The Coq Proof Assistant

# Library Coq.ZArith.Zdiv

# Euclidean Division

Initial Contribution by Claude Marché and Xavier Urbain
Require Export ZArith_base.
Require Import Zbool ZArithRing Zcomplements Setoid Morphisms.
Local Open Scope Z_scope.
The definition of the division is now in BinIntDef, the initial specifications and properties are in BinInt.
Notation Zdiv_eucl_POS := Z.pos_div_eucl (only parsing).
Notation Zmod := Z.modulo (only parsing).
Notation Zmod_POS_bound := Z.pos_div_eucl_bound (only parsing).
Notation Zmod_pos_bound := Z.mod_pos_bound (only parsing).
Notation Zmod_neg_bound := Z.mod_neg_bound (only parsing).

# Main division theorems

NB: many things are stated twice for compatibility reasons
Lemma Z_div_mod_POS :
forall b:Z,
b > 0 ->
forall a:positive,
let (q, r) := Z.pos_div_eucl a b in Zpos a = b * q + r /\ 0 <= r < b.
Theorem Z_div_mod a b :
b > 0 ->
let (q, r) := Z.div_eucl a b in a = b * q + r /\ 0 <= r < b.
For stating the fully general result, let's give a short name to the condition on the remainder.
Definition Remainder r b := 0 <= r < b \/ b < r <= 0.
Another equivalent formulation:
Definition Remainder_alt r b := Z.abs r < Z.abs b /\ Z.sgn r <> - Z.sgn b.
Lemma Remainder_equiv : forall r b, Remainder r b <-> Remainder_alt r b.
#[global]
Hint Unfold Remainder : core.
Now comes the fully general result about Euclidean division.
Theorem Z_div_mod_full a b :
b <> 0 ->
let (q, r) := Z.div_eucl a b in a = b * q + r /\ Remainder r b.
The same results as before, stated separately in terms of Z.div and Z.modulo
Lemma Z_mod_remainder a b : b<>0 -> Remainder (a mod b) b.
Lemma Z_mod_lt a b : b > 0 -> 0 <= a mod b < b.
Lemma Z_mod_neg a b : b < 0 -> b < a mod b <= 0.
Lemma Z_div_mod_eq_full a b : a = b*(a/b) + (a mod b).
Lemma Z_div_mod_eq_deprecated a b : b > 0 -> a = b*(a/b) + (a mod b).
#[deprecated(since="8.14",note="Use Z_div_mod_eq_full instead")]
Notation Z_div_mod_eq := Z_div_mod_eq_deprecated (only parsing).
Lemma Zmod_eq_full a b : b<>0 -> a mod b = a - (a/b)*b.
Lemma Zmod_eq a b : b>0 -> a mod b = a - (a/b)*b.
Existence theorem
Theorem Zdiv_eucl_exist : forall (b:Z)(Hb:b>0)(a:Z),
{qr : Z * Z | let (q, r) := qr in a = b * q + r /\ 0 <= r < b}.
Arguments Zdiv_eucl_exist : default implicits.
Uniqueness theorems
Theorem Zdiv_mod_unique b q1 q2 r1 r2 :
0 <= r1 < Z.abs b -> 0 <= r2 < Z.abs b ->
b*q1+r1 = b*q2+r2 -> q1=q2 /\ r1=r2.
Theorem Zdiv_mod_unique_2 :
forall b q1 q2 r1 r2:Z,
Remainder r1 b -> Remainder r2 b ->
b*q1+r1 = b*q2+r2 -> q1=q2 /\ r1=r2.
Theorem Zdiv_unique_full:
forall a b q r, Remainder r b ->
a = b*q + r -> q = a/b.
Theorem Zdiv_unique:
forall a b q r, 0 <= r < b ->
a = b*q + r -> q = a/b.
Theorem Zmod_unique_full:
forall a b q r, Remainder r b ->
a = b*q + r -> r = a mod b.
Theorem Zmod_unique:
forall a b q r, 0 <= r < b ->
a = b*q + r -> r = a mod b.

