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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.forall a b : t, b ~= 0 -> a mod b < b
Proof.forall a b : t, b ~= 0 -> a mod b < b intros.a, b:t
H:b ~= 0
a mod b < b apply mod_bound_pos; auto'. Qed.

Another formulation of the main equation

Lemma mod_eq :
 forall a b, b~=0 -> a mod b == a - b*(a/b).forall a b : t, b ~= 0 -> a mod b == a - b * (a / b)
Proof.forall a b : t, b ~= 0 -> a mod b == a - b * (a / b)
intros.a, b:t
H:b ~= 0
a mod b == a - b * (a / b)
symmetry.a, b:t
H:b ~= 0
a - b * (a / b) == a mod b apply add_sub_eq_l.a, b:t
H:b ~= 0
b * (a / b) + a mod b == a symmetry.a, b:t
H:b ~= 0
a == b * (a / b) + a mod b
now apply div_mod.
Qed.

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.forall b q1 q2 r1 r2 : t,
r1 < b ->
r2 < b ->
b * q1 + r1 == b * q2 + r2 -> q1 == q2 /\ r1 == r2
Proof.forall b q1 q2 r1 r2 : t,
r1 < b ->
r2 < b ->
b * q1 + r1 == b * q2 + r2 -> q1 == q2 /\ r1 == r2 intros.b, q1, q2, r1, r2:t
H:r1 < b
H0:r2 < b
H1:b * q1 + r1 == b * q2 + r2
q1 == q2 /\ r1 == r2 apply div_mod_unique with b; auto'. Qed.

Theorem div_unique:
 forall a b q r, r<b -> a == b*q + r -> q == a/b.forall a b q r : t,
r < b -> a == b * q + r -> q == a / b
Proof.forall a b q r : t,
r < b -> a == b * q + r -> q == a / b intros; apply div_unique with r; auto'. Qed.

Theorem mod_unique:
 forall a b q r, r<b -> a == b*q + r -> r == a mod b.forall a b q r : t,
r < b -> a == b * q + r -> r == a mod b
Proof.forall a b q r : t,
r < b -> a == b * q + r -> r == a mod b intros.a, b, q, r:t
H:r < b
H0:a == b * q + r
r == a mod b apply mod_unique with q; auto'. Qed.

Theorem div_unique_exact: forall a b q, b~=0 -> a == b*q -> q == a/b.forall a b q : t, b ~= 0 -> a == b * q -> q == a / b
Proof.forall a b q : t, b ~= 0 -> a == b * q -> q == a / b intros.a, b, q:t
H:b ~= 0
H0:a == b * q
q == a / b apply div_unique_exact; auto'. Qed.

A division by itself returns 1

Lemma div_same : forall a, a~=0 -> a/a == 1.forall a : t, a ~= 0 -> a / a == 1
Proof.forall a : t, a ~= 0 -> a / a == 1 intros.a:t
H:a ~= 0
a / a == 1 apply div_same; auto'. Qed.

Lemma mod_same : forall a, a~=0 -> a mod a == 0.forall a : t, a ~= 0 -> a mod a == 0
Proof.forall a : t, a ~= 0 -> a mod a == 0 intros.a:t
H:a ~= 0
a mod a == 0 apply mod_same; auto'. Qed.

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

Theorem div_small: forall a b, a<b -> a/b == 0.forall a b : t, a < b -> a / b == 0
Proof.forall a b : t, a < b -> a / b == 0 intros.a, b:t
H:a < b
a / b == 0 apply div_small; auto'. Qed.

Same situation, in term of modulo:

Theorem mod_small: forall a b, a<b -> a mod b == a.forall a b : t, a < b -> a mod b == a
Proof.forall a b : t, a < b -> a mod b == a intros.a, b:t
H:a < b
a mod b == a apply mod_small; auto'. Qed.

Basic values of divisions and modulo.

