# Library Coq.Numbers.Integer.Abstract.ZDivEucl

Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv.

# Euclidean Division for integers, Euclid convention

We use here the "usual" formulation of the Euclid Theorem forall a b, b<>0 -> exists r q, a = b*q+r /\ 0 <= r < |b|
The outcome of the modulo function is hence always positive. This corresponds to convention "E" in the following paper:
R. Boute, "The Euclidean definition of the functions div and mod", ACM Transactions on Programming Languages and Systems, Vol. 14, No.2, pp. 127-144, April 1992.
See files ZDivTrunc and ZDivFloor for others conventions.
We simply extend NZDiv with a bound for modulo that holds regardless of the sign of a and b. This new specification subsume mod_bound_pos, which nonetheless stays there for subtyping. Note also that ZAxiomSig now already contain a div and a modulo (that follow the Floor convention). We just ignore them here.

Module Type EuclidSpec (Import A : ZAxiomsSig')(Import B : DivMod A).
Axiom mod_always_pos : forall a b, b ~= 0 -> 0 <= B.modulo a b < abs b.
End EuclidSpec.

Module Type ZEuclid (Z:ZAxiomsSig) := NZDiv.NZDiv Z <+ EuclidSpec Z.

Module ZEuclidProp
(Import A : ZAxiomsSig')
(Import B : ZMulOrderProp A)
(Import C : ZSgnAbsProp A B)
(Import D : ZEuclid A).

We put notations in a scope, to avoid warnings about redefinitions of notations
Infix "/" := D.div : euclid.
Infix "mod" := D.modulo : euclid.
Local Open Scope euclid.

Module Import Private_NZDiv := Nop <+ NZDivProp A D B.

Another formulation of the main equation

Lemma mod_eq :
forall a b, b~=0 -> a mod b == a - b*(a/b).

Ltac pos_or_neg a :=
let LT := fresh "LT" in
let LE := fresh "LE" in
destruct (le_gt_cases 0 a) as [LE|LT]; [|rewrite <- opp_pos_neg in LT].

Uniqueness theorems

Theorem div_mod_unique : forall b q1 q2 r1 r2 : t,
0<=r1<abs b -> 0<=r2<abs b ->
b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2.

Theorem div_unique:
forall a b q r, 0<=r<abs b -> a == b*q + r -> q == a/b.

Theorem mod_unique:
forall a b q r, 0<=r<abs b -> a == b*q + r -> r == a mod b.

Sign rules

Lemma div_opp_r : forall a b, b~=0 -> a/(-b) == -(a/b).

Lemma mod_opp_r : forall a b, b~=0 -> a mod (-b) == a mod b.

Lemma div_opp_l_z : forall a b, b~=0 -> a mod b == 0 ->
(-a)/b == -(a/b).

Lemma div_opp_l_nz : forall a b, b~=0 -> a mod b ~= 0 ->
(-a)/b == -(a/b)-sgn b.

Lemma mod_opp_l_z : forall a b, b~=0 -> a mod b == 0 ->
(-a) mod b == 0.

Lemma mod_opp_l_nz : forall a b, b~=0 -> a mod b ~= 0 ->
(-a) mod b == abs b - (a mod b).

Lemma div_opp_opp_z : forall a b, b~=0 -> a mod b == 0 ->
(-a)/(-b) == a/b.

Lemma div_opp_opp_nz : forall a b, b~=0 -> a mod b ~= 0 ->
(-a)/(-b) == a/b + sgn(b).

Lemma mod_opp_opp_z : forall a b, b~=0 -> a mod b == 0 ->
(-a) mod (-b) == 0.

Lemma mod_opp_opp_nz : forall a b, b~=0 -> a mod b ~= 0 ->
(-a) mod (-b) == abs b - a mod b.

A division by itself returns 1

Lemma div_same : forall a, a~=0 -> a/a == 1.

Lemma mod_same : forall a, a~=0 -> a mod a == 0.

A division of a small number by a bigger one yields zero.

Theorem div_small: forall a b, 0<=a<b -> a/b == 0.

Same situation, in term of modulo:

Theorem mod_small: forall a b, 0<=a<b -> a mod b == a.

# Basic values of divisions and modulo.

Lemma div_0_l: forall a, a~=0 -> 0/a == 0.

Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0.

Lemma div_1_r: forall a, a/1 == a.

Lemma mod_1_r: forall a, a mod 1 == 0.

Lemma div_1_l: forall a, 1<a -> 1/a == 0.

Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1.

Lemma div_mul : forall a b, b~=0 -> (a*b)/b == a.

Lemma mod_mul : forall a b, b~=0 -> (a*b) mod b == 0.

Theorem div_unique_exact a b q: b~=0 -> a == b*q -> q == a/b.

# Order results about mod and div

A modulo cannot grow beyond its starting point.