# Basic values of divisions and modulo.

Lemma Zmod_0_l: forall a, 0 mod a = 0.
Lemma Zmod_0_r: forall a, a mod 0 = a.
Lemma Zdiv_0_l: forall a, 0/a = 0.
Lemma Zdiv_0_r: forall a, a/0 = 0.
Ltac zero_or_not a :=
destruct (Z.eq_dec a 0);
[subst; rewrite ?Zmod_0_l, ?Zdiv_0_l, ?Zmod_0_r, ?Zdiv_0_r;
auto with zarith|].
Lemma Zmod_1_r: forall a, a mod 1 = 0.
Lemma Zdiv_1_r: forall a, a/1 = a.
#[global]
Hint Resolve Zmod_0_l Zmod_0_r Zdiv_0_l Zdiv_0_r Zdiv_1_r Zmod_1_r
: zarith.
Lemma Zdiv_1_l: forall a, 1 < a -> 1/a = 0.
Lemma Zmod_1_l: forall a, 1 < a -> 1 mod a = 1.
Lemma Z_div_same_full : forall a:Z, a<>0 -> a/a = 1.
Lemma Z_mod_same_full : forall a, a mod a = 0.
Lemma Z_mod_mult : forall a b, (a*b) mod b = 0.
Lemma Z_div_mult_full : forall a b:Z, b <> 0 -> (a*b)/b = a.

# Order results about Z.modulo and Z.div

Lemma Z_div_pos: forall a b, b > 0 -> 0 <= a -> 0 <= a/b.
Lemma Z_div_ge0: forall a b, b > 0 -> a >= 0 -> a/b >=0.
Lemma Z_div_nonneg_nonneg : forall a b, 0 <= a -> 0 <= b -> 0 <= a / b.
Lemma Z_mod_nonneg_nonneg : forall a b, 0 <= a -> 0 <= b -> 0 <= a mod b.
As soon as the divisor is greater or equal than 2, the division is strictly decreasing.
Lemma Z_div_lt : forall a b:Z, b >= 2 -> a > 0 -> a/b < a.
A division of a small number by a bigger one yields zero.
Theorem Zdiv_small: forall a b, 0 <= a < b -> a/b = 0.
Same situation, in term of modulo:
Theorem Zmod_small: forall a n, 0 <= a < n -> a mod n = a.
Z.ge is compatible with a positive division.
Lemma Z_div_ge : forall a b c:Z, c > 0 -> a >= b -> a/c >= b/c.
Same, with Z.le.
Lemma Z_div_le : forall a b c:Z, c > 0 -> a <= b -> a/c <= b/c.
With our choice of division, rounding of (a/b) is always done toward bottom:
Lemma Z_mult_div_ge : forall a b:Z, b > 0 -> b*(a/b) <= a.
Lemma Z_mult_div_ge_neg : forall a b:Z, b < 0 -> b*(a/b) >= a.
The previous inequalities are exact iff the modulo is zero.
Lemma Z_div_exact_full_1 : forall a b:Z, a = b*(a/b) -> a mod b = 0.
Lemma Z_div_exact_full_2 : forall a b:Z, b <> 0 -> a mod b = 0 -> a = b*(a/b).
A modulo cannot grow beyond its starting point.
Theorem Zmod_le: forall a b, 0 < b -> 0 <= a -> a mod b <= a.
Some additional inequalities about Z.div.
Theorem Zdiv_lt_upper_bound:
forall a b q, 0 < b -> a < q*b -> a/b < q.
Theorem Zdiv_le_upper_bound:
forall a b q, 0 < b -> a <= q*b -> a/b <= q.
Theorem Zdiv_le_lower_bound:
forall a b q, 0 < b -> q*b <= a -> q <= a/b.
A division of respect opposite monotonicity for the divisor
Lemma Zdiv_le_compat_l: forall p q r, 0 <= p -> 0 < q < r ->
p / r <= p / q.
Theorem Zdiv_sgn: forall a b,
0 <= Z.sgn (a/b) * Z.sgn a * Z.sgn b.

