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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.forall b q1 q2 r1 r2 : t,
0 <= r1 < b ->
0 <= r2 < b ->
b * q1 + r1 == b * q2 + r2 -> q1 == q2 /\ r1 == r2
Proof.forall b q1 q2 r1 r2 : t,
0 <= r1 < b ->
0 <= r2 < b ->
b * q1 + r1 == b * q2 + r2 -> q1 == q2 /\ r1 == r2
intros b.b:t
forall q1 q2 r1 r2 : t,
0 <= r1 < b ->
0 <= r2 < b ->
b * q1 + r1 == b * q2 + r2 -> q1 == q2 /\ r1 == r2
assert (U : forall q1 q2 r1 r2,
            b*q1+r1 == b*q2+r2 -> 0<=r1<b -> 0<=r2 -> q1<q2 -> False).b:t
forall q1 q2 r1 r2 : t,
b * q1 + r1 == b * q2 + r2 ->
0 <= r1 < b -> 0 <= r2 -> q1 < q2 -> False
b:t
U:forall q1 q2 r1 r2 : t,
b * q1 + r1 == b * q2 + r2 ->
0 <= r1 < b -> 0 <= r2 -> q1 < q2 -> False
forall q1 q2 r1 r2 : t,
0 <= r1 < b ->
0 <= r2 < b ->
b * q1 + r1 == b * q2 + r2 -> q1 == q2 /\ r1 == r2
-b:t
forall q1 q2 r1 r2 : t,
b * q1 + r1 == b * q2 + r2 ->
0 <= r1 < b -> 0 <= r2 -> q1 < q2 -> False intros q1 q2 r1 r2 EQ LT Hr1 Hr2.b, q1, q2, r1, r2:t
EQ:b * q1 + r1 == b * q2 + r2
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
False
  contradict EQ.b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q1 + r1 ~= b * q2 + r2
  apply lt_neq.b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q1 + r1 < b * q2 + r2
  apply lt_le_trans with (b*q1+b).b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q1 + r1 < b * q1 + b
b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q1 + b <= b * q2 + r2
  +b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q1 + r1 < b * q1 + b rewrite <- add_lt_mono_l.b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
r1 < b tauto.
  +b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q1 + b <= b * q2 + r2 apply le_trans with (b*q2).b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q1 + b <= b * q2
b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q2 <= b * q2 + r2
    *b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q1 + b <= b * q2 rewrite mul_comm, <- mul_succ_l, mul_comm.b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * S q1 <= b * q2
      apply mul_le_mono_nonneg_l; intuition; try order.b, q1, q2, r1, r2:t
Hr1:0 <= r2
Hr2:q1 < q2
H:0 <= r1
H0:r1 < b
S q1 <= q2
      rewrite le_succ_l; auto.
    *b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q2 <= b * q2 + r2 rewrite <- (add_0_r (b*q2)) at 1.b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
b * q2 + 0 <= b * q2 + r2
      rewrite <- add_le_mono_l.b, q1, q2, r1, r2:t
LT:0 <= r1 < b
Hr1:0 <= r2
Hr2:q1 < q2
0 <= r2 tauto.

