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diff --git a/src/Assembly/Wordize.v b/src/Assembly/Wordize.v new file mode 100644 index 000000000..e5ad3f066 --- /dev/null +++ b/src/Assembly/Wordize.v @@ -0,0 +1,507 @@ + +Require Export Bedrock.Word Bedrock.Nomega. +Require Import NArith PArith Ndigits Nnat NPow NPeano Ndec Compare_dec. +Require Import FunctionalExtensionality ProofIrrelevance. +Require Import QhasmUtil QhasmEvalCommon. + +Hint Rewrite wordToN_nat Nat2N.inj_add N2Nat.inj_add + Nat2N.inj_mul N2Nat.inj_mul Npow2_nat : N. + +Open Scope nword_scope. + +Section WordizeUtil. + Lemma break_spec: forall (m n: nat) (x: word n) low high, + (low, high) = break m x + -> &x = (&high * Npow2 m + &low)%N. + Proof. + intros; unfold break in *; destruct (le_dec m n); + inversion H; subst; clear H; simpl. + Admitted. + + Lemma mask_wand : forall (n: nat) (x: word n) m b, + (& (mask (N.to_nat m) x) < b)%N + -> (& (x ^& (@NToWord n (N.ones m))) < b)%N. + Proof. + Admitted. + + Lemma convS_id: forall A x p, x = (@convS A A x p). + Proof. + intros; unfold convS; vm_compute. + replace p with (eq_refl A); intuition. + apply proof_irrelevance. + Qed. + + Lemma word_param_eq: forall n m, word n = word m -> n = m. + Proof. (* TODO: How do we prove this *) Admitted. + + Lemma word_conv_eq: forall {n m} (y: word m) p, + &y = &(@convS (word m) (word n) y p). + Proof. + intros. + revert p. + destruct (Nat.eq_dec n m). + + - rewrite e; intros; apply f_equal; apply convS_id. + + - intros; contradict n0. + apply word_param_eq; rewrite p; intuition. + Qed. + + 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. + + Lemma natToWord_convS: forall {n m} (x: word n) p, + & x = & @convS (word n) (word m) x p. + Proof. admit. Qed. + + Lemma natToWord_combine: forall {n} (x: word n) k, + & x = & combine x (wzero k). + Proof. admit. Qed. + + Lemma natToWord_split1: forall {n} (x: word n) p, + & x = & split1 n 0 (convS x p). + Proof. admit. Qed. + + Lemma extend_bound: forall k n (p: (k <= n)%nat) (w: word k), + (& (extend p w) < Npow2 k)%N. + Proof. + intros. + assert (n = k + (n - k)) by abstract omega. + replace (& (extend p w)) with (& w); try apply word_size_bound. + unfold extend. + rewrite <- word_conv_eq. + unfold zext. + clear; revert w; induction (n - k). + + - intros. + assert (word k = word (k + 0)) as Z by intuition. + replace w with (split1 k 0 (convS w Z)). + replace (wzero 0) with (split2 k 0 (convS w Z)). + rewrite <- natToWord_split1 with (p := Z). + rewrite combine_split. + apply natToWord_convS. + + + admit. + + admit. + + - intros; rewrite <- natToWord_combine; intuition. + Qed. + + Lemma Npow2_split: forall a b, + (Npow2 (a + b) = (Npow2 a) * (Npow2 b))%N. + Proof. + intros; revert a. + induction b. + + - intros; simpl; replace (a + 0) with a; nomega. + + - intros. + replace (a + S b) with (S a + b) by intuition. + rewrite (IHb (S a)); simpl; clear IHb. + induction (Npow2 a), (Npow2 b); simpl; intuition. + rewrite Pos.mul_xO_r; intuition. + Qed. + + Lemma Npow2_ignore: forall {n} (x: word n), + x = NToWord _ (& x + Npow2 n). + Proof. intros. Admitted. + +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_plus: forall {n} (x y: word n), + if (overflows n (&x + &y)%N) + then (&x + &y)%N = (& (x ^+ y) + Npow2 n)%N + else (&x + &y)%N = & (x ^+ y). + Proof. Admitted. + + Lemma wordize_awc: forall {n} (x y: word n) (c: bool), + if (overflows n (&x + &y + (if c then 1 else 0))%N) + then (&x + &y + (if c then 1 else 0))%N = (&(addWithCarry x y c) + Npow2 n)%N + else (&x + &y + (if c then 1 else 0))%N = &(addWithCarry x y c). + Proof. Admitted. + + 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_mult: forall {n} (x y: word n) (b: N), + (&x * &y)%N = (&(x ^* y) + + &((EvalUtil.highBits (n/2) x) ^* (EvalUtil.highBits (n/2) y)) * Npow2 n)%N. + Proof. intros. Admitted. + + 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. + + Theorem constant_bound_nat : forall (n k: nat), + (N.of_nat k < Npow2 n)%N + -> (& (@natToWord n k) < (N.of_nat k) + 1)%N. + Proof. + intros. + rewrite wordToN_nat. + rewrite wordToNat_natToWord_idempotent; + try assumption; 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); + abstract (unfold c in *; try first [ + apply N.compare_eq_iff in _H + | apply N.compare_lt_iff in _H + | pose proof (N.compare_antisym