# Library Coq.Numbers.Natural.Abstract.NDiv

Require Import NAxioms NSub NZDiv.

Properties of Euclidean Division

Module Type NDivProp (Import N : NAxiomsSig')(Import NP : NSubProp N).

We benefit from what already exists for NZ
Module Import Private_NZDiv := Nop <+ NZDivProp N N NP.

Ltac auto' := try rewrite <- neq_0_lt_0; auto using le_0_l.

Let's now state again theorems, but without useless hypothesis.

Lemma mod_upper_bound : forall a b, b ~= 0 -> a mod b < b.

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, r1<b -> r2<b ->
b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2.

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

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

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

Same situation, in term of modulo:

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

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 <-> a<b).

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

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

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

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

The previous inequality 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, b~=0 -> a < b*q -> a/b < q.

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

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

A division respects opposite monotonicity for the divisor

Lemma div_le_compat_l: forall p q r, 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_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.

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.

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

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

A last inequality:

Theorem div_mul_le:
forall a b c, b~=0 -> 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 NDivProp.