Lemma div_0_l: forall a, a~=0 -> 0/a == 0.forall a : t, a ~= 0 -> 0 / a == 0
Proof.forall a : t, a ~= 0 -> 0 / a == 0 intros.a:t
H:a ~= 0
0 / a == 0 apply div_0_l; auto'. Qed.

Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0.forall a : t, a ~= 0 -> 0 mod a == 0
Proof.forall a : t, a ~= 0 -> 0 mod a == 0 intros.a:t
H:a ~= 0
0 mod a == 0 apply mod_0_l; auto'. Qed.

Lemma div_1_r: forall a, a/1 == a.forall a : t, a / 1 == a
Proof.forall a : t, a / 1 == a intros.a:t
a / 1 == a apply div_1_r; auto'. Qed.

Lemma mod_1_r: forall a, a mod 1 == 0.forall a : t, a mod 1 == 0
Proof.forall a : t, a mod 1 == 0 intros.a:t
a mod 1 == 0 apply mod_1_r; auto'. Qed.

Lemma div_1_l: forall a, 1<a -> 1/a == 0.forall a : t, 1 < a -> 1 / a == 0
Proof.forall a : t, 1 < a -> 1 / a == 0 exact div_1_l. Qed.

Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1.forall a : t, 1 < a -> 1 mod a == 1
Proof.forall a : t, 1 < a -> 1 mod a == 1 exact mod_1_l. Qed.

Lemma div_mul : forall a b, b~=0 -> (a*b)/b == a.forall a b : t, b ~= 0 -> a * b / b == a
Proof.forall a b : t, b ~= 0 -> a * b / b == a intros.a, b:t
H:b ~= 0
a * b / b == a apply div_mul; auto'. Qed.

Lemma mod_mul : forall a b, b~=0 -> (a*b) mod b == 0.forall a b : t, b ~= 0 -> (a * b) mod b == 0
Proof.forall a b : t, b ~= 0 -> (a * b) mod b == 0 intros.a, b:t
H:b ~= 0
(a * b) mod b == 0 apply mod_mul; auto'. Qed.

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.forall a b : t, b ~= 0 -> a mod b <= a
Proof.forall a b : t, b ~= 0 -> a mod b <= a intros.a, b:t
H:b ~= 0
a mod b <= a apply mod_le; auto'. Qed.

Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b.forall a b : t, 0 < b <= a -> 0 < a / b
Proof.forall a b : t, 0 < b <= a -> 0 < a / b exact div_str_pos. Qed.

Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> a<b).forall a b : t, b ~= 0 -> a / b == 0 <-> a < b
Proof.forall a b : t, b ~= 0 -> a / b == 0 <-> a < b intros.a, b:t
H:b ~= 0
a / b == 0 <-> a < b apply div_small_iff; auto'. Qed.

Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> a<b).forall a b : t, b ~= 0 -> a mod b == a <-> a < b
Proof.forall a b : t, b ~= 0 -> a mod b == a <-> a < b intros.a, b:t
H:b ~= 0
a mod b == a <-> a < b apply mod_small_iff; auto'. Qed.

Lemma div_str_pos_iff : forall a b, b~=0 -> (0<a/b <-> b<=a).forall a b : t, b ~= 0 -> 0 < a / b <-> b <= a
Proof.forall a b : t, b ~= 0 -> 0 < a / b <-> b <= a intros.a, b:t
H:b ~= 0
0 < a / b <-> b <= a apply div_str_pos_iff; auto'. Qed.

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.forall a b : t, 0 < a -> 1 < b -> a / b < a
Proof.forall a b : t, 0 < a -> 1 < b -> a / b < a exact div_lt. Qed.

le is compatible with a positive division.

Lemma div_le_mono : forall a b c, c~=0 -> a<=b -> a/c <= b/c.forall a b c : t, c ~= 0 -> a <= b -> a / c <= b / c
Proof.forall a b c : t, c ~= 0 -> a <= b -> a / c <= b / c intros.a, b, c:t
H:c ~= 0
H0:a <= b
a / c <= b / c apply div_le_mono; auto'. Qed.

