# Library Coq.Numbers.NatInt.NZDiv

Euclidean Division
Require Import NZAxioms NZMulOrder.
The first signatures will be common to all divisions over NZ, N and Z
Module Type DivMod (Import A : Typ).
Parameters Inline div modulo : t -> t -> t.
End DivMod.
Module Type DivModNotation (A : Typ)(Import B : DivMod A).
Infix "/" := div.
Infix "mod" := modulo (at level 40, no associativity).
End DivModNotation.
Module Type DivMod' (A : Typ) := DivMod A <+ DivModNotation A.
Module Type NZDivSpec (Import A : NZOrdAxiomsSig')(Import B : DivMod' A).
Declare Instance div_wd : Proper (eq ==> eq ==> eq) div.
Declare Instance mod_wd : Proper (eq ==> eq ==> eq) modulo.
Axiom div_mod : forall a b, b ~= 0 -> a == b *( a / b ) + ( a mod b ).
Axiom mod_bound_pos : forall a b, 0<= a -> 0< b -> 0 <= a mod b < b.
End NZDivSpec.
The different divisions will only differ in the conditions they impose on modulo. For NZ, we have only described the behavior on positive numbers.
Module Type NZDiv (A : NZOrdAxiomsSig) := DivMod A <+ NZDivSpec A.
Module Type NZDiv' (A : NZOrdAxiomsSig) := NZDiv A <+ DivModNotation A.
Module Type NZDivProp
(Import A : NZOrdAxiomsSig')
(Import B : NZDiv' A)
(Import C : NZMulOrderProp A).
Uniqueness theorems
Theorem div_mod_unique :
forall b q1 q2 r1 r2, 0<= r1 < b -> 0<= r2 < b ->
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.
Theorem div_unique_exact a b q:
0<= a -> 0< b -> a == b * q -> q == a / b.
A division by itself returns 1
Lemma div_same : forall a, 0< a -> a / a == 1.
Lemma mod_same : forall a, 0< a -> 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, 0< a -> 0/ a == 0.
Lemma mod_0_l: forall a, 0< a -> 0 mod a == 0.
Lemma div_1_r: forall a, 0<= a -> a /1 == a.
Lemma mod_1_r: forall a, 0<= 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, 0<= a -> 0< b -> ( a * b )/ b == a.
Lemma mod_mul : forall a b, 0<= a -> 0< b -> ( 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.
Lemma 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, 0<= a -> 0< b -> (a / b ==0 <-> a < b).
Lemma mod_small_iff : forall a b, 0<= a -> 0< b -> (a mod b == a <-> a < b).
Lemma div_str_pos_iff : forall a b, 0<= a -> 0< b -> (0< a / b <-> b <= a).
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 -> 0<= a <= b -> a / c <= b / c.
The following two properties could be used as specification of div
Lemma mul_div_le : forall a b, 0<= a -> 0< b -> b *( a / b ) <= a.
Lemma mul_succ_div_gt : forall a b, 0<= a -> 0< b -> a < b *( S (a / b)).
The previous inequality is exact iff the modulo is zero.
Lemma div_exact : forall a b, 0<= a -> 0< b -> (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<= a -> 0< b -> a <= b * q -> a / b <= q.
Theorem div_le_lower_bound:
forall a b q, 0<= a -> 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, 0<= a -> 0<= a + b * c -> 0< c ->
( a + b * c ) mod c == a mod c.
Lemma div_add : forall a b c, 0<= a -> 0<= a + b * c -> 0< c ->
( a + b * c ) / c == a / c + b.
Lemma div_add_l: forall a b c, 0<= c -> 0<= a * b + c -> 0< b ->
( a * b + c ) / b == a + c / b.
Cancellations.
Lemma div_mul_cancel_r : forall a b c, 0<= a -> 0< b -> 0< c ->
( a * c )/( b * c ) == a / b.
Lemma div_mul_cancel_l : forall a b c, 0<= a -> 0< b -> 0< c ->
( c * a )/( c * b ) == a / b.
Lemma mul_mod_distr_l: forall a b c, 0<= a -> 0< b -> 0< c ->
( c * a ) mod ( c * b ) == c * ( a mod b ).
Lemma mul_mod_distr_r: forall a b c, 0<= a -> 0< b -> 0< c ->
( a * c ) mod ( b * c ) == ( a mod b ) * c.
Operations modulo.
Theorem mod_mod: forall a n, 0<= a -> 0< n ->
( a mod n ) mod n == a mod n.
Lemma mul_mod_idemp_l : forall a b n, 0<= a -> 0<= b -> 0< n ->
( ( a mod n )* b ) mod n == ( a * b ) mod n.
Lemma mul_mod_idemp_r : forall a b n, 0<= a -> 0<= b -> 0< n ->
( a *( b mod n ) ) mod n == ( a * b ) mod n.
Theorem mul_mod: forall a b n, 0<= a -> 0<= b -> 0< n ->
( a * b ) mod n == ( ( a mod n ) * ( b mod n ) ) mod n.
Lemma add_mod_idemp_l : forall a b n, 0<= a -> 0<= b -> 0< n ->
( ( a mod n )+ b ) mod n == ( a + b ) mod n.
Lemma add_mod_idemp_r : forall a b n, 0<= a -> 0<= b -> 0< n ->
( a +( b mod n ) ) mod n == ( a + b ) mod n.
Theorem add_mod: forall a b n, 0<= a -> 0<= b -> 0< n ->
( a + b ) mod n == ( a mod n + b mod n ) mod n.
Lemma div_div : forall a b c, 0<= a -> 0< b -> 0< c ->
( a / b )/ c == a /( b * c ).
Lemma mod_mul_r : forall a b c, 0<= a -> 0< b -> 0< c ->
a mod ( b * c ) == a mod b + b *( ( a / b ) mod 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, 0<= a -> 0< b ->
(a mod b == 0 <-> exists c , a == b * c).
End NZDivProp.