-b:t
U:forall q1 q2 r1 r2 : t,
b * q1 + r1 == b * q2 + r2 ->
0 <= r1 < b -> 0 <= r2 -> q1 < q2 -> False
forall q1 q2 r1 r2 : t,
0 <= r1 < b ->
0 <= r2 < b ->
b * q1 + r1 == b * q2 + r2 -> q1 == q2 /\ r1 == r2 intros q1 q2 r1 r2 Hr1 Hr2 EQ; destruct (lt_trichotomy q1 q2) as [LT|[EQ'|GT]].b:t
U:forall q0 q3 r0 r3 : t,
b * q0 + r0 == b * q3 + r3 ->
0 <= r0 < b -> 0 <= r3 -> q0 < q3 -> False
q1, q2, r1, r2:t
Hr1:0 <= r1 < b
Hr2:0 <= r2 < b
EQ:b * q1 + r1 == b * q2 + r2
LT:q1 < q2
q1 == q2 /\ r1 == r2
b:t
U:forall q0 q3 r0 r3 : t,
b * q0 + r0 == b * q3 + r3 ->
0 <= r0 < b -> 0 <= r3 -> q0 < q3 -> False
q1, q2, r1, r2:t
Hr1:0 <= r1 < b
Hr2:0 <= r2 < b
EQ:b * q1 + r1 == b * q2 + r2
EQ':q1 == q2
q1 == q2 /\ r1 == r2
b:t
U:forall q0 q3 r0 r3 : t,
b * q0 + r0 == b * q3 + r3 ->
0 <= r0 < b -> 0 <= r3 -> q0 < q3 -> False
q1, q2, r1, r2:t
Hr1:0 <= r1 < b
Hr2:0 <= r2 < b
EQ:b * q1 + r1 == b * q2 + r2
GT:q2 < q1
q1 == q2 /\ r1 == r2
  +b:t
U:forall q0 q3 r0 r3 : t,
b * q0 + r0 == b * q3 + r3 ->
0 <= r0 < b -> 0 <= r3 -> q0 < q3 -> False
q1, q2, r1, r2:t
Hr1:0 <= r1 < b
Hr2:0 <= r2 < b
EQ:b * q1 + r1 == b * q2 + r2
LT:q1 < q2
q1 == q2 /\ r1 == r2 elim (U q1 q2 r1 r2); intuition.
  +b:t
U:forall q0 q3 r0 r3 : t,
b * q0 + r0 == b * q3 + r3 ->
0 <= r0 < b -> 0 <= r3 -> q0 < q3 -> False
q1, q2, r1, r2:t
Hr1:0 <= r1 < b
Hr2:0 <= r2 < b
EQ:b * q1 + r1 == b * q2 + r2
EQ':q1 == q2
q1 == q2 /\ r1 == r2 split; auto.b:t
U:forall q0 q3 r0 r3 : t,
b * q0 + r0 == b * q3 + r3 ->
0 <= r0 < b -> 0 <= r3 -> q0 < q3 -> False
q1, q2, r1, r2:t
Hr1:0 <= r1 < b
Hr2:0 <= r2 < b
EQ:b * q1 + r1 == b * q2 + r2
EQ':q1 == q2
r1 == r2 rewrite EQ' in EQ.b:t
U:forall q0 q3 r0 r3 : t,
b * q0 + r0 == b * q3 + r3 ->
0 <= r0 < b -> 0 <= r3 -> q0 < q3 -> False
q1, q2, r1, r2:t
Hr1:0 <= r1 < b
Hr2:0 <= r2 < b
EQ:b * q2 + r1 == b * q2 + r2
EQ':q1 == q2
r1 == r2 rewrite add_cancel_l in EQ; auto.
  +b:t
U:forall q0 q3 r0 r3 : t,
b * q0 + r0 == b * q3 + r3 ->
0 <= r0 < b -> 0 <= r3 -> q0 < q3 -> False
q1, q2, r1, r2:t
Hr1:0 <= r1 < b
Hr2:0 <= r2 < b
EQ:b * q1 + r1 == b * q2 + r2
GT:q2 < q1
q1 == q2 /\ r1 == r2 elim (U q2 q1 r2 r1); intuition.
Qed.

Theorem div_unique:
 forall a b q r, 0<=a -> 0<=r<b ->
   a == b*q + r -> q == a/b.forall a b q r : t,
0 <= a -> 0 <= r < b -> a == b * q + r -> q == a / b
Proof.forall a b q r : t,
0 <= a -> 0 <= r < b -> a == b * q + r -> q == a / b
intros a b q r Ha (Hb,Hr) EQ.a, b, q, r:t
Ha:0 <= a
Hb:0 <= r
Hr:r < b
EQ:a == b * q + r
q == a / b
destruct (div_mod_unique b q (a/b) r (a mod b)); auto.a, b, q, r:t
Ha:0 <= a
Hb:0 <= r
Hr:r < b
EQ:a == b * q + r
0 <= a mod b < b
a, b, q, r:t
Ha:0 <= a
Hb:0 <= r
Hr:r < b
EQ:a == b * q + r
b * q + r == b * (a / b) + a mod b
-a, b, q, r:t
Ha:0 <= a
Hb:0 <= r
Hr:r < b
EQ:a == b * q + r
0 <= a mod b < b apply mod_bound_pos; order.
-a, b, q, r:t
Ha:0 <= a
Hb:0 <= r
Hr:r < b
EQ:a == b * q + r
b * q + r == b * (a / b) + a mod b rewrite <- div_mod; order.
Qed.

