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Require Export Bedrock.Word Bedrock.Nomega.
Require Import NArith PArith Ndigits Nnat NPow Compare_dec.
Require Import FunctionalExtensionality.
Hint Rewrite wordToN_nat Nat2N.inj_add N2Nat.inj_add
Nat2N.inj_mul N2Nat.inj_mul Npow2_nat : N.
Delimit Scope wordize_scope with wordize.
Open Scope wordize_scope.
Notation "& x" := (wordToN x) (at level 30) : wordize_scope.
Notation "** x" := (NToWord _ x) (at level 30) : wordize_scope.
Notation "$ x" := (natToWord x) (at level 30) : wordize_scope.
Section Definitions.
Definition take (k n: nat) (p: (k <= n)%nat) (w: word n): word k.
replace n with (k + (n - k)) in * by abstract omega.
refine (split1 k _ w).
Defined.
Definition shiftr {n} (w: word n) (k: nat): word n.
destruct (le_dec k n).
- replace n with (k + (n - k)) in * by abstract intuition.
refine (sext (take k (k + (n - k)) l w) _).
- exact (wzero n).
Defined.
Definition mask {n} (m: nat) (w: word n): word n.
destruct (le_dec m n).
- replace n with (m + (n - m)) in * by (abstract intuition).
refine (w ^& (zext (wones m) (n - m))).
- exact w.
Defined.
End Definitions.
Section WordizeUtil.
Lemma to_nat_lt: forall x b, (x < b)%N <-> (N.to_nat x < N.to_nat b)%nat.
Proof. (* via Nat2N.inj_compare *) Admitted.
Lemma of_nat_lt: forall x b, (x < b)%nat <-> (N.of_nat x < N.of_nat b)%N.
Proof. (* via N2Nat.inj_compare *) Admitted.
Lemma Npow2_spec : forall n, Npow2 n = N.pow 2 (N.of_nat n).
Proof. (* by induction and omega *) Admitted.
Lemma NToWord_wordToN: forall sz x, NToWord sz (wordToN x) = x.
Proof.
intros.
rewrite NToWord_nat.
rewrite wordToN_nat.
rewrite Nat2N.id.
rewrite natToWord_wordToNat.
intuition.
Qed.
Lemma wordToN_NToWord: forall sz x, (x < Npow2 sz)%N -> wordToN (NToWord sz x) = x.
Proof.
intros.
rewrite NToWord_nat.
rewrite wordToN_nat.
rewrite <- (N2Nat.id x).
apply Nat2N.inj_iff.
rewrite Nat2N.id.
apply natToWord_inj with (sz:=sz);
try rewrite natToWord_wordToNat;
intuition.
- apply wordToNat_bound.
- rewrite <- Npow2_nat; apply to_nat_lt; assumption.
Qed.
Lemma word_size_bound : forall {n} (w: word n), (&w < Npow2 n)%N.
Proof.
intros; pose proof (wordToNat_bound w) as B;
rewrite of_nat_lt in B;
rewrite <- Npow2_nat in B;
rewrite N2Nat.id in B;
rewrite <- wordToN_nat in B;
assumption.
Qed.
Lemma Npow2_gt0 : forall x, (0 < Npow2 x)%N.
Proof.
intros; induction x.
- simpl; apply N.lt_1_r; intuition.
- replace (Npow2 (S x)) with (2 * (Npow2 x))%N by intuition.
apply (N.lt_0_mul 2 (Npow2 x)); left; split; apply N.neq_0_lt_0.
+ intuition; inversion H.
+ apply N.neq_0_lt_0 in IHx; intuition.
Qed.
End WordizeUtil.
(** Wordization Lemmas **)
Section Wordization.
Lemma wordize_plus: forall {n} (x y: word n) (b: N),
(b <= Npow2 n)%N
-> (&x < b)%N
-> (&y < (Npow2 n - b))%N
-> (&x + &y)%N = & (x ^+ y).
Proof.
intros.
unfold wplus, wordBin.
rewrite wordToN_NToWord; intuition.
apply (N.lt_le_trans _ (b + &y)%N _).
- apply N.add_lt_le_mono; try assumption; intuition.
- replace (Npow2 n) with (b + Npow2 n - b)%N by nomega.
replace (b + Npow2 n - b)%N with (b + (Npow2 n - b))%N by (
replace (b + (Npow2 n - b))%N with ((Npow2 n - b) + b)%N by nomega;
rewrite (N.sub_add b (Npow2 n)) by assumption;
nomega).
apply N.add_le_mono_l; try nomega.
apply N.lt_le_incl; assumption.
Qed.
Lemma wordize_mult: forall {n} (x y: word n) (b: N),
(1 < n)%nat -> (0 < b)%N
-> (&x < b)%N
-> (&y < (Npow2 n) / b)%N
-> (&x * &y)%N = & (x ^* y).