# Relations between usual operations and Z.modulo and Z.div

Lemma Z_mod_plus_full : forall a b c:Z, (a + b * c) mod c = a mod c.
Lemma Z_div_plus_full : forall a b c:Z, c <> 0 -> (a + b * c) / c = a / c + b.
Theorem Z_div_plus_full_l: forall a b c : Z, b <> 0 -> (a * b + c) / b = a + c / b.
Z.opp and Z.div, Z.modulo. Due to the choice of convention for our Euclidean division, some of the relations about Z.opp and divisions are rather complex.
Lemma Zdiv_opp_opp : forall a b:Z, (-a)/(-b) = a/b.
Lemma Zmod_opp_opp : forall a b:Z, (-a) mod (-b) = - (a mod b).
Lemma Z_mod_zero_opp_full : forall a b:Z, a mod b = 0 -> (-a) mod b = 0.
Lemma Z_mod_nz_opp_full : forall a b:Z, a mod b <> 0 ->
(-a) mod b = b - (a mod b).
Lemma Z_mod_zero_opp_r : forall a b:Z, a mod b = 0 -> a mod (-b) = 0.
Lemma Z_mod_nz_opp_r : forall a b:Z, a mod b <> 0 ->
a mod (-b) = (a mod b) - b.
Lemma Z_div_zero_opp_full : forall a b:Z, a mod b = 0 -> (-a)/b = -(a/b).
Lemma Z_div_nz_opp_full : forall a b:Z, b <> 0 -> a mod b <> 0 ->
(-a)/b = -(a/b)-1.
Lemma Z_div_zero_opp_r : forall a b:Z, a mod b = 0 -> a/(-b) = -(a/b).
Lemma Z_div_nz_opp_r : forall a b:Z, b <> 0 -> a mod b <> 0 ->
a/(-b) = -(a/b)-1.
Cancellations.
Lemma Zdiv_mult_cancel_r : forall a b c:Z,
c <> 0 -> (a*c)/(b*c) = a/b.
Lemma Zdiv_mult_cancel_l : forall a b c:Z,
c<>0 -> (c*a)/(c*b) = a/b.
Lemma Zmult_mod_distr_l: forall a b c,
(c*a) mod (c*b) = c * (a mod b).
Lemma Zmult_mod_distr_r: forall a b c,
(a*c) mod (b*c) = (a mod b) * c.
Operations modulo.
Theorem Zmod_mod: forall a n, (a mod n) mod n = a mod n.
Theorem Zmult_mod: forall a b n,
(a * b) mod n = ((a mod n) * (b mod n)) mod n.
Theorem Zplus_mod: forall a b n,
(a + b) mod n = (a mod n + b mod n) mod n.
Theorem Zminus_mod: forall a b n,
(a - b) mod n = (a mod n - b mod n) mod n.
Lemma Zplus_mod_idemp_l: forall a b n, (a mod n + b) mod n = (a + b) mod n.
Lemma Zplus_mod_idemp_r: forall a b n, (b + a mod n) mod n = (b + a) mod n.
Lemma Zminus_mod_idemp_l: forall a b n, (a mod n - b) mod n = (a - b) mod n.
Lemma Zminus_mod_idemp_r: forall a b n, (a - b mod n) mod n = (a - b) mod n.
Lemma Zmult_mod_idemp_l: forall a b n, (a mod n * b) mod n = (a * b) mod n.
Lemma Zmult_mod_idemp_r: forall a b n, (b * (a mod n)) mod n = (b * a) mod n.
For a specific number N, equality modulo N is hence a nice setoid equivalence, compatible with +, - and *.
Section EqualityModulo.
Variable N:Z.
Definition eqm a b := (a mod N = b mod N).
Infix "==" := eqm (at level 70).
Lemma eqm_refl : forall a, a == a.
Lemma eqm_sym : forall a b, a == b -> b == a.
Lemma eqm_trans : forall a b c,
a == b -> b == c -> a == c.
Instance eqm_setoid : Equivalence eqm.
Instance Zplus_eqm : Proper (eqm ==> eqm ==> eqm) Z.add.
Instance Zminus_eqm : Proper (eqm ==> eqm ==> eqm) Z.sub.
Instance Zmult_eqm : Proper (eqm ==> eqm ==> eqm) Z.mul.
Instance Zopp_eqm : Proper (eqm ==> eqm) Z.opp.
Lemma Zmod_eqm : forall a, (a mod N) == a.
End EqualityModulo.
Lemma Zdiv_Zdiv : forall a b c, 0<=b -> 0<=c -> (a/b)/c = a/(b*c).
Unfortunately, the previous result isn't always true on negative numbers. For instance: 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2)
Lemma Zmod_div : forall a b, a mod b / b = 0.
A last inequality:
Theorem Zdiv_mult_le:
forall a b c, 0<=a -> 0<=b -> 0<=c -> c*(a/b) <= (c*a)/b.
Z.modulo is related to divisibility (see more in Znumtheory)
Lemma Zmod_divides : forall a b, b<>0 ->
(a mod b = 0 <-> exists c, a = b*c).
Particular case : dividing by 2 is related with parity
Lemma Zdiv2_div : forall a, Z.div2 a = a/2.
Lemma Zmod_odd : forall a, a mod 2 = if Z.odd a then 1 else 0.
Lemma Zmod_even : forall a, a mod 2 = if Z.even a then 0 else 1.
Lemma Zodd_mod : forall a, Z.odd a = Zeq_bool (a mod 2) 1.
Lemma Zeven_mod : forall a, Z.even a = Zeq_bool (a mod 2) 0.