Theorem mod_le: forall a b, 0<=a -> b~=0 -> a mod b <= a.

Theorem div_pos : forall a b, 0<=a -> 0<b -> 0<= a/b.

Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b.

Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> 0<=a<abs b).

Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> 0<=a<abs b).

As soon as the divisor is strictly greater than 1, the division is strictly decreasing.

Lemma div_lt : forall a b, 0<a -> 1<b -> a/b < a.

le is compatible with a positive division.

Lemma div_le_mono : forall a b c, 0<c -> a<=b -> a/c <= b/c.

In this convention, div performs Rounding-Toward-Bottom when divisor is positive, and Rounding-Toward-Top otherwise.
Since we cannot speak of rational values here, we express this fact by multiplying back by b, and this leads to a nice unique statement.

Lemma mul_div_le : forall a b, b~=0 -> b*(a/b) <= a.

Giving a reversed bound is slightly more complex

Lemma mul_succ_div_gt: forall a b, 0<b -> a < b*(S (a/b)).

Lemma mul_pred_div_gt: forall a b, b<0 -> a < b*(P (a/b)).

NB: The three previous properties could be used as specifications for div.
Inequality mul_div_le is exact iff the modulo is zero.

Lemma div_exact : forall a b, b~=0 -> (a == b*(a/b) <-> a mod b == 0).

Theorem div_lt_upper_bound:
forall a b q, 0<b -> a < b*q -> a/b < q.

Theorem div_le_upper_bound:
forall a b q, 0<b -> a <= b*q -> a/b <= q.

Theorem div_le_lower_bound:
forall a b q, 0<b -> b*q <= a -> q <= a/b.

A division respects opposite monotonicity for the divisor

Lemma div_le_compat_l: forall p q r, 0<=p -> 0<q<=r -> p/r <= p/q.

# Relations between usual operations and mod and div

Lemma mod_add : forall a b c, c~=0 ->
(a + b * c) mod c == a mod c.

Lemma div_add : forall a b c, c~=0 ->
(a + b * c) / c == a / c + b.

Lemma div_add_l: forall a b c, b~=0 ->
(a * b + c) / b == a + c / b.

Cancellations.
With the current convention, the following isn't always true when c<0: -3*-1 / -2*-1 = 3/2 = 1 while -3/-2 = 2

Lemma div_mul_cancel_r : forall a b c, b~=0 -> 0<c ->
(a*c)/(b*c) == a/b.

Lemma div_mul_cancel_l : forall a b c, b~=0 -> 0<c ->
(c*a)/(c*b) == a/b.

Lemma mul_mod_distr_l: forall a b c, b~=0 -> 0<c ->
(c*a) mod (c*b) == c * (a mod b).

Lemma mul_mod_distr_r: forall a b c, b~=0 -> 0<c ->
(a*c) mod (b*c) == (a mod b) * c.

Operations modulo.

Theorem mod_mod: forall a n, n~=0 ->
(a mod n) mod n == a mod n.

Lemma mul_mod_idemp_l : forall a b n, n~=0 ->
((a mod n)*b) mod n == (a*b) mod n.

Lemma mul_mod_idemp_r : forall a b n, n~=0 ->
(a*(b mod n)) mod n == (a*b) mod n.

Theorem mul_mod: forall a b n, n~=0 ->
(a * b) mod n == ((a mod n) * (b mod n)) mod n.

Lemma add_mod_idemp_l : forall a b n, n~=0 ->
((a mod n)+b) mod n == (a+b) mod n.

Lemma add_mod_idemp_r : forall a b n, n~=0 ->
(a+(b mod n)) mod n == (a+b) mod n.

Theorem add_mod: forall a b n, n~=0 ->
(a+b) mod n == (a mod n + b mod n) mod n.

With the current convention, the following result isn't always true with a negative intermediate divisor. For instance 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2) and 3/(-2)/2 = -1 <> 0 = 3 / (-2*2) .

Lemma div_div : forall a b c, 0<b -> c~=0 ->
(a/b)/c == a/(b*c).

Similarly, the following result doesn't always hold when b<0. For instance 3 mod (-2*-2)) = 3 while 3 mod (-2) + (-2)*((3/-2) mod -2) = -1.

Lemma mod_mul_r : forall a b c, 0<b -> c~=0 ->
a mod (b*c) == a mod b + b*((a/b) mod c).

Lemma mod_div: forall a b, b~=0 ->
a mod b / b == 0.

A last inequality:

Theorem div_mul_le:
forall a b c, 0<=a -> 0<b -> 0<=c -> c*(a/b) <= (c*a)/b.

mod is related to divisibility

Lemma mod_divides : forall a b, b~=0 ->
(a mod b == 0 <-> (b|a)).

End ZEuclidProp.