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BigNumPrelude.v
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BigNumPrelude.v
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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *)
(************************************************************************)
(** * BigNumPrelude *)
(** Auxiliary functions & theorems used for arbitrary precision efficient
numbers. *)
From Stdlib Require Import ArithRing.
From Stdlib Require Export ZArith.
From Stdlib Require Export Znumtheory.
From Stdlib Require Export Zpow_facts.
From Stdlib Require Uint63.
From Stdlib Require Import Lia.
Declare ML Module "coq-bignums.plugin".
(* *** Nota Bene ***
All results that were general enough have been moved in ZArith.
Only remain here specialized lemmas and compatibility elements.
(P.L. 5/11/2007).
*)
Local Open Scope Z_scope.
(* For compatibility of scripts, weaker version of some lemmas of Z.div *)
Lemma Zlt0_not_eq : forall n, 0<n -> n<>0.
Proof.
auto with zarith.
Qed.
Definition Zdiv_mult_cancel_r a b c H := Zdiv.Zdiv_mult_cancel_r a b c (Zlt0_not_eq _ H).
Definition Zdiv_mult_cancel_l a b c H := Zdiv.Zdiv_mult_cancel_r a b c (Zlt0_not_eq _ H).
Definition Z_div_plus_l a b c H := Zdiv.Z_div_plus_full_l a b c (Zlt0_not_eq _ H).
(* Automation *)
#[global]
Hint Extern 2 (Z.le _ _) =>
(match goal with
|- Zpos _ <= Zpos _ => exact (eq_refl _)
| H: _ <= ?p |- _ <= ?p => apply Z.le_trans with (2 := H)
| H: _ < ?p |- _ <= ?p => apply Z.lt_le_incl; apply Z.le_lt_trans with (2 := H)
end) : core.
#[global]
Hint Extern 2 (Z.lt _ _) =>
(match goal with
|- Zpos _ < Zpos _ => exact (eq_refl _)
| H: _ <= ?p |- _ <= ?p => apply Z.lt_le_trans with (2 := H)
| H: _ < ?p |- _ <= ?p => apply Z.le_lt_trans with (2 := H)
end) : core.
#[global]
Hint Resolve Z.lt_gt Z.le_ge Z_div_pos: zarith.
(**************************************
Properties of order and product
**************************************)
Theorem beta_lex: forall a b c d beta,
a * beta + b <= c * beta + d ->
0 <= b < beta -> 0 <= d < beta ->
a <= c.
Proof. nia. Qed.
Theorem beta_lex_inv: forall a b c d beta,
a < c -> 0 <= b < beta ->
0 <= d < beta ->
a * beta + b < c * beta + d.
Proof. nia. Qed.
Lemma beta_mult : forall h l beta,
0 <= h < beta -> 0 <= l < beta -> 0 <= h*beta+l < beta^2.
Proof. nia. Qed.
Lemma Zmult_lt_b :
forall b x y, 0 <= x < b -> 0 <= y < b -> 0 <= x * y <= b^2 - 2*b + 1.
Proof. nia. Qed.
Lemma sum_mul_carry : forall xh xl yh yl wc cc beta,
1 < beta ->
0 <= wc < beta ->
0 <= xh < beta ->
0 <= xl < beta ->
0 <= yh < beta ->
0 <= yl < beta ->
0 <= cc < beta^2 ->
wc*beta^2 + cc = xh*yl + xl*yh ->
0 <= wc <= 1.
Proof. nia. Qed.
Theorem mult_add_ineq: forall x y cross beta,
0 <= x < beta ->
0 <= y < beta ->
0 <= cross < beta ->
0 <= x * y + cross < beta^2.
Proof. nia. Qed.
Theorem mult_add_ineq2: forall x y c cross beta,
0 <= x < beta ->
0 <= y < beta ->
0 <= c*beta + cross <= 2*beta - 2 ->
0 <= x * y + (c*beta + cross) < beta^2.
Proof. nia. Qed.
Theorem mult_add_ineq3: forall x y c cross beta,
0 <= x < beta ->
0 <= y < beta ->
0 <= cross <= beta - 2 ->
0 <= c <= 1 ->
0 <= x * y + (c*beta + cross) < beta^2.
Proof. nia. Qed.
#[global]
Hint Rewrite Z.mul_1_r Z.mul_0_r Z.mul_1_l Z.mul_0_l Z.add_0_l Z.add_0_r Z.sub_0_r: rm10.
(**************************************
Properties of Z.div and Z.modulo
**************************************)
Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a.
Proof.
intros a b H H1;case (Z_mod_lt a b);auto with zarith;intros H2 H3;split;auto.
case (Z.le_gt_cases b a); intros H4; auto with zarith.
rewrite Zmod_small; auto with zarith.
Qed.