Lemma mul_div_le : forall a b, b~=0 -> b*(a/b) <= a.forall a b : t, b ~= 0 -> b * (a / b) <= a
Proof.forall a b : t, b ~= 0 -> b * (a / b) <= a intros.a, b:t
H:b ~= 0
b * (a / b) <= a apply mul_div_le; auto'. Qed.

Lemma mul_succ_div_gt: forall a b, b~=0 -> a < b*(S (a/b)).forall a b : t, b ~= 0 -> a < b * S (a / b)
Proof.forall a b : t, b ~= 0 -> a < b * S (a / b) intros; apply mul_succ_div_gt; auto'. Qed.

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).forall a b : t,
b ~= 0 -> a == b * (a / b) <-> a mod b == 0
Proof.forall a b : t,
b ~= 0 -> a == b * (a / b) <-> a mod b == 0 intros.a, b:t
H:b ~= 0
a == b * (a / b) <-> a mod b == 0 apply div_exact; auto'. Qed.

Some additional inequalities about div.

Theorem div_lt_upper_bound:
  forall a b q, b~=0 -> a < b*q -> a/b < q.forall a b q : t, b ~= 0 -> a < b * q -> a / b < q
Proof.forall a b q : t, b ~= 0 -> a < b * q -> a / b < q intros.a, b, q:t
H:b ~= 0
H0:a < b * q
a / b < q apply div_lt_upper_bound; auto'. Qed.

Theorem div_le_upper_bound:
  forall a b q, b~=0 -> a <= b*q -> a/b <= q.forall a b q : t, b ~= 0 -> a <= b * q -> a / b <= q
Proof.forall a b q : t, b ~= 0 -> a <= b * q -> a / b <= q intros; apply div_le_upper_bound; auto'. Qed.

Theorem div_le_lower_bound:
  forall a b q, b~=0 -> b*q <= a -> q <= a/b.forall a b q : t, b ~= 0 -> b * q <= a -> q <= a / b
Proof.forall a b q : t, b ~= 0 -> b * q <= a -> q <= a / b intros; apply div_le_lower_bound; auto'. Qed.

A division respects opposite monotonicity for the divisor

Lemma div_le_compat_l: forall p q r, 0<q<=r -> p/r <= p/q.forall p q r : t, 0 < q <= r -> p / r <= p / q
Proof.forall p q r : t, 0 < q <= r -> p / r <= p / q intros.p, q, r:t
H:0 < q <= r
p / r <= p / q apply div_le_compat_l.p, q, r:t
H:0 < q <= r
0 <= p
p, q, r:t
H:0 < q <= r
0 < q <= r auto'.p, q, r:t
H:0 < q <= r
0 < q <= r auto. Qed.

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.forall a b c : t,
c ~= 0 -> (a + b * c) mod c == a mod c
Proof.forall a b c : t,
c ~= 0 -> (a + b * c) mod c == a mod c intros.a, b, c:t
H:c ~= 0
(a + b * c) mod c == a mod c apply mod_add; auto'. Qed.

Lemma div_add : forall a b c, c~=0 ->
 (a + b * c) / c == a / c + b.forall a b c : t,
c ~= 0 -> (a + b * c) / c == a / c + b
Proof.forall a b c : t,
c ~= 0 -> (a + b * c) / c == a / c + b intros.a, b, c:t
H:c ~= 0
(a + b * c) / c == a / c + b apply div_add; auto'. Qed.

Lemma div_add_l: forall a b c, b~=0 ->
 (a * b + c) / b == a + c / b.forall a b c : t,
b ~= 0 -> (a * b + c) / b == a + c / b
Proof.forall a b c : t,
b ~= 0 -> (a * b + c) / b == a + c / b intros.a, b, c:t
H:b ~= 0
(a * b + c) / b == a + c / b apply div_add_l; auto'. Qed.