Theorem mod_unique:
 forall a b q r, 0<=a -> 0<=r<b ->
  a == b*q + r -> r == a mod b.forall a b q r : t,
0 <= a -> 0 <= r < b -> a == b * q + r -> r == a mod b
Proof.forall a b q r : t,
0 <= a -> 0 <= r < b -> a == b * q + r -> r == a mod b
intros a b q r Ha (Hb,Hr) EQ.a, b, q, r:t
Ha:0 <= a
Hb:0 <= r
Hr:r < b
EQ:a == b * q + r
r == a mod b
destruct (div_mod_unique b q (a/b) r (a mod b)); auto.a, b, q, r:t
Ha:0 <= a
Hb:0 <= r
Hr:r < b
EQ:a == b * q + r
0 <= a mod b < b
a, b, q, r:t
Ha:0 <= a
Hb:0 <= r
Hr:r < b
EQ:a == b * q + r
b * q + r == b * (a / b) + a mod b
-a, b, q, r:t
Ha:0 <= a
Hb:0 <= r
Hr:r < b
EQ:a == b * q + r
0 <= a mod b < b apply mod_bound_pos; order.
-a, b, q, r:t
Ha:0 <= a
Hb:0 <= r
Hr:r < b
EQ:a == b * q + r
b * q + r == b * (a / b) + a mod b rewrite <- div_mod; order.
Qed.

Theorem div_unique_exact a b q:
 0<=a -> 0<b -> a == b*q -> q == a/b.a, b, q:t
0 <= a -> 0 < b -> a == b * q -> q == a / b
Proof.a, b, q:t
0 <= a -> 0 < b -> a == b * q -> q == a / b
 intros Ha Hb H.a, b, q:t
Ha:0 <= a
Hb:0 < b
H:a == b * q
q == a / b apply div_unique with 0; nzsimpl; now try split.
Qed.

A division by itself returns 1

Lemma div_same : forall a, 0<a -> a/a == 1.forall a : t, 0 < a -> a / a == 1
Proof.forall a : t, 0 < a -> a / a == 1
intros.a:t
H:0 < a
a / a == 1 symmetry.a:t
H:0 < a
1 == a / a apply div_unique_exact; nzsimpl; order.
Qed.

Lemma mod_same : forall a, 0<a -> a mod a == 0.forall a : t, 0 < a -> a mod a == 0
Proof.forall a : t, 0 < a -> a mod a == 0
intros.a:t
H:0 < a
a mod a == 0 symmetry.a:t
H:0 < a
0 == a mod a
apply mod_unique with 1; intuition; try order.a:t
H:0 < a
a == a * 1 + 0
now nzsimpl.
Qed.

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

Theorem div_small: forall a b, 0<=a<b -> a/b == 0.forall a b : t, 0 <= a < b -> a / b == 0
Proof.forall a b : t, 0 <= a < b -> a / b == 0
intros.a, b:t
H:0 <= a < b
a / b == 0 symmetry.a, b:t
H:0 <= a < b
0 == a / b
apply div_unique with a; intuition; try order.a, b:t
H0:0 <= a
H1:a < b
a == b * 0 + a
now nzsimpl.
Qed.

Same situation, in term of modulo:

Theorem mod_small: forall a b, 0<=a<b -> a mod b == a.forall a b : t, 0 <= a < b -> a mod b == a
Proof.forall a b : t, 0 <= a < b -> a mod b == a
intros.a, b:t
H:0 <= a < b
a mod b == a symmetry.a, b:t
H:0 <= a < b
a == a mod b
apply mod_unique with 0; intuition; try order.a, b:t
H0:0 <= a
H1:a < b
a == b * 0 + a
now nzsimpl.
Qed.

