# Library Coq.Numbers.Integer.Abstract.ZDivFloor

# Euclidean Division for integers (Floor convention)

We use here the convention known as Floor, or Round-Toward-Bottom, where a/b is the closest integer below the exact fraction. It can be summarized by:
a = bq+r /\ 0 <= |r| < |b| /\ Sign(r) = Sign(b)
This is the convention followed historically by Zdiv in Coq, and corresponds to convention "F" 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 ZDivEucl for others conventions.
Require Import ZAxioms ZProperties NZDiv.
Module Type ZDivSpecific (Import Z:ZAxiomsSig')(Import DM : DivMod' Z).
Axiom mod_pos_bound : forall a b, 0 < b -> 0 <= a mod b < b.
Axiom mod_neg_bound : forall a b, b < 0 -> b < a mod b <= 0.
End ZDivSpecific.
Module Type ZDiv (Z:ZAxiomsSig)
:= DivMod Z <+ NZDivCommon Z <+ ZDivSpecific Z.
Module Type ZDivSig := ZAxiomsExtSig <+ ZDiv.
Module Type ZDivSig' := ZAxiomsExtSig' <+ ZDiv <+ DivModNotation.
Module ZDivPropFunct (Import Z : ZDivSig')(Import ZP : ZPropSig Z).
We benefit from what already exists for NZ
Module ZD <: NZDiv Z.
Definition div := div.
Definition modulo := modulo.
Definition div_wd := div_wd.
Definition mod_wd := mod_wd.
Definition div_mod := div_mod.
Lemma mod_bound : forall a b, 0<= a -> 0< b -> 0 <= a mod b < b.
End ZD.
Module Import NZDivP := NZDivPropFunct Z ZP ZD.
Another formulation of the main equation
Lemma mod_eq :
forall a b, b ~=0 -> a mod b == a - b *( a / b ).
Uniqueness theorems
Theorem div_mod_unique : forall b q1 q2 r1 r2 : t,
(0<= r1 < b \/ b < r1 <=0) -> (0<= r2 < b \/ b < r2 <=0) ->
b * q1 + r1 == b * q2 + r2 -> q1 == q2 /\ r1 == r2.
Theorem div_unique:
forall a b q r, (0<= r < b \/ b < r <=0) -> a == b * q + r -> q == a / b.
Theorem div_unique_pos:
forall a b q r, 0<= r < b -> a == b * q + r -> q == a / b.
Theorem div_unique_neg:
forall a b q r, 0<= r < b -> a == b * q + r -> q == a / b.
Theorem mod_unique:
forall a b q r, (0<= r < b \/ b < r <=0) -> a == b * q + r -> r == a mod b.
Theorem mod_unique_pos:
forall a b q r, 0<= r < b -> a == b * q + r -> r == a mod b.
Theorem mod_unique_neg:
forall a b q r, b < r <=0 -> a == b * q + r -> r == a mod b.
Sign rules
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].
Fact mod_bound_or : forall a b, b ~=0 -> 0<= a mod b < b \/ b < a mod b <=0.
Fact opp_mod_bound_or : forall a b, b ~=0 ->
0 <= -( a mod b ) < - b \/ - b < -( a mod b ) <= 0.
Lemma div_opp_opp : forall a b, b ~=0 -> - a / - b == a / b.
Lemma mod_opp_opp : forall a b, b ~=0 -> ( - a ) mod ( - b ) == - ( a mod b ).
With the current conventions, the other sign rules are rather complex.
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 ) -1.
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 == b - a mod b.
Lemma div_opp_r_z :
forall a b, b ~=0 -> a mod b == 0 -> a /( - b ) == -( a / b ).
Lemma div_opp_r_nz :
forall a b, b ~=0 -> a mod b ~= 0 -> a /( - b ) == -( a / b ) -1.
Lemma mod_opp_r_z :
forall a b, b ~=0 -> a mod b == 0 -> a mod ( - b ) == 0.
Lemma mod_opp_r_nz :
forall a b, b ~=0 -> a mod b ~= 0 -> a mod ( - b ) == ( a mod b ) - b.
The sign of a mod b is the one of b
Lemma mod_sign : forall a b, b ~=0 -> (0 <= ( a mod b ) * 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.

# Order results about mod and div

A modulo cannot grow beyond its starting point.
Theorem mod_le: forall a b, 0<= a -> 0< b -> 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 < b \/ b < a <=0).
Lemma mod_small_iff : forall a b, b ~=0 -> (a mod b == a <-> 0<= a < b \/ b < a <=0).
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.
Since we cannot speak of rational values here, we express this fact by multiplying back by b, and this leads to separates statements according to the sign of b.
First, a/b is below the exact fraction ...
Lemma mul_div_le : forall a b, 0< b -> b *( a / b ) <= a.
Lemma mul_div_ge : forall a b, b <0 -> a <= b *( a / b ).
... and moreover it is the larger such integer, since S(a/b) is strictly above the exact fraction.
Lemma mul_succ_div_gt: forall a b, 0< b -> a < b *( S (a / b)).
Lemma mul_succ_div_lt: forall a b, b <0 -> b *( S (a / b)) < a.
NB: The four 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).
Some additionnal inequalities about div.
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.
Lemma div_mul_cancel_r : forall a b c, b ~=0 -> c ~=0 ->
( a * c )/( b * c ) == a / b.
Lemma div_mul_cancel_l : forall a b c, b ~=0 -> c ~=0 ->
( c * a )/( c * b ) == a / b.
Lemma mul_mod_distr_l: forall a b c, b ~=0 -> c ~=0 ->
( c * a ) mod ( c * b ) == c * ( a mod b ).
Lemma mul_mod_distr_r: forall a b c, b ~=0 -> c ~=0 ->
( 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 for negative divisors. For instance 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2) .
Lemma div_div : forall a b c, 0< b -> 0< c ->
( a / b )/ c == a /( b * c ).
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 <-> exists c , a == b * c).
End ZDivPropFunct.