Theorem Zmod_distr: forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a ->
(2 ^a * r + t) mod (2 ^ b) = (2 ^a * r) mod (2 ^ b) + t.
Proof.
intros a b r t (H1, H2) H3 (H4, H5).
assert (t < 2 ^ b).
apply Z.lt_le_trans with (1:= H5); auto with zarith.
apply Zpower_le_monotone; auto with zarith.
rewrite Zplus_mod; auto with zarith.
rewrite Zmod_small with (a := t); auto with zarith.
apply Zmod_small; auto with zarith.
split; auto with zarith.
assert (0 <= 2 ^a * r); auto with zarith.
apply Z.add_nonneg_nonneg; auto with zarith.
match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end;
auto with zarith.
pattern (2 ^ b) at 2; replace (2 ^ b) with ((2 ^ b - 2 ^a) + 2 ^ a);
try ring.
apply Z.add_le_lt_mono; auto with zarith.
replace b with ((b - a) + a); try ring.
rewrite Zpower_exp; auto with zarith.
pattern (2 ^a) at 4; rewrite <- (Z.mul_1_l (2 ^a));
try rewrite <- Z.mul_sub_distr_r.
rewrite (Z.mul_comm (2 ^(b - a))); rewrite Zmult_mod_distr_l;
auto with zarith.
rewrite (Z.mul_comm (2 ^a)); apply Z.mul_le_mono_nonneg_r; auto with zarith.
match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end;
auto with zarith.
Qed.
Theorem Zmod_shift_r:
forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a ->
(r * 2 ^a + t) mod (2 ^ b) = (r * 2 ^a) mod (2 ^ b) + t.
Proof.
intros a b r t (H1, H2) H3 (H4, H5).
assert (t < 2 ^ b).
apply Z.lt_le_trans with (1:= H5); auto with zarith.
apply Zpower_le_monotone; auto with zarith.
rewrite Zplus_mod; auto with zarith.
rewrite Zmod_small with (a := t); auto with zarith.
apply Zmod_small; auto with zarith.
split; auto with zarith.
assert (0 <= 2 ^a * r); auto with zarith.
apply Z.add_nonneg_nonneg; auto with zarith.
match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end;
auto with zarith.
pattern (2 ^ b) at 2;replace (2 ^ b) with ((2 ^ b - 2 ^a) + 2 ^ a); try ring.
apply Z.add_le_lt_mono; auto with zarith.
replace b with ((b - a) + a); try ring.
rewrite Zpower_exp; auto with zarith.
pattern (2 ^a) at 4; rewrite <- (Z.mul_1_l (2 ^a));
try rewrite <- Z.mul_sub_distr_r.
repeat rewrite (fun x => Z.mul_comm x (2 ^ a)); rewrite Zmult_mod_distr_l;
auto with zarith.
apply Z.mul_le_mono_nonneg_l; auto with zarith.
match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end;
auto with zarith.
Qed.
Theorem Zdiv_shift_r:
forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a ->
(r * 2 ^a + t) / (2 ^ b) = (r * 2 ^a) / (2 ^ b).
Proof.
intros a b r t (H1, H2) H3 (H4, H5).
assert (Eq: t < 2 ^ b); auto with zarith.
apply Z.lt_le_trans with (1 := H5); auto with zarith.
apply Zpower_le_monotone; auto with zarith.
pattern (r * 2 ^ a) at 1; rewrite Z_div_mod_eq_full with (b := 2 ^ b);
auto with zarith.
rewrite <- Z.add_assoc.
rewrite <- Zmod_shift_r; auto with zarith.
rewrite (Z.mul_comm (2 ^ b)); rewrite Z_div_plus_full_l; auto with zarith.
rewrite (fun x y => @Zdiv_small (x mod y)); auto with zarith.
match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end;
auto with zarith.
Qed.
Lemma shift_unshift_mod : forall n p a,
0 <= a < 2^n ->
0 <= p <= n ->
a * 2^p = a / 2^(n - p) * 2^n + (a*2^p) mod 2^n.
Proof.
intros n p a H1 H2.
pattern (a*2^p) at 1;replace (a*2^p) with
(a*2^p/2^n * 2^n + a*2^p mod 2^n).
2:symmetry;rewrite (Z.mul_comm (a*2^p/2^n));apply Z_div_mod_eq_full.
replace (a * 2 ^ p / 2 ^ n) with (a / 2 ^ (n - p));trivial.
replace (2^n) with (2^(n-p)*2^p).
symmetry;apply Zdiv_mult_cancel_r.
destruct H1;trivial.
cut (0 < 2^p); auto with zarith.
rewrite <- Zpower_exp.
replace (n-p+p) with n;trivial. ring.
lia. lia.
Qed.
Lemma shift_unshift_mod_2 : forall n p a, 0 <= p <= n ->
((a * 2 ^ (n - p)) mod (2^n) / 2 ^ (n - p)) mod (2^n) =
a mod 2 ^ p.
