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Require Export Bedrock.Word Bedrock.Nomega.
Require Import NPeano NArith PArith Ndigits Nnat Pnat.
Require Import Arith Ring List Compare_dec Bool.
Require Import FunctionalExtensionality.
Delimit Scope wordize_scope with wordize.
Local Open Scope wordize_scope.
Notation "& x" := (wordToNat x) (at level 30) : wordize_scope.
Lemma word_size_bound : forall {n} (w: word n),
(& w <= pow2 n - 1)%nat.
Proof. intros; assert (B := wordToNat_bound w); omega. Qed.
Lemma pow2_gt0 : forall x, (pow2 x > 0)%nat.
Proof. intros; induction x; simpl; intuition. Qed.
Lemma wordize_plus: forall {n} (x y: word n) (b: nat),
(&x <= b)%nat
-> (&y <= (pow2 n - b - 1))%nat
-> (&x + &y) = wordToNat (x ^+ y).
Proof.
intros; unfold wplus, wordBin;
repeat rewrite wordToN_nat in *;
repeat rewrite NToWord_nat in *;
rewrite <- Nat2N.inj_add; rewrite Nat2N.id.
destruct (wordToNat_natToWord n (&x + &y)) as [k HW];
destruct HW as [HW HK]; rewrite HW; clear HW.
assert ((&x) + (&y) <= (pow2 n - 1))%nat by (
pose proof (word_size_bound x);
pose proof (word_size_bound y);
omega).
assert (k * pow2 n <= pow2 n - 1)%nat by omega.
destruct k; subst; simpl in *; intuition;
clear H H0 H1 HK;
contradict H2.
pose proof (pow2_gt0 n).
induction k; simpl; intuition.
Qed.
Lemma wordize_mult: forall {n} (x y: word n) (b: nat),
(1 <= n)%nat
-> (&x <= b)%nat
-> (&y <= (pow2 n - 1) / b)%nat
-> (&x * &y)%nat = wordToNat (x ^* y).
Proof.
intros; unfold wmult, wordBin;
repeat rewrite wordToN_nat in *;
repeat rewrite NToWord_nat in *;
rewrite <- Nat2N.inj_mul; rewrite Nat2N.id.
destruct (wordToNat_natToWord n (&x * &y)) as [k HW];
destruct HW as [HW HK]; rewrite HW; clear HW.
assert ((&x) * (&y) <= (pow2 n - 1))%nat. {
destruct (Nat.eq_dec b 0); subst;
try abstract (replace (&x) with 0 by intuition; omega).
transitivity ((&x * (pow2 n - 1)) / b). {
transitivity (&x * ((pow2 n - 1) / b)).
+ apply Nat.mul_le_mono_nonneg_l; intuition.
+ apply Nat.div_mul_le; assumption.
}
transitivity ((b * (pow2 n - 1)) / (b * 1)). {
rewrite Nat.div_mul_cancel_l; intuition.
rewrite Nat.div_1_r.
apply Nat.div_le_upper_bound; intuition.
apply Nat.mul_le_mono_pos_r; intuition.
induction n; try inversion H; simpl; intuition.
pose proof (pow2_gt0 n).
omega.
}
rewrite Nat.div_mul_cancel_l; intuition.
rewrite Nat.div_1_r; intuition.
}
assert (k * (pow2 n) <= pow2 n - 1)%nat by omega.
induction k; try omega.
assert (pow2 n - 1 < S k * (pow2 n))%nat. {
destruct n; try inversion H; simpl; try omega.
pose proof (pow2_gt0 n).
induction k; try omega.
transitivity (pow2 n + pow2 n + pow2 n); try omega.
replace (pow2 n + (pow2 n + 0) + S k * _)
with (pow2 n + pow2 n + pow2 n +
(pow2 n + k * (pow2 n + (pow2 n + 0)))).
apply Nat.lt_add_pos_r.
apply Nat.add_pos_nonneg; try omega; intuition.
simpl; omega.
}
contradict H3; omega.
Qed.
Lemma testbit_pos0: forall p k,
testbit (Pos.to_nat p~0) k =
match k with | O => false | S k' => testbit (Pos.to_nat p) k' end.
intros; destruct k.