Basic values of divisions and modulo.

Lemma div_0_l: forall a, 0<a -> 0/a == 0.forall a : t, 0 < a -> 0 / a == 0
Proof.forall a : t, 0 < a -> 0 / a == 0
intros; apply div_small; split; order.
Qed.

Lemma mod_0_l: forall a, 0<a -> 0 mod a == 0.forall a : t, 0 < a -> 0 mod a == 0
Proof.forall a : t, 0 < a -> 0 mod a == 0
intros; apply mod_small; split; order.
Qed.

Lemma div_1_r: forall a, 0<=a -> a/1 == a.forall a : t, 0 <= a -> a / 1 == a
Proof.forall a : t, 0 <= a -> a / 1 == a
intros.a:t
H:0 <= a
a / 1 == a symmetry.a:t
H:0 <= a
a == a / 1 apply div_unique_exact; nzsimpl; order'.
Qed.

Lemma mod_1_r: forall a, 0<=a -> a mod 1 == 0.forall a : t, 0 <= a -> a mod 1 == 0
Proof.forall a : t, 0 <= a -> a mod 1 == 0
intros.a:t
H:0 <= a
a mod 1 == 0 symmetry.a:t
H:0 <= a
0 == a mod 1
apply mod_unique with a; try split; try order; try apply lt_0_1.a:t
H:0 <= a
a == 1 * a + 0
now nzsimpl.
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
intros; apply div_small; split; auto.a:t
H:1 < a
0 <= 1 apply le_0_1.
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
intros; apply mod_small; split; auto.a:t
H:1 < a
0 <= 1 apply le_0_1.
Qed.

Lemma div_mul : forall a b, 0<=a -> 0<b -> (a*b)/b == a.forall a b : t, 0 <= a -> 0 < b -> a * b / b == a
Proof.forall a b : t, 0 <= a -> 0 < b -> a * b / b == a
intros; symmetry.a, b:t
H:0 <= a
H0:0 < b
a == a * b / b apply div_unique_exact; trivial.a, b:t
H:0 <= a
H0:0 < b
0 <= a * b
a, b:t
H:0 <= a
H0:0 < b
a * b == b * a
-a, b:t
H:0 <= a
H0:0 < b
0 <= a * b apply mul_nonneg_nonneg; order.
-a, b:t
H:0 <= a
H0:0 < b
a * b == b * a apply mul_comm.
Qed.

Lemma mod_mul : forall a b, 0<=a -> 0<b -> (a*b) mod b == 0.forall a b : t, 0 <= a -> 0 < b -> (a * b) mod b == 0
Proof.forall a b : t, 0 <= a -> 0 < b -> (a * b) mod b == 0
intros; symmetry.a, b:t
H:0 <= a
H0:0 < b
0 == (a * b) mod b
apply mod_unique with a; try split; try order.a, b:t
H:0 <= a
H0:0 < b
0 <= a * b
a, b:t
H:0 <= a
H0:0 < b
a * b == b * a + 0
-a, b:t
H:0 <= a
H0:0 < b
0 <= a * b apply mul_nonneg_nonneg; order.
-a, b:t
H:0 <= a
H0:0 < b
a * b == b * a + 0 nzsimpl; apply mul_comm.
Qed.

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.forall a b : t, 0 <= a -> 0 < b -> a mod b <= a
Proof.forall a b : t, 0 <= a -> 0 < b -> a mod b <= a
intros.a, b:t
H:0 <= a
H0:0 < b
a mod b <= a destruct (le_gt_cases b a).a, b:t
H:0 <= a
H0:0 < b
H1:b <= a
a mod b <= a
a, b:t
H:0 <= a
H0:0 < b
H1:a < b
a mod b <= a
-a, b:t
H:0 <= a
H0:0 < b
H1:b <= a
a mod b <= a apply le_trans with b; auto.a, b:t
H:0 <= a
H0:0 < b
H1:b <= a
a mod b <= b
  apply lt_le_incl.a, b:t
H:0 <= a
H0:0 < b
H1:b <= a
a mod b < b destruct (mod_bound_pos a b); auto.
-a, b:t
H:0 <= a
H0:0 < b
H1:a < b
a mod b <= a rewrite lt_eq_cases; right.a, b:t
H:0 <= a
H0:0 < b
H1:a < b
a mod b == a
  apply mod_small; auto.
Qed.


