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Require Import Coq.ZArith.ZArith Coq.micromega.Lia Coq.ZArith.Znumtheory Coq.ZArith.Zpow_facts.
Require Import Crypto.Util.ZUtil.Hints.Core.
Require Import Crypto.Util.ZUtil.ZSimplify.Core.
Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem.
Require Import Crypto.Util.ZUtil.Tactics.LtbToLt.
Require Import Crypto.Util.ZUtil.Tactics.ReplaceNegWithPos.
Require Import Crypto.Util.Tactics.BreakMatch.
Require Import Crypto.Util.Tactics.DestructHead.
Local Open Scope Z_scope.
Module Z.
Lemma elim_mod : forall a b m, a = b -> a mod m = b mod m.
Proof. intros; subst; auto. Qed.
Hint Resolve elim_mod : zarith.
Lemma mod_add_full : forall a b c, (a + b * c) mod c = a mod c.
Proof. intros a b c; destruct (Z_zerop c); try subst; autorewrite with zsimplify; reflexivity. Qed.
Hint Rewrite mod_add_full : zsimplify.
Lemma mod_add_l_full : forall a b c, (a * b + c) mod b = c mod b.
Proof. intros a b c; rewrite (Z.add_comm _ c); autorewrite with zsimplify; reflexivity. Qed.
Hint Rewrite mod_add_l_full : zsimplify.
Lemma mod_add'_full : forall a b c, (a + b * c) mod b = a mod b.
Proof. intros a b c; rewrite (Z.mul_comm _ c); autorewrite with zsimplify; reflexivity. Qed.
Lemma mod_add_l'_full : forall a b c, (a * b + c) mod a = c mod a.
Proof. intros a b c; rewrite (Z.mul_comm _ b); autorewrite with zsimplify; reflexivity. Qed.
Hint Rewrite mod_add'_full mod_add_l'_full : zsimplify.
Lemma mod_add_l : forall a b c, b <> 0 -> (a * b + c) mod b = c mod b.
Proof. intros a b c H; rewrite (Z.add_comm _ c); autorewrite with zsimplify; reflexivity. Qed.
Lemma mod_add' : forall a b c, b <> 0 -> (a + b * c) mod b = a mod b.
Proof. intros a b c H; rewrite (Z.mul_comm _ c); autorewrite with zsimplify; reflexivity. Qed.
Lemma mod_add_l' : forall a b c, a <> 0 -> (a * b + c) mod a = c mod a.
Proof. intros a b c H; rewrite (Z.mul_comm _ b); autorewrite with zsimplify; reflexivity. Qed.
Lemma add_pow_mod_l : forall a b c, a <> 0 -> 0 < b ->
((a ^ b) + c) mod a = c mod a.
Proof.
intros a b c H H0; replace b with (b - 1 + 1) by ring;
rewrite Z.pow_add_r, Z.pow_1_r by omega; auto using Z.mod_add_l.
Qed.
Lemma mod_exp_0 : forall a x m, x > 0 -> m > 1 -> a mod m = 0 ->
a ^ x mod m = 0.
Proof.
intros a x m H H0 H1.
replace x with (Z.of_nat (Z.to_nat x)) in * by (apply Z2Nat.id; omega).
induction (Z.to_nat x). {
simpl in *; omega.
} {
rewrite Nat2Z.inj_succ in *.
rewrite Z.pow_succ_r by omega.
rewrite Z.mul_mod by omega.
case_eq n; intros. {
subst. simpl.
rewrite Zmod_1_l by omega.
rewrite H1.
apply Zmod_0_l.
} {
subst.
rewrite IHn by (rewrite Nat2Z.inj_succ in *; omega).
rewrite H1.
auto.
}
}
Qed.
Lemma mod_pow : forall (a m b : Z), (0 <= b) -> (m <> 0) ->
a ^ b mod m = (a mod m) ^ b mod m.
Proof.
intros a m b H H0; rewrite <- (Z2Nat.id b) by auto.
induction (Z.to_nat b) as [|n IHn]; auto.
rewrite Nat2Z.inj_succ.
do 2 rewrite Z.pow_succ_r by apply Nat2Z.is_nonneg.
rewrite Z.mul_mod by auto.
rewrite (Z.mul_mod (a mod m) ((a mod m) ^ Z.of_nat n) m) by auto.
rewrite <- IHn by auto.
rewrite Z.mod_mod by auto.
reflexivity.
Qed.
Lemma mod_to_nat x m (Hm:(0 < m)%Z) (Hx:(0 <= x)%Z) : (Z.to_nat x mod Z.to_nat m = Z.to_nat (x mod m))%nat.
pose proof Zdiv.mod_Zmod (Z.to_nat x) (Z.to_nat m) as H;
rewrite !Z2Nat.id in H by omega.
rewrite <-H by (change 0%nat with (Z.to_nat 0); rewrite Z2Nat.inj_iff; omega).
rewrite !Nat2Z.id; reflexivity.
