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(*
* Vericert: Verified high-level synthesis.
* Copyright (C) 2020 Yann Herklotz <yann@yannherklotz.com>
* 2020 James Pollard <j@mes.dev>
*
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <https://www.gnu.org/licenses/>.
*)
Require Import Coq.ZArith.BinInt.
Require Import Coq.micromega.Lia.
Require Import Coq.ZArith.ZArith.
Local Open Scope Z_scope.
Module ZLib.
Lemma mod2_cases: forall (n: Z), n mod 2 = 0 \/ n mod 2 = 1.
Proof.
intros. pose proof (Z.mod_pos_bound n 2). lia.
Qed.
Lemma div_mul_undo: forall a b,
b <> 0 ->
a mod b = 0 ->
a / b * b = a.
Proof.
intros.
pose proof Z.div_mul_cancel_l as A. specialize (A a 1 b).
replace (b * 1) with b in A by lia.
rewrite Z.div_1_r in A.
rewrite Z.mul_comm.
rewrite <- Z.divide_div_mul_exact; try assumption.
- apply A; congruence.
- apply Z.mod_divide; assumption.
Qed.
Lemma sub_mod_0: forall (a b m: Z),
a mod m = 0 ->
b mod m = 0 ->
(a - b) mod m = 0.
Proof.
intros *. intros E1 E2.
rewrite Zminus_mod.
rewrite E1. rewrite E2.
reflexivity.
Qed.
Lemma add_mod_0: forall a b m : Z,
a mod m = 0 ->
b mod m = 0 ->
(a + b) mod m = 0.
Proof.
intros *. intros E1 E2.
rewrite Zplus_mod.
rewrite E1. rewrite E2.
reflexivity.
Qed.
Lemma Z_mod_mult': forall a b : Z,
(a * b) mod a = 0.
Proof.
intros. rewrite Z.mul_comm. apply Z_mod_mult.
Qed.
Lemma mod_add_r: forall a b,
b <> 0 ->
(a + b) mod b = a mod b.
Proof.
intros. rewrite <- Z.add_mod_idemp_r by lia.
rewrite Z.mod_same by lia.
rewrite Z.add_0_r.
reflexivity.
Qed.
Lemma mod_pow2_same_bounds: forall a n,
a mod 2 ^ n = a ->
0 <= n ->
0 <= a < 2 ^ n.
Proof.
intros. rewrite <- H. apply Z.mod_pos_bound.
apply Z.pow_pos_nonneg; lia.
Qed.
Lemma pow2_nonneg: forall n,
0 <= 2 ^ n.
Proof.
intros. apply Z.pow_nonneg. lia.
Qed.
Lemma pow2_pos: forall n,
0 <= n ->
0 < 2 ^ n.
Proof.
intros. apply Z.pow_pos_nonneg; lia.
Qed.
Lemma pow2_times2: forall i,
0 < i ->
2 ^ i = 2 * 2 ^ (i - 1).
Proof.
intros.
rewrite <- Z.pow_succ_r by lia.
f_equal.
lia.
Qed.
Lemma pow2_div2: forall i,
0 <= i ->
2 ^ (i - 1) = 2 ^ i / 2.
Proof.
intros.
assert (i = 0 \/ 0 < i) as C by lia. destruct C as [C | C].
- subst. reflexivity.
- rewrite Z.pow_sub_r by lia.
reflexivity.
Qed.
Lemma div_mul_undo_le: forall a b,
0 <= a ->
0 < b ->
a / b * b <= a.
Proof.
intros.
pose proof (Zmod_eq_full a b) as P.
pose proof (Z.mod_bound_pos a b) as Q.
lia.
Qed.
Lemma testbit_true_nonneg: forall a i,
0 <= a ->
0 <= i ->
Z.testbit a i = true ->
2 ^ i <= a.
Proof.
intros.
apply Z.testbit_true in H1; [|assumption].
pose proof (pow2_pos i H0).
eapply Z.le_trans; [| apply (div_mul_undo_le a (2 ^ i)); lia].
replace (2 ^ i) with (1 * 2 ^ i) at 1 by lia.
apply Z.mul_le_mono_nonneg_r; [lia|].
pose proof (Z.div_pos a (2 ^ i)).
assert (a / 2 ^ i <> 0); [|lia].
intro E. rewrite E in H1. cbv in H1. discriminate H1.
Qed.
