# 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.