Cancellations.

Lemma div_mul_cancel_r : forall a b c, b~=0 -> c~=0 ->
 (a*c)/(b*c) == a/b.forall a b c : t,
b ~= 0 -> c ~= 0 -> a * c / (b * c) == a / b
Proof.forall a b c : t,
b ~= 0 -> c ~= 0 -> a * c / (b * c) == a / b intros.a, b, c:t
H:b ~= 0
H0:c ~= 0
a * c / (b * c) == a / b apply div_mul_cancel_r; auto'. Qed.

Lemma div_mul_cancel_l : forall a b c, b~=0 -> c~=0 ->
 (c*a)/(c*b) == a/b.forall a b c : t,
b ~= 0 -> c ~= 0 -> c * a / (c * b) == a / b
Proof.forall a b c : t,
b ~= 0 -> c ~= 0 -> c * a / (c * b) == a / b intros.a, b, c:t
H:b ~= 0
H0:c ~= 0
c * a / (c * b) == a / b apply div_mul_cancel_l; auto'. Qed.

Lemma mul_mod_distr_r: forall a b c, b~=0 -> c~=0 ->
  (a*c) mod (b*c) == (a mod b) * c.forall a b c : t,
b ~= 0 -> c ~= 0 -> (a * c) mod (b * c) == a mod b * c
Proof.forall a b c : t,
b ~= 0 -> c ~= 0 -> (a * c) mod (b * c) == a mod b * c intros.a, b, c:t
H:b ~= 0
H0:c ~= 0
(a * c) mod (b * c) == a mod b * c apply mul_mod_distr_r; auto'. Qed.

Lemma mul_mod_distr_l: forall a b c, b~=0 -> c~=0 ->
  (c*a) mod (c*b) == c * (a mod b).forall a b c : t,
b ~= 0 ->
c ~= 0 -> (c * a) mod (c * b) == c * (a mod b)
Proof.forall a b c : t,
b ~= 0 ->
c ~= 0 -> (c * a) mod (c * b) == c * (a mod b) intros.a, b, c:t
H:b ~= 0
H0:c ~= 0
(c * a) mod (c * b) == c * (a mod b) apply mul_mod_distr_l; auto'. Qed.

Operations modulo.

Theorem mod_mod: forall a n, n~=0 ->
 (a mod n) mod n == a mod n.forall a n : t, n ~= 0 -> (a mod n) mod n == a mod n
Proof.forall a n : t, n ~= 0 -> (a mod n) mod n == a mod n intros.a, n:t
H:n ~= 0
(a mod n) mod n == a mod n apply mod_mod; auto'. Qed.

Lemma mul_mod_idemp_l : forall a b n, n~=0 ->
 ((a mod n)*b) mod n == (a*b) mod n.forall a b n : t,
n ~= 0 -> (a mod n * b) mod n == (a * b) mod n
Proof.forall a b n : t,
n ~= 0 -> (a mod n * b) mod n == (a * b) mod n intros.a, b, n:t
H:n ~= 0
(a mod n * b) mod n == (a * b) mod n apply mul_mod_idemp_l; auto'. Qed.

Lemma mul_mod_idemp_r : forall a b n, n~=0 ->
 (a*(b mod n)) mod n == (a*b) mod n.forall a b n : t,
n ~= 0 -> (a * (b mod n)) mod n == (a * b) mod n
Proof.forall a b n : t,
n ~= 0 -> (a * (b mod n)) mod n == (a * b) mod n intros.a, b, n:t
H:n ~= 0
(a * (b mod n)) mod n == (a * b) mod n apply mul_mod_idemp_r; auto'. Qed.