Proof.
intros.
rewrite Zmod_small.
rewrite Zmod_eq by (auto with zarith).
unfold Z.sub at 1.
rewrite Z_div_plus_l by (auto with zarith).
assert (2^n = 2^(n-p)*2^p).
rewrite <- Zpower_exp by (auto with zarith).
replace (n-p+p) with n; auto with zarith.
rewrite H0.
rewrite <- Zdiv_Zdiv, Z_div_mult by (auto with zarith).
rewrite (Z.mul_comm (2^(n-p))), Z.mul_assoc.
rewrite <- Z.mul_opp_l.
rewrite Z_div_mult by (auto with zarith).
symmetry; apply Zmod_eq; auto with zarith.
remember (a * 2 ^ (n - p)) as b.
destruct (Z_mod_lt b (2^n)); auto with zarith.
split.
apply Z_div_pos; auto with zarith.
apply Zdiv_lt_upper_bound; auto with zarith.
apply Z.lt_le_trans with (2^n); auto with zarith.
rewrite <- (Z.mul_1_r (2^n)) at 1.
assert (0 < 2^ (n-p)) by auto with zarith.
apply Z.mul_le_mono_nonneg; auto with zarith.
Qed.
Lemma div_le_0 : forall p x, 0 <= x -> 0 <= x / 2 ^ p.
Proof.
intros p x Hle;destruct (Z_le_gt_dec 0 p).
apply Zdiv_le_lower_bound;auto with zarith.
replace (2^p) with 0.
destruct x;compute;intro;discriminate.
destruct p;trivial;discriminate.
Qed.
Lemma div_lt : forall p x y, 0 <= x < y -> x / 2^p < y.
Proof.
intros p x y H;destruct (Z_le_gt_dec 0 p).
apply Zdiv_lt_upper_bound;auto with zarith.
apply Z.lt_le_trans with y;auto with zarith.
assert (0 < 2^p) by auto with zarith.
rewrite <- (Z.mul_1_r y);apply Z.mul_le_mono_nonneg;auto with zarith.
replace (2^p) with 0.
destruct x;change (0<y);auto with zarith.
destruct p;trivial;discriminate.
Qed.
Theorem Zgcd_div_pos a b:
0 < b -> 0 < Z.gcd a b -> 0 < b / Z.gcd a b.
Proof.
intros Hb Hg.
assert (H : 0 <= b / Z.gcd a b) by (apply Z.div_pos; auto with zarith).
Z.le_elim H; trivial.
rewrite (Zdivide_Zdiv_eq (Z.gcd a b) b), <- H, Z.mul_0_r in Hb;
auto using Z.gcd_divide_r with zarith.
Qed.
Theorem Zdiv_neg a b:
a < 0 -> 0 < b -> a / b < 0.
Proof.
intros Ha Hb.
assert (b > 0) by lia.
generalize (Z_mult_div_ge a _ H); intros.
assert (b * (a / b) < 0)%Z.
apply Z.le_lt_trans with a; auto with zarith.
destruct b; try (compute in Hb; discriminate).
destruct (a/Zpos p)%Z.
compute in H1; discriminate.
compute in H1; discriminate.
compute; auto.
Qed.
Lemma Zdiv_gcd_zero : forall a b, b / Z.gcd a b = 0 -> b <> 0 ->
Z.gcd a b = 0.
Proof.
intros.
generalize (Zgcd_is_gcd a b); destruct 1.
destruct H2 as (k,Hk).
generalize H; rewrite Hk at 1.
destruct (Z.eq_dec (Z.gcd a b) 0) as [H'|H']; auto.
rewrite Z_div_mult_full; auto.
intros; subst k; simpl in *; subst b; elim H0; auto.
Qed.
Lemma Zgcd_mult_rel_prime : forall a b c,
Z.gcd a c = 1 -> Z.gcd b c = 1 -> Z.gcd (a*b) c = 1.
Proof.
intros.
rewrite Zgcd_1_rel_prime in *.
apply rel_prime_sym; apply rel_prime_mult; apply rel_prime_sym; auto.
Qed.
Lemma Zcompare_gt : forall (A:Type)(a a':A)(p q:Z),
match (p?=q)%Z with Gt => a | _ => a' end =
if Z_le_gt_dec p q then a' else a.
Proof.
intros.
destruct Z_le_gt_dec as [H|H].
red in H.
destruct (p?=q)%Z; auto; elim H; auto.
rewrite H; auto.
Qed.
Lemma Zsquare_le x : x <= x*x.
Proof.
destruct (Z.lt_ge_cases 0 x).
- rewrite <- Z.mul_1_l at 1.
rewrite <- Z.mul_le_mono_pos_r; auto with zarith.
- pose proof (Z.square_nonneg x); auto with zarith.
Qed.