(* Division of positive numbers is positive. *)

Lemma div_pos: forall a b, 0<=a -> 0<b -> 0 <= a/b.forall a b : t, 0 <= a -> 0 < b -> 0 <= a / b
Proof.forall a b : t, 0 <= a -> 0 < b -> 0 <= a / b
intros.a, b:t
H:0 <= a
H0:0 < b
0 <= a / b
rewrite (mul_le_mono_pos_l _ _ b); auto; nzsimpl.a, b:t
H:0 <= a
H0:0 < b
0 <= b * (a / b)
rewrite (add_le_mono_r _ _ (a mod b)).a, b:t
H:0 <= a
H0:0 < b
0 + a mod b <= b * (a / b) + a mod b
rewrite <- div_mod by order.a, b:t
H:0 <= a
H0:0 < b
0 + a mod b <= a
nzsimpl.a, b:t
H:0 <= a
H0:0 < b
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
intros a b (Hb,Hab).a, b:t
Hb:0 < b
Hab:b <= a
0 < a / b
assert (LE : 0 <= a/b) by (apply div_pos; order).a, b:t
Hb:0 < b
Hab:b <= a
LE:0 <= a / b
0 < a / b
assert (MOD : a mod b < b) by (destruct (mod_bound_pos a b); order).a, b:t
Hb:0 < b
Hab:b <= a
LE:0 <= a / b
MOD:a mod b < b
0 < a / b
rewrite lt_eq_cases in LE; destruct LE as [LT|EQ]; auto.a, b:t
Hb:0 < b
Hab:b <= a
EQ:0 == a / b
MOD:a mod b < b
0 < a / b
exfalso; revert Hab.a, b:t
Hb:0 < b
EQ:0 == a / b
MOD:a mod b < b
b <= a -> False
rewrite (div_mod a b), <-EQ; nzsimpl; order.
Qed.

Lemma div_small_iff : forall a b, 0<=a -> 0<b -> (a/b==0 <-> a<b).forall a b : t,
0 <= a -> 0 < b -> a / b == 0 <-> a < b
Proof.forall a b : t,
0 <= a -> 0 < b -> a / b == 0 <-> a < b
intros a b Ha Hb; split; intros Hab.a, b:t
Ha:0 <= a
Hb:0 < b
Hab:a / b == 0
a < b
a, b:t
Ha:0 <= a
Hb:0 < b
Hab:a < b
a / b == 0
-a, b:t
Ha:0 <= a
Hb:0 < b
Hab:a / b == 0
a < b destruct (lt_ge_cases a b); auto.a, b:t
Ha:0 <= a
Hb:0 < b
Hab:a / b == 0
H:b <= a
a < b
  symmetry in Hab.a, b:t
Ha:0 <= a
Hb:0 < b
Hab:0 == a / b
H:b <= a
a < b contradict Hab.a, b:t
Ha:0 <= a
Hb:0 < b
H:b <= a
0 ~= a / b apply lt_neq, div_str_pos; auto.
-a, b:t
Ha:0 <= a
Hb:0 < b
Hab:a < b
a / b == 0 apply div_small; auto.
Qed.