- unfold testbit; simpl; intuition.
unfold odd; replace (even _) with true; intuition.
symmetry; apply even_spec.
exists (Pos.to_nat p).
replace (p~0)%positive with (2*p)%positive by intuition.
apply Pos2Nat.inj_mul.
- rewrite (Pos2Nat.inj_xO); intuition.
rewrite Nat.testbit_even_succ; intuition.
Qed.
Lemma testbit_pos1: forall p k,
testbit (Pos.to_nat p~1) k =
match k with | O => true | S k' => testbit (Pos.to_nat p) k' end.
intros; destruct k.
- unfold testbit; simpl; intuition.
apply odd_spec; exists (Pos.to_nat p).
replace (p~1)%positive with (2*p + 1)%positive by intuition.
rewrite Pos2Nat.inj_add; rewrite Pos2Nat.inj_mul; intuition.
- rewrite (Pos2Nat.inj_xI); intuition.
replace (S (2 * Pos.to_nat p)) with ((2 * Pos.to_nat p) + 1) by intuition.
rewrite Nat.testbit_odd_succ; intuition.
Qed.
Lemma testbit_zero: testbit 0 = (fun (_: nat) => false).
apply functional_extensionality; intros;
rewrite testbit_0_l; intuition.
Qed.
Definition even_dec (x: nat): {exists x', x = 2*x'} + {exists x', x = 2*x' + 1}.
refine (if (bool_dec (odd x) true) then right _ else left _).
- apply odd_spec in _H; destruct _H; exists x0. abstract intuition.
- unfold odd in _H.
assert (even x = true) by abstract (destruct (even x); intuition).
apply even_spec in H; destruct H; exists x0; abstract intuition.
Defined.
Lemma testbit_spec: forall (n x y: nat), (x < pow2 n)%nat -> (y < pow2 n)%nat ->
(forall k, (k < n)%nat -> testbit x k = testbit y k) -> x = y.
Proof.
intro n; induction n; intros. simpl in *; omega.
destruct (even_dec x) as [px|px], (even_dec y) as [py|py];
destruct px as [x' px], py as [y' py];
rewrite px in *; rewrite py in *;
clear x y px py;
replace (pow2 (S n)) with (2 * (pow2 n)) in * by intuition;
assert (x' < pow2 n)%nat by intuition;
assert (y' < pow2 n)%nat by intuition.
- apply Nat.mul_cancel_l; intuition.
apply IHn; intuition.
assert (S k < S n)%nat as Z by intuition.
pose proof (H1 (S k) Z); intuition.
repeat rewrite testbit_even_succ in H5; intuition.
- assert (0 < S n)%nat as Z by intuition.
apply (H1 0) in Z.
rewrite testbit_even_0 in Z;
rewrite testbit_odd_0 in Z;
inversion Z.
- assert (0 < S n)%nat as Z by intuition.
apply (H1 0) in Z.
rewrite testbit_even_0 in Z;
rewrite testbit_odd_0 in Z;
inversion Z.
- apply Nat.add_cancel_r; apply Nat.mul_cancel_l; intuition.
apply IHn; intuition.
assert (S k < S n)%nat as Z by intuition.
pose proof (H1 (S k) Z); intuition.
repeat rewrite testbit_odd_succ in H5; intuition.
Qed.
Lemma odd_def0: forall x, odd (S (x*2)) = true.
Proof. intros; intuition. apply odd_spec. exists x. omega. Qed.
Lemma odd_def1: forall x, odd (x * 2) = false.
Proof.
intros; intuition; unfold odd.
replace (even _) with true; intuition; symmetry.
rewrite even_spec.
exists x. omega.
Qed.
Inductive BinF := | binF: forall (a b c d: bool), BinF.
Definition applyBinF (f: BinF) (x y: bool) :=
match f as f' return f = f' -> _ with
| binF a b c d => fun _ =>
if x
then if y
then a
else b
else if y
then c
else d
end eq_refl.
Definition boolToBinF (f: bool -> bool -> bool): {g: BinF | f = applyBinF g}.
intros; exists (binF (f true true) (f true false) (f false true) (f false false));
abstract (
apply functional_extensionality; intro x;
apply functional_extensionality; intro y;
destruct x, y; unfold applyBinF; simpl; intuition).