Qed.
Lemma mul_div_eq_full : forall a m, m <> 0 -> m * (a / m) = (a - a mod m).
Proof.
intros a m H. rewrite (Z_div_mod_eq_full a m) at 2 by auto. ring.
Qed.
Hint Rewrite mul_div_eq_full using zutil_arith : zdiv_to_mod.
Hint Rewrite <-mul_div_eq_full using zutil_arith : zmod_to_div.
Lemma f_equal_mul_mod x y x' y' m : x mod m = x' mod m -> y mod m = y' mod m -> (x * y) mod m = (x' * y') mod m.
Proof.
intros H0 H1; rewrite Zmult_mod, H0, H1, <- Zmult_mod; reflexivity.
Qed.
Hint Resolve f_equal_mul_mod : zarith.
Lemma f_equal_add_mod x y x' y' m : x mod m = x' mod m -> y mod m = y' mod m -> (x + y) mod m = (x' + y') mod m.
Proof.
intros H0 H1; rewrite Zplus_mod, H0, H1, <- Zplus_mod; reflexivity.
Qed.
Hint Resolve f_equal_add_mod : zarith.
Lemma f_equal_opp_mod x x' m : x mod m = x' mod m -> (-x) mod m = (-x') mod m.
Proof.
intro H.
destruct (Z_zerop (x mod m)) as [H'|H'], (Z_zerop (x' mod m)) as [H''|H''];
try congruence.
{ rewrite !Z_mod_zero_opp_full by assumption; reflexivity. }
{ rewrite Z_mod_nz_opp_full, H, <- Z_mod_nz_opp_full by assumption; reflexivity. }
Qed.
Hint Resolve f_equal_opp_mod : zarith.
Lemma f_equal_sub_mod x y x' y' m : x mod m = x' mod m -> y mod m = y' mod m -> (x - y) mod m = (x' - y') mod m.
Proof.
rewrite <- !Z.add_opp_r; auto with zarith.
Qed.
Hint Resolve f_equal_sub_mod : zarith.
Lemma mul_div_eq : forall a m, m > 0 -> m * (a / m) = (a - a mod m).
Proof.
intros a m H.
rewrite (Z_div_mod_eq a m) at 2 by auto.
ring.
Qed.
Lemma mul_div_eq' : (forall a m, m > 0 -> (a / m) * m = (a - a mod m))%Z.
Proof.
intros a m H.
rewrite (Z_div_mod_eq a m) at 2 by auto.
ring.
Qed.
Hint Rewrite mul_div_eq mul_div_eq' using zutil_arith : zdiv_to_mod.
Hint Rewrite <- mul_div_eq' using zutil_arith : zmod_to_div.
Lemma mod_div_eq0 : forall a b, 0 < b -> (a mod b) / b = 0.
Proof.
intros.
apply Z.div_small.
auto using Z.mod_pos_bound.
Qed.
Hint Rewrite mod_div_eq0 using zutil_arith : zsimplify.
Local Lemma mod_pull_div_helper a b c X
(HX : forall a b c d e f g,
X a b c d e f g = if a =? 0 then c else 0)
: 0 <> b
-> 0 <> c
-> (a / b) mod c
= (a mod (c * b)) / b
+ if c <? 0 then - X ((a / b) mod c) (a mod (c * b)) ((a mod (c * b)) / b) a b c (a / b) else 0.
Proof.
intros; break_match; Z.ltb_to_lt; rewrite ?Z.sub_0_r, ?Z.add_0_r;
assert (0 <> c * b) by nia; Z.div_mod_to_quot_rem_in_goal; subst;
destruct_head'_or; destruct_head'_and;
try assert (b < 0) by omega;
try assert (c < 0) by omega;
Z.replace_all_neg_with_pos;
try match goal with
| [ H : ?c * ?b * ?q1 + ?r1 = ?b * (?c * ?q2 + _) + _ |- _ ]
=> assert (q1 = q2) by nia; progress subst
end;
rewrite ?HX; clear HX X;
try nia;
repeat match goal with
| [ |- - ?x = ?y ] => is_var y; assert (y <= 0) by nia; Z.replace_all_neg_with_pos
| [ |- - ?x = ?y + -_ ] => is_var y; assert (y <= 0) by nia; Z.replace_all_neg_with_pos
| [ H : -?x + (-?y + ?z) = -?w + ?v |- _ ]
=> assert (x + (y + -z) = w + -v) by omega; clear H
| [ H : ?c * ?b * ?q1 + (?b * ?q2 + ?r) = ?b * (?c * ?q1' + ?q2') + ?r' |- _ ]
=> assert (c * q1 + q2 = c * q1' + q2') by nia;
assert (r = r') by nia;
clear H
| [ H : -?x < -?y + ?z |- _ ] => assert (y + -z < x) by omega; clear H
| [ H : -?x + ?y <= 0 |- _ ] => assert (0 <= x + -y) by omega; clear H
| _ => progress Z.clean_neg
| _ => progress subst
end.