Theorem mul_mod: forall a b n, n~=0 ->
 (a * b) mod n == ((a mod n) * (b mod n)) mod n.forall a b n : t,
n ~= 0 -> (a * b) mod n == (a mod n * (b mod n)) mod n
Proof.forall a b n : t,
n ~= 0 -> (a * b) mod n == (a mod n * (b mod n)) mod n intros.a, b, n:t
H:n ~= 0
(a * b) mod n == (a mod n * (b mod n)) mod n apply mul_mod; auto'. Qed.

Lemma add_mod_idemp_l : forall a b n, n~=0 ->
 ((a mod n)+b) mod n == (a+b) mod n.forall a b n : t,
n ~= 0 -> (a mod n + b) mod n == (a + b) mod n
Proof.forall a b n : t,
n ~= 0 -> (a mod n + b) mod n == (a + b) mod n intros.a, b, n:t
H:n ~= 0
(a mod n + b) mod n == (a + b) mod n apply add_mod_idemp_l; auto'. Qed.

Lemma add_mod_idemp_r : forall a b n, n~=0 ->
 (a+(b mod n)) mod n == (a+b) mod n.forall a b n : t,
n ~= 0 -> (a + b mod n) mod n == (a + b) mod n
Proof.forall a b n : t,
n ~= 0 -> (a + b mod n) mod n == (a + b) mod n intros.a, b, n:t
H:n ~= 0
(a + b mod n) mod n == (a + b) mod n apply add_mod_idemp_r; auto'. Qed.

Theorem add_mod: forall a b n, n~=0 ->
 (a+b) mod n == (a mod n + b mod n) mod n.forall a b n : t,
n ~= 0 -> (a + b) mod n == (a mod n + b mod n) mod n
Proof.forall a b n : t,
n ~= 0 -> (a + b) mod n == (a mod n + b mod n) mod n intros.a, b, n:t
H:n ~= 0
(a + b) mod n == (a mod n + b mod n) mod n apply add_mod; auto'. Qed.

Lemma div_div : forall a b c, b~=0 -> c~=0 ->
 (a/b)/c == a/(b*c).forall a b c : t,
b ~= 0 -> c ~= 0 -> a / b / c == a / (b * c)
Proof.forall a b c : t,
b ~= 0 -> c ~= 0 -> a / b / c == a / (b * c) intros.a, b, c:t
H:b ~= 0
H0:c ~= 0
a / b / c == a / (b * c) apply div_div; auto'. Qed.

Lemma mod_mul_r : forall a b c, b~=0 -> c~=0 ->
 a mod (b*c) == a mod b + b*((a/b) mod c).forall a b c : t,
b ~= 0 ->
c ~= 0 ->
a mod (b * c) == a mod b + b * ((a / b) mod c)
Proof.forall a b c : t,
b ~= 0 ->
c ~= 0 ->
a mod (b * c) == a mod b + b * ((a / b) mod c) intros.a, b, c:t
H:b ~= 0
H0:c ~= 0
a mod (b * c) == a mod b + b * ((a / b) mod c) apply mod_mul_r; auto'. Qed.

A last inequality:

Theorem div_mul_le:
 forall a b c, b~=0 -> c*(a/b) <= (c*a)/b.forall a b c : t, b ~= 0 -> c * (a / b) <= c * a / b
Proof.forall a b c : t, b ~= 0 -> c * (a / b) <= c * a / b intros.a, b, c:t
H:b ~= 0
c * (a / b) <= c * a / b apply div_mul_le; auto'. Qed.

mod is related to divisibility

Lemma mod_divides : forall a b, b~=0 ->
 (a mod b == 0 <-> exists c, a == b*c).forall a b : t,
b ~= 0 -> a mod b == 0 <-> (exists c : t, a == b * c)
Proof.forall a b : t,
b ~= 0 -> a mod b == 0 <-> (exists c : t, a == b * c) intros.a, b:t
H:b ~= 0
a mod b == 0 <-> (exists c : t, a == b * c) apply mod_divides; auto'. Qed.

End NDivProp.