Proof.
intros; unfold wmult, wordBin.
rewrite wordToN_NToWord; intuition.
apply (N.lt_le_trans _ (b * ((Npow2 n) / b))%N _).
- apply N.mul_lt_mono; assumption.
- apply N.mul_div_le; nomega.
Qed.
Lemma wordize_and: forall {n} (x y: word n),
N.land (&x) (&y) = & (x ^& y).
Proof.
intros; pose proof (Npow2_gt0 n).
pose proof (word_size_bound x).
pose proof (word_size_bound y).
induction n.
- rewrite (shatter_word_0 x) in *.
rewrite (shatter_word_0 y) in *.
simpl; intuition.
- rewrite (shatter_word x) in *.
rewrite (shatter_word y) in *.
induction (whd x), (whd y).
+ admit.
+ admit.
+ admit.
+ admit.
Qed.
Lemma wordize_shiftr: forall {n} (x: word n) (k: nat),
(N.shiftr_nat (&x) k) = & (shiftr x k).
Proof.
intros.
admit.
Qed.
End Wordization.
Section Bounds.
Theorem constant_bound_N : forall {n} (k: word n),
(& k < & k + 1)%N.
Proof. intros; nomega. Qed.
Lemma let_bound : forall {n} (x: word n) (f: word n -> word n) xb fb,
(& x < xb)%N
-> (forall x', & x' < xb -> & (f x') < fb)%N
-> ((let k := x in &(f k)) < fb)%N.
Proof. intros; eauto. Qed.
Definition Nlt_dec (x y: N): {(x < y)%N} + {(x >= y)%N}.
refine (
let c := N.compare x y in
match c as c' return c = c' -> _ with
| Lt => fun _ => left _
| _ => fun _ => right _
end eq_refl); admit.
Defined.
Theorem wplus_bound : forall {n} (w1 w2 : word n) b1 b2,
(&w1 < b1)%N
-> (&w2 < b2)%N
-> (&(w1 ^+ w2) < b1 + b2)%N.
Proof.
intros.
destruct (N.compare (b1 + b2)%N (Npow2 n));
rewrite <- wordize_plus with (b := b1);
try apply N.add_lt_mono;
try assumption.
- apply N.add_lt_mono; assumption.
- admit.
- admit.
- apply N.add_lt_mono; assumption.
- apply N.add_lt_mono; assumption.
Qed.
Theorem wmult_bound : forall (w1 w2 : word n) b1 b2,
w1 <= b1
-> w2 <= b2
-> w1 ^* w2 <= b1 * b2.
Proof.
preomega.
destruct (le_lt_dec (pow2 n) (N.to_nat b1 * N.to_nat b2)).
specialize (wordToNat_bound (w1 ^* w2)); nomega.
rewrite wmult_alt.
unfold wmultN, wordBinN.
rewrite wordToNat_natToWord_idempotent.
ge_to_le_nat.
apply Mult.mult_le_compat; nomega.
pre_nomega.
apply Lt.le_lt_trans with (N.to_nat b1 * N.to_nat b2); auto.
apply Mult.mult_le_compat; nomega.
Qed.
Theorem shiftr_bound : forall (w : word n) b bits,
w <= b
-> shiftr w bits <= N.shiftr b (N.of_nat bits).
Proof.
admit.
Qed.
Theorem mask_bound : forall (w : word n) m,
mask m w <= Npow2 m - 1.
Proof.
admit.
Qed.
Theorem mask_update_bound : forall (w : word n) b m,
w <= b
-> mask m w <= (N.min b (Npow2 m - 1)).
Proof.
admit.
Qed.
End Bounds.
(** Wordization Tactics **)
Ltac wordize_ast :=
repeat match goal with
| [ H: (& ?x < ?b)%N |- context[((& ?x) + (& ?y))%N] ] => rewrite (wordize_plus x y b)
| [ H: (& ?x < ?b)%N |- context[((& ?x) * (& ?y))%N] ] => rewrite (wordize_mult x y b)
| [ |- context[N.land (& ?x) (& ?y)] ] => rewrite (wordize_and x y)
| [ |- context[N.shiftr (& ?x) ?k] ] => rewrite (wordize_shiftr x k)
| [ |- (_ < _ / _)%N ] => unfold N.div; simpl
| [ |- context[Npow2 _] ] => simpl
| [ |- (?x < ?c)%N ] => assumption
| [ |- _ = _ ] => reflexivity
end.
Ltac lt_crush := try abstract (clear; vm_compute; intuition).
(** Bounding Tactics **)
(** Examples **)
Module WordizationExamples.
Lemma wordize_example0: forall (x y z: word 16),
(wordToN x < 10)%N ->
(wordToN y < 10)%N ->
(wordToN z < 10)%N ->
& (x ^* y) = (&x * &y)%N.
Proof.
intros.
wordize_ast; lt_crush.
transitivity 10%N; try assumption; lt_crush.
Qed.
End WordizationExamples.
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