Lemma mod_small_iff : forall a b, 0<=a -> 0<b -> (a mod b == a <-> a<b).forall a b : t,
0 <= a -> 0 < b -> a mod b == a <-> a < b
Proof.forall a b : t,
0 <= a -> 0 < b -> a mod b == a <-> a < b
intros a b Ha Hb.a, b:t
Ha:0 <= a
Hb:0 < b
a mod b == a <-> a < b split; intros H; auto using mod_small.a, b:t
Ha:0 <= a
Hb:0 < b
H:a mod b == a
a < b
rewrite <- div_small_iff; auto.a, b:t
Ha:0 <= a
Hb:0 < b
H:a mod b == a
a / b == 0
rewrite <- (mul_cancel_l _ _ b) by order.a, b:t
Ha:0 <= a
Hb:0 < b
H:a mod b == a
b * (a / b) == b * 0
rewrite <- (add_cancel_r _ _ (a mod b)).a, b:t
Ha:0 <= a
Hb:0 < b
H:a mod b == a
b * (a / b) + a mod b == b * 0 + a mod b
rewrite <- div_mod, H by order.a, b:t
Ha:0 <= a
Hb:0 < b
H:a mod b == a
a == b * 0 + a now nzsimpl.
Qed.

Lemma div_str_pos_iff : forall a b, 0<=a -> 0<b -> (0<a/b <-> b<=a).forall a b : t,
0 <= a -> 0 < b -> 0 < a / b <-> b <= a
Proof.forall a b : t,
0 <= a -> 0 < b -> 0 < a / b <-> b <= a
intros a b Ha Hb; split; intros Hab.a, b:t
Ha:0 <= a
Hb:0 < b
Hab:0 < a / b
b <= a
a, b:t
Ha:0 <= a
Hb:0 < b
Hab:b <= a
0 < a / b
-a, b:t
Ha:0 <= a
Hb:0 < b
Hab:0 < a / b
b <= a destruct (lt_ge_cases a b) as [LT|LE]; auto.a, b:t
Ha:0 <= a
Hb:0 < b
Hab:0 < a / b
LT:a < b
b <= a
  rewrite <- div_small_iff in LT; order.
-a, b:t
Ha:0 <= a
Hb:0 < b
Hab:b <= a
0 < a / b apply div_str_pos; 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
intros.a, b:t
H:0 < a
H0:1 < b
a / b < a
assert (0 < b) by (apply lt_trans with 1; auto using lt_0_1).a, b:t
H:0 < a
H0:1 < b
H1:0 < b
a / b < a
destruct (lt_ge_cases a b).a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:a < b
a / b < a
a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
a / b < a
-a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:a < b
a / b < a rewrite div_small; try split; order.
-a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
a / b < a rewrite (div_mod a b) at 2 by order.a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
a / b < b * (a / b) + a mod b
  apply lt_le_trans with (b*(a/b)).a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
a / b < b * (a / b)
a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
b * (a / b) <= b * (a / b) + a mod b
  +a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
a / b < b * (a / b) rewrite <- (mul_1_l (a/b)) at 1.a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
1 * (a / b) < b * (a / b)
    rewrite <- mul_lt_mono_pos_r; auto.a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
0 < a / b
    apply div_str_pos; auto.
  +a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
b * (a / b) <= b * (a / b) + a mod b rewrite <- (add_0_r (b*(a/b))) at 1.a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
b * (a / b) + 0 <= b * (a / b) + a mod b
    rewrite <- add_le_mono_l.a, b:t
H:0 < a
H0:1 < b
H1:0 < b
H2:b <= a
0 <= a mod b destruct (mod_bound_pos a b); order.
Qed.

le is compatible with a positive division.