all:match goal with
| [ H : ?c * ?q + ?r = ?c * ?q' + ?r' |- _ ]
=> first [ constr_eq q q'; assert (r = r') by nia; clear H
| assert (q = q') by nia; assert (r = r') by nia; clear H
| lazymatch goal with
| [ H' : r' < c |- _ ]
=> destruct (Z_dec' r c) as [[?|?]|?]
| [ H' : r < c |- _ ]
=> destruct (Z_dec' r' c) as [[?|?]|?]
end;
subst;
[ assert (q = q') by nia; assert (r = r') by nia; clear H
| nia
| first [ assert (1 + q = q') by nia | assert (q = 1 + q') by nia ];
first [ assert (r' = 0) by nia | assert (r = 0) by nia ] ] ]
end.
all:try omega.
all:break_match; Z.ltb_to_lt; omega.
Qed.
Lemma mod_pull_div_full a b c
: (a / b) mod c
= if ((c <? 0) && ((a / b) mod c =? 0))%bool
then 0
else (a mod (c * b)) / b.
Proof.
destruct (Z_zerop b), (Z_zerop c); subst;
autorewrite with zsimplify; try reflexivity.
{ break_match; Z.ltb_to_lt; omega. }
{ erewrite mod_pull_div_helper at 1 by (omega || reflexivity); cbv beta.
destruct (c <? 0) eqn:?; simpl; [ | omega ].
break_innermost_match; omega. }
Qed.
Lemma mod_pull_div a b c
: 0 <= c -> (a / b) mod c = a mod (c * b) / b.
Proof. rewrite mod_pull_div_full; destruct (c <? 0) eqn:?; Z.ltb_to_lt; simpl; omega. Qed.
Lemma small_mod_eq a b n: a mod n = b mod n -> 0 <= a < n -> a = b mod n.
Proof. intros; rewrite <-(Z.mod_small a n); auto. Qed.
Lemma mod_bound_min_max l x u d (H : l <= x <= u)
: (if l / d =? u / d then Z.min (l mod d) (u mod d) else Z.min 0 (d + 1))
<= x mod d
<= if l / d =? u / d then Z.max (l mod d) (u mod d) else Z.max 0 (d - 1).
Proof.
destruct (Z_dec d 0) as [ [?|?] | ? ];
try solve [ subst; autorewrite with zsimplify; simpl; split; reflexivity
| repeat first [ progress Z.div_mod_to_quot_rem_in_goal
| progress subst
| progress break_innermost_match
| progress Z.ltb_to_lt
| progress destruct_head'_or
| progress destruct_head'_and
| progress apply Z.min_case_strong
| progress apply Z.max_case_strong
| progress intros
| omega
| match goal with
| [ H : ?x <= ?y, H' : ?y <= ?x |- _ ] => assert (x = y) by omega; clear H H'
| _ => progress subst
| [ H : ?d * ?q0 + ?r0 = ?d * ?q1 + ?r1 |- _ ]
=> assert (q0 = q1) by nia; subst q0
| [ H : ?d * ?q0 + ?r0 <= ?d * ?q1 + ?r1 |- _ ]
=> assert (q0 = q1) by nia; subst q0
end ] ].
Qed.
Lemma mod_mod_0_0_eq x y : x mod y = 0 -> y mod x = 0 -> x = y \/ x = - y \/ x = 0 \/ y = 0.
Proof.
destruct (Z_zerop x), (Z_zerop y); eauto.
Z.div_mod_to_quot_rem_in_goal; subst.
rewrite ?Z.add_0_r in *.
match goal with
| [ H : ?x = ?x * ?q * ?q' |- _ ]
=> assert (q * q' = 1) by nia;
destruct_head'_or;
first [ assert (q < 0) by nia
| assert (0 < q) by nia ];
first [ assert (q' < 0) by nia
| assert (0 < q') by nia ]
end;
nia.
Qed.
Lemma mod_mod_0_0_eq_pos x y : 0 < x -> 0 < y -> x mod y = 0 -> y mod x = 0 -> x = y.
Proof. intros ?? H0 H1; pose proof (mod_mod_0_0_eq x y H0 H1); omega. Qed.
Lemma mod_mod_trans x y z : y <> 0 -> x mod y = 0 -> y mod z = 0 -> x mod z = 0.
Proof.
destruct (Z_zerop x), (Z_zerop z); subst; autorewrite with zsimplify_const; auto; intro.
Z.generalize_div_eucl x y.
Z.generalize_div_eucl y z.
intros; subst.
rewrite ?Z.add_0_r in *.
rewrite <- Z.mul_assoc.
rewrite <- Zmult_mod_idemp_l, Z_mod_same_full.
autorewrite with zsimplify_const.
reflexivity.
Qed.
End Z.
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