, state[Cons x2 (Cons (x1*x2) ls), 0, zeroes, stdin, stdout])).
rewrite <- (plus_0_r (x1*x2)).
apply (H x1 x2 0).
repeat bf_step; simpl.
assert
(forall ls x rs stdin stdout c,
iter
([-END]c, state[ls, x, rs, stdin, stdout])
(c, state[ls, 0, rs, stdin, stdout]))
as Hreset.
intros; induction x; repeat bf_step; apply IHx.
apply (iter_trans _ ([-END]< END, state[Cons (x1*x2) ls, x2, zeroes, stdin, stdout])).
bf_step.
apply (iter_trans _ (< END, state[Cons (x1*x2) ls, 0, zeroes, stdin, stdout])).
apply (Hreset (Cons (x1*x2) ls) x2 zeroes stdin stdout (< END)).
repeat bf_step.
Qed.
Corollary add_two_numbers :
forall n m stdin,
iter (push n; (push m; add), init stdin)
(END, state[zeroes, m+n, zeroes, stdin, nil]).
Proof.
unfold init.
intros.
apply (about_sequence
(push n) (push m; add) _
state[zeroes, n, zeroes, stdin, nil]).
apply (iter_trans _
(END, state[Cons 0 zeroes, n, zeroes, stdin, nil])).
apply about_push.
bf_step.
apply (about_sequence
(push m) add _
state[Cons n zeroes, m, zeroes, stdin, nil]).
apply about_push.
apply about_add.
Qed.