# Library Coq.Numbers.Integer.Abstract.ZDivTrunc

# Euclidean Division for integers (Trunc convention)

We use here the convention known as Trunc, or Round-Toward-Zero, where a/b is the integer with the largest absolute value to be between zero and the exact fraction. It can be summarized by:
a = bq+r /\ 0 <= |r| < |b| /\ Sign(r) = Sign(a)
This is the convention of Ocaml and many other systems (C, ASM, ...). This convention is named "T" 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 ZDivFloor and ZDivEucl for others conventions.
Require Import ZAxioms ZProperties NZDiv.
Module Type ZDivSpecific (Import Z:ZAxiomsSig')(Import DM : DivMod' Z).
Axiom mod_bound : forall a b, 0<= a -> 0< b -> 0 <= a mod b < b.
Axiom mod_opp_l : forall a b, b ~= 0 -> ( - a ) mod b == - ( a mod b ).
Axiom mod_opp_r : forall a b, b ~= 0 -> a mod ( - b ) == a mod b.
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 Import NZDivP := NZDivPropFunct Z ZP Z.
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].
Another formulation of the main equation
Lemma mod_eq :
forall a b, b ~=0 -> a mod b == a - b *( a / b ).
A few sign rules (simple ones)
Lemma mod_opp_opp : forall a b, b ~= 0 -> ( - a ) mod ( - b ) == - ( a mod b ).
Lemma div_opp_l : forall a b, b ~= 0 -> ( - a )/ b == -( a / b ).
Lemma div_opp_r : forall a b, b ~= 0 -> a /( - b ) == -( a / b ).
Lemma div_opp_opp : forall a b, b ~= 0 -> ( - a )/( - b ) == a / b.
The sign of a mod b is the one of a
Lemma mod_sign : forall a b, b ~=0 -> 0 <= ( a mod b ) * a.
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<= a -> 0<= r < b -> a == b * q + r -> q == a / b.
Theorem mod_unique:
forall a b q r, 0<= a -> 0<= r < b -> a == b * q + r -> r == 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.

# 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 <-> abs a < abs b).
Lemma mod_small_iff : forall a b, b ~=0 -> (a mod b == a <-> abs 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.
With this choice of division, rounding of div is always done toward zero:
Lemma mul_div_le : forall a b, 0<= a -> b ~=0 -> 0 <= b *( a / b ) <= a.
Lemma mul_div_ge : forall a b, a <=0 -> b ~=0 -> a <= b *( a / b ) <= 0.
For positive numbers, considering S (a/b) leads to an upper bound for a
Lemma mul_succ_div_gt: forall a b, 0<= a -> 0< b -> a < b *( S (a / b)).
Similar results with negative numbers
Lemma mul_pred_div_lt: forall a b, a <=0 -> 0< b -> b *( P (a / b)) < a.
Lemma mul_pred_div_gt: forall a b, 0<= a -> b <0 -> a < b *( P (a / b)).
Lemma mul_succ_div_lt: forall a b, a <=0 -> b <0 -> b *( S (a / b)) < a.
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<= a -> 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

Unlike with other division conventions, some results here aren't always valid, and need to be restricted. For instance (a+b*c) mod c <> a mod c for a=9,b=-5,c=2
Lemma mod_add : forall a b c, c ~=0 -> 0 <= ( a + b * c )* a ->
( a + b * c ) mod c == a mod c.
Lemma div_add : forall a b c, c ~=0 -> 0 <= ( a + b * c )* a ->
( a + b * c ) / c == a / c + b.
Lemma div_add_l: forall a b c, b ~=0 -> 0 <= ( a * b + c )* c ->
( 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_r: forall a b c, b ~=0 -> c ~=0 ->
( a * c ) mod ( b * c ) == ( a mod b ) * c.
Lemma mul_mod_distr_l: forall a b c, b ~=0 -> c ~=0 ->
( c * a ) mod ( c * b ) == c * ( a mod b ).
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.
addition and modulo
Generally speaking, unlike with other conventions, we don't have (a+b) mod n = (a mod n + b mod n) mod n for any a and b. For instance, take (8 + (-10)) mod 3 = -2 whereas (8 mod 3 + (-10 mod 3)) mod 3 = 1.
Lemma add_mod_idemp_l : forall a b n, n ~=0 -> 0 <= a * b ->
( ( a mod n )+ b ) mod n == ( a + b ) mod n.
Lemma add_mod_idemp_r : forall a b n, n ~=0 -> 0 <= a * b ->
( a +( b mod n ) ) mod n == ( a + b ) mod n.
Theorem add_mod: forall a b n, n ~=0 -> 0 <= a * b ->
( a + b ) mod n == ( a mod n + b mod n ) mod n.
Conversely, the following result needs less restrictions here.
Lemma div_div : forall a b c, b ~=0 -> c ~=0 ->
( 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.