# Compatibility

Weaker results kept only for compatibility
Lemma Z_mod_same : forall a, a > 0 -> a mod a = 0.
Lemma Z_div_same : forall a, a > 0 -> a/a = 1.
Lemma Z_div_plus : forall a b c:Z, c > 0 -> (a + b * c) / c = a / c + b.
Lemma Z_div_mult : forall a b:Z, b > 0 -> (a*b)/b = a.
Lemma Z_mod_plus : forall a b c:Z, c > 0 -> (a + b * c) mod c = a mod c.
Lemma Z_div_exact_1 : forall a b:Z, b > 0 -> a = b*(a/b) -> a mod b = 0.
Lemma Z_div_exact_2 : forall a b:Z, b > 0 -> a mod b = 0 -> a = b*(a/b).
Lemma Z_mod_zero_opp : forall a b:Z, b > 0 -> a mod b = 0 -> (-a) mod b = 0.

# A direct way to compute Z.modulo

Fixpoint Zmod_POS (a : positive) (b : Z) : Z :=
match a with
| xI a' =>
let r := Zmod_POS a' b in
let r' := (2 * r + 1) in
if r' <? b then r' else (r' - b)
| xO a' =>
let r := Zmod_POS a' b in
let r' := (2 * r) in
if r' <? b then r' else (r' - b)
| xH => if 2 <=? b then 1 else 0
end.
Definition Zmod' a b :=
match a with
| Z0 => 0
| Zpos a' =>
match b with
| Z0 => a
| Zpos _ => Zmod_POS a' b
| Zneg b' =>
let r := Zmod_POS a' (Zpos b') in
match r with Z0 => 0 | _ => b + r end
end
| Zneg a' =>
match b with
| Z0 => a
| Zpos _ =>
let r := Zmod_POS a' b in
match r with Z0 => 0 | _ => b - r end
| Zneg b' => - (Zmod_POS a' (Zpos b'))
end
end.
Theorem Zmod_POS_correct a b : Zmod_POS a b = snd (Z.pos_div_eucl a b).
Theorem Zmod'_correct: forall a b, Zmod' a b = a mod b.
Another convention is possible for division by negative numbers:

# quotient is always the biggest integer smaller than or equal to a/b

# remainder is hence always positive or null.

Theorem Zdiv_eucl_extended :
forall b:Z,
b <> 0 ->
forall a:Z,
{qr : Z * Z | let (q, r) := qr in a = b * q + r /\ 0 <= r < Z.abs b}.
Arguments Zdiv_eucl_extended : default implicits.

# Division and modulo in Z agree with same in nat:

Lemma div_Zdiv_deprecated (n m: nat): m <> O ->
Z.of_nat (n / m) = Z.of_nat n / Z.of_nat m.
#[deprecated(since="8.14",note="Use Nat2Z.inj_div instead.")]
Notation div_Zdiv := div_Zdiv_deprecated (only parsing).
Lemma mod_Zmod_deprecated (n m: nat): m <> O ->
Z.of_nat (n mod m) = (Z.of_nat n) mod (Z.of_nat m).
#[deprecated(since="8.14",note="Use Nat2Z.inj_mod instead.")]
Notation mod_Zmod := mod_Zmod_deprecated (only parsing).