x y) as _H0; + rewrite _H in _H0; simpl in _H0; + apply N.compare_lt_iff in _H0 ]; nomega). + 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 (Nlt_dec (b1 + b2)%N (Npow2 n)); + rewrite <- wordize_plus' with (b := b1); + try apply N.add_lt_mono; + try assumption. + + (* A couple inequality subgoals *) + Admitted. + + Theorem wmult_bound : forall {n} (w1 w2 : word n) b1 b2, + (1 < n)%nat + -> (&w1 < b1)%N + -> (&w2 < b2)%N + -> (&(w1 ^* w2) < b1 * b2)%N. + Proof. + intros. + destruct (Nlt_dec (b1 * b2)%N (Npow2 n)); + rewrite <- wordize_mult' with (b := b1); + try apply N.mul_lt_mono; + try assumption; + try nomega. + + (* A couple inequality subgoals *) + Admitted. + + Theorem shiftr_bound : forall {n} (w : word n) b bits, + (&w < b)%N + -> (&(shiftr w bits) < N.succ (N.shiftr_nat b bits))%N. + Proof. + intros. + assert (& shiftr w bits <= N.shiftr_nat b bits)%N. { + rewrite <- wordize_shiftr. + induction bits; unfold N.shiftr_nat in *; simpl; intuition. + + - unfold N.le, N.lt in *; rewrite H; intuition; inversion H0. + + - revert IHbits; + generalize (nat_iter bits N.div2 (& w)), + (nat_iter bits N.div2 b); + clear; intros x y H. + + admit. (* Monotonicity of N.div2 *) + } + + apply N.le_lteq in H0; destruct H0; nomega. + Qed. + + Theorem mask_bound : forall {n} (w : word n) m, + (n > 1)%nat -> + (&(mask m w) < Npow2 m)%N. + Proof. + intros. + unfold mask in *; destruct (le_dec m n); simpl; + try apply extend_bound. + + pose proof (word_size_bound w). + apply (N.le_lt_trans _ (Npow2 n) _). + + - unfold N.le, N.lt in *; rewrite H0; intuition; inversion H1. + + - clear H H0. + replace m with ((m - n) + n) by nomega. + replace (Npow2 n) with (1 * (Npow2 n))%N + by (rewrite N.mul_comm; nomega). + rewrite Npow2_split. + apply N.mul_lt_mono_pos_r. + + + apply Npow2_gt0. + + + assert (0 < m - n)%nat by omega. + induction (m - n); try inversion H; try abstract ( + simpl; replace 2 with (S 1) by omega; + apply N.lt_1_2). + + assert (0 < n1)%nat as Z by omega; apply IHn1 in Z. + apply (N.le_lt_trans _ (Npow2 n1) _). + + * admit. + * admit. + Qed. + + Theorem mask_update_bound : forall {n} (w : word n) b m, + (n > 1)%nat + -> (&w < b)%N + -> (&(mask m w) < (N.min b (Npow2 m)))%N. + Proof. + intros; unfold mask, N.min; + destruct (le_dec m n), + (N.compare b (Npow2 m)); + simpl; try assumption. + + Admitted. + +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 **) + +Ltac multi_apply0 A L := pose proof (L A). + +Ltac multi_apply1 A L := + match goal with + | [ H: A < ?b |- _] => pose proof (L A b H) + end. + +Ltac multi_apply2 A B L := + match goal with + | [ H1: A < ?b1, H2: B < ?b2 |- _] => pose proof (L A B b1 b2 H1 H2) + end. + +Ltac multi_recurse n T := + match goal with + | [ H: (T < _)%N |- _] => idtac + | _ => + is_var T; + let T' := (eval cbv delta [T] in T) in multi_recurse n T'; + match goal with + | [ H : T' < ?x |- _ ] => + pose proof (H : T < x) + end + + | _ => + match T with + | ?W1 ^+ ?W2 => + multi_recurse n W1; multi_recurse n W2; + multi_apply2 W1 W2 (@wplus_bound n) + + | ?W1 ^* ?W2 => + multi_recurse n W1; multi_recurse n W2; + multi_apply2 W1 W2 (@wmult_bound n) + + | mask ?m ?w => + multi_recurse n w; + multi_apply1 w (fun b => @mask_update_bound n w b) + + | mask ?m ?w => + multi_recurse n w; + pose proof (@mask_bound n w m) + + | ?x ^& (@NToWord _ (N.ones ?m)) => + multi_recurse n (mask (N.to_nat m) x); + match goal with + | [ H: (& (mask (N.to_nat m) x) < ?b)%N |- _] => + pose proof (@mask_wand n x m b H) + end + + | shiftr ?w ?bits => + multi_recurse n w; + match goal with + | [ H: (w < ?b)%N |- _] => + pose proof (@shiftr_bound n w b bits H) + end + + | NToWord _ ?k => pose proof (@constant_bound_N n k) + | natToWord _ ?k => pose proof (@constant_bound_nat n k) + | _ => pose proof (@word_size_bound n T) + end + end. + +Lemma unwrap_let: forall {n} (y: word n) (f: word n -> word n) (b: N), + (&(let x := y in f x) < b)%N <-> let x := y in (&(f x) < b)%N. +Proof. intuition. Qed. + +Ltac multi_bound n := + match goal with + | [|- let A := ?B in _] => + multi_recurse n B; intro; multi_bound n + | [|- ((let A := _ in _) < _)%N] => + apply unwrap_let; multi_bound n + | [|- (?W < _)%N ] => + multi_recurse n W + end. + +(** 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. + +Close Scope nword_scope. |