Qed.
Lemma testbit_odd_succ': forall x k, testbit (S (x * 2)) (S k) = testbit x k.
intros.
replace (S (x * 2)) with ((2 * x) + 1) by omega.
apply testbit_odd_succ.
Qed.
Lemma testbit_even_succ': forall x k, testbit (x * 2) (S k) = testbit x k.
intros; replace (x * 2) with (2 * x) by omega; apply testbit_even_succ.
Qed.
Lemma testbit_odd_zero': forall x, testbit (S (x * 2)) 0 = true.
intros.
replace (S (x * 2)) with ((2 * x) + 1) by omega.
apply testbit_odd_0.
Qed.
Lemma testbit_even_zero': forall x, testbit (x * 2) 0 = false.
intros; replace (x * 2) with (2 * x) by omega; apply testbit_even_0.
Qed.
Lemma testbit_bitwp: forall {n} (x y: word n) f k, (k < n)%nat ->
testbit (wordToNat (bitwp f x y)) k = f (testbit (&x) k) (testbit (&y) k).
Proof.
intros.
pose proof (shatter_word x);
pose proof (shatter_word y);
simpl in *.
induction n. inversion H. clear IHn; rewrite H0, H1; clear H0 H1; simpl.
replace f with (applyBinF (proj1_sig (boolToBinF f))) in *
by (destruct (boolToBinF f); simpl; intuition);
generalize (boolToBinF f) as g; intro g;
destruct g as [g pg]; simpl; clear pg f.
revert x y H; generalize k n; clear k n; induction k, n;
intros; try omega.
- destruct g as [a b c d], (whd x), (whd y);
destruct a, b, c, d; unfold applyBinF in *; clear H;
repeat rewrite testbit_odd_zero';
repeat rewrite testbit_even_zero';
reflexivity.
- destruct g as [a b c d], (whd x), (whd y);
destruct a, b, c, d; unfold applyBinF in *; clear H;
repeat rewrite testbit_odd_zero';
repeat rewrite testbit_even_zero';
reflexivity.
- assert (k < S n)%nat as HB by omega;
pose proof (IHk n (wtl x) (wtl y) HB) as Z; clear HB IHk.
assert (forall {m} (w: word (S m)),
&w = if whd w
then S (& wtl w * 2)
else & wtl w * 2) as r0. {
clear H Z x y; intros.
pose proof (shatter_word w); simpl in H; rewrite H; clear H; simpl.
destruct (whd w); intuition.
} repeat rewrite <- r0 in Z; clear r0.
assert (forall {m} (a b: word (S m)),
& bitwp (applyBinF g) a b
= if applyBinF g (whd a) (whd b)
then S (& bitwp (applyBinF g) (wtl a) (wtl b) * 2)
else & bitwp (applyBinF g) (wtl a) (wtl b) * 2) as r1. {
clear H Z x y; intros.
pose proof (shatter_word a); pose proof (shatter_word b);
simpl in *; rewrite H; rewrite H0; clear H H0; simpl.
destruct (applyBinF g (whd a) (whd b)); intuition.
} repeat rewrite <- r1 in Z; clear r1.
destruct g as [a b c d], (whd x), (whd y);
destruct a, b, c, d; unfold applyBinF in *; clear H;
repeat rewrite testbit_odd_succ';
repeat rewrite testbit_even_succ';
assumption.
Qed.
(* (forall k, testbit x k = testbit y k) <-> x = y. *)
Lemma wordize_and: forall {n} (x y: word n),
(Nat.land (&x) (&y))%nat = & (x ^& y).
Proof.
intros n x y.
pose proof (pow2_gt0 n).
assert (&x < pow2 n)%nat by (pose proof (word_size_bound x); intuition).
assert (&y < pow2 n)%nat by (pose proof (word_size_bound y); intuition).
apply (testbit_spec n).
- induction n.
+ simpl in *; intuition.
replace (&x) with 0 by intuition.
replace (&y) with 0 by intuition.
simpl; intuition.
+ admit.
- pose (word_size_bound (x ^& y)); intuition.
- intros; rewrite land_spec.
unfold wand; rewrite testbit_bitwp; intuition.
Qed.
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