Lemma range_div_pos: forall a b c d,
0 < d ->
a <= b <= c ->
a / d <= b / d <= c / d.
Proof.
intuition idtac.
- apply (Z.div_le_mono _ _ _ H H1).
- apply (Z.div_le_mono _ _ _ H H2).
Qed.
Lemma testbit_true_nonneg': forall a i,
0 <= i ->
2 ^ i <= a < 2 ^ (i + 1) ->
Z.testbit a i = true.
Proof.
intros.
apply Z.testbit_true; [assumption|].
destruct H0 as [A B].
pose proof (pow2_pos i H) as Q.
apply (Z.div_le_mono _ _ _ Q) in A.
rewrite Z_div_same in A by lia.
pose proof (Z.div_lt_upper_bound a (2 ^ i) 2 Q) as P.
rewrite Z.mul_comm in P.
replace i with (i + 1 - 1) in P by lia.
rewrite <- pow2_times2 in P by lia.
specialize (P B).
replace (i + 1 - 1) with i in P by lia.
replace (a / 2 ^ i) with 1 by lia.
reflexivity.
Qed.
Lemma testbit_false_nonneg: forall a i,
0 <= a < 2 ^ i ->
0 < i ->
Z.testbit a (i - 1) = false ->
a < 2 ^ (i - 1).
Proof.
intros.
assert (2 ^ (i - 1) <= a < 2 ^ i \/ a < 2 ^ (i - 1)) as C by lia.
destruct C as [C | C]; [exfalso|assumption].
assert (Z.testbit a (i - 1) = true); [|congruence].
replace i with (i - 1 + 1) in C at 2 by lia.
apply testbit_true_nonneg'; lia.
Qed.
Lemma signed_bounds_to_sz_pos: forall sz n,
- 2 ^ (sz - 1) <= n < 2 ^ (sz - 1) ->
0 < sz.
Proof.
intros.
assert (0 < sz \/ sz - 1 < 0) as C by lia.
destruct C as [C | C]; [assumption|exfalso].
rewrite Z.pow_neg_r in H by assumption.
lia.
Qed.
Lemma two_digits_encoding_inj_lo: forall base a b c d: Z,
0 <= b < base ->
0 <= d < base ->
base * a + b = base * c + d ->
b = d.
Proof.
intros.
pose proof Z.mod_unique as P.
specialize P with (b := base) (q := c) (r := d).
specialize P with (2 := H1).
rewrite P by lia.
rewrite <- Z.add_mod_idemp_l by lia.
rewrite Z.mul_comm.
rewrite Z.mod_mul by lia.
rewrite Z.add_0_l.
rewrite Z.mod_small by lia.
reflexivity.
Qed.
Lemma two_digits_encoding_inj_hi: forall base a b c d: Z,
0 <= b < base ->
0 <= d < base ->
base * a + b = base * c + d ->
a = c.
Proof.
intros. nia.
Qed.
Lemma Z_to_nat_neg: forall (n: Z),
n < 0 ->
Z.to_nat n = 0%nat.
Proof.
intros. destruct n; (lia||reflexivity).
Qed.
End ZLib.
Module ZExtra.
Lemma mod_0_bounds :
forall x y m,
0 < m ->
x mod m = 0 ->
y mod m = 0 ->
x <> y ->
x + m > y ->
y + m <= x.
Proof.
intros x y m.
intros POS XMOD YMOD NEQ H.
destruct (Z_le_gt_dec (y + m) x); eauto.
apply Znumtheory.Zmod_divide in YMOD; try lia.
apply Znumtheory.Zmod_divide in XMOD; try lia.
inversion XMOD as [x']; subst; clear XMOD.
inversion YMOD as [y']; subst; clear YMOD.
assert (x' <> y') as NEQ' by lia; clear NEQ.
rewrite <- Z.mul_succ_l in H.
rewrite <- Z.mul_succ_l in g.
apply Zmult_gt_reg_r in H;
apply Zmult_gt_reg_r in g; lia.
Qed.
Lemma Ple_not_eq :
forall x y,
(x < y)%positive -> x <> y.
Proof. lia. Qed.
Lemma Pge_not_eq :
forall x y,
(y < x)%positive -> x <> y.
Proof. lia. Qed.
Lemma Ple_Plt_Succ :
forall x y n,
(x <= y)%positive -> (x < y + n)%positive.
Proof. lia. Qed.
End ZExtra.
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