Lemma div_le_mono : forall a b c, 0<c -> 0<=a<=b -> a/c <= b/c.forall a b c : t,
0 < c -> 0 <= a <= b -> a / c <= b / c
Proof.forall a b c : t,
0 < c -> 0 <= a <= b -> a / c <= b / c
intros a b c Hc (Ha,Hab).a, b, c:t
Hc:0 < c
Ha:0 <= a
Hab:a <= b
a / c <= b / c
rewrite lt_eq_cases in Hab.a, b, c:t
Hc:0 < c
Ha:0 <= a
Hab:a < b \/ a == b
a / c <= b / c destruct Hab as [LT|EQ];
 [|rewrite EQ; order].a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
a / c <= b / c
rewrite <- lt_succ_r.a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
a / c < S (b / c)
rewrite (mul_lt_mono_pos_l c) by order.a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
c * (a / c) < c * S (b / c)
nzsimpl.a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
c * (a / c) < c * (b / c) + c
rewrite (add_lt_mono_r _ _ (a mod c)).a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
c * (a / c) + a mod c < c * (b / c) + c + a mod c
rewrite <- div_mod by order.a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
a < c * (b / c) + c + a mod c
apply lt_le_trans with b; auto.a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
b <= c * (b / c) + c + a mod c
rewrite (div_mod b c) at 1 by order.a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
c * (b / c) + b mod c <= c * (b / c) + c + a mod c
rewrite <- add_assoc, <- add_le_mono_l.a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
b mod c <= c + a mod c
apply le_trans with (c+0).a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
b mod c <= c + 0
a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
c + 0 <= c + a mod c
-a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
b mod c <= c + 0 nzsimpl; destruct (mod_bound_pos b c); order.
-a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
c + 0 <= c + a mod c rewrite <- add_le_mono_l.a, b, c:t
Hc:0 < c
Ha:0 <= a
LT:a < b
0 <= a mod c destruct (mod_bound_pos a c); order.
Qed.

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.forall a b : t, 0 <= a -> 0 < b -> b * (a / b) <= a
Proof.forall a b : t, 0 <= a -> 0 < b -> b * (a / b) <= a
intros.a, b:t
H:0 <= a
H0:0 < b
b * (a / b) <= a
rewrite (add_le_mono_r _ _ (a mod b)), <- div_mod by order.a, b:t
H:0 <= a
H0:0 < b
a <= a + a mod b
rewrite <- (add_0_r a) at 1.a, b:t
H:0 <= a
H0:0 < b
a + 0 <= a + a mod b
rewrite <- add_le_mono_l.a, b:t
H:0 <= a
H0:0 < b
0 <= a mod b destruct (mod_bound_pos a b); order.
Qed.

Lemma mul_succ_div_gt : forall a b, 0<=a -> 0<b -> a < b*(S (a/b)).forall a b : t, 0 <= a -> 0 < b -> a < b * S (a / b)
Proof.forall a b : t, 0 <= a -> 0 < b -> a < b * S (a / b)
intros.a, b:t
H:0 <= a
H0:0 < b
a < b * S (a / b)
rewrite (div_mod a b) at 1 by order.a, b:t
H:0 <= a
H0:0 < b
b * (a / b) + a mod b < b * S (a / b)
rewrite (mul_succ_r).a, b:t
H:0 <= a
H0:0 < b
b * (a / b) + a mod b < b * (a / b) + b
rewrite <- add_lt_mono_l.a, b:t
H:0 <= a
H0:0 < b
a mod b < b
destruct (mod_bound_pos a b); auto.
Qed.

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).forall a b : t,
0 <= a -> 0 < b -> a == b * (a / b) <-> a mod b == 0
Proof.forall a b : t,
0 <= a -> 0 < b -> a == b * (a / b) <-> a mod b == 0
intros.a, b:t
H:0 <= a
H0:0 < b
a == b * (a / b) <-> a mod b == 0 rewrite (div_mod a b) at 1 by order.a, b:t
H:0 <= a
H0:0 < b
b * (a / b) + a mod b == b * (a / b) <-> a mod b == 0
rewrite <- (add_0_r (b*(a/b))) at 2.a, b:t
H:0 <= a
H0:0 < b
b * (a / b) + a mod b == b * (a / b) + 0 <->
a mod b == 0
apply add_cancel_l.
Qed.

Some additional inequalities about div.

Theorem div_lt_upper_bound:
  forall a b q, 0<=a -> 0<b -> a < b*q -> a/b < q.forall a b q : t,
0 <= a -> 0 < b -> a < b * q -> a / b < q
Proof.forall a b q : t,
0 <= a -> 0 < b -> a < b * q -> a / b < q
intros.a, b, q:t
H:0 <= a
H0:0 < b
H1:a < b * q
a / b < q
rewrite (mul_lt_mono_pos_l b) by order.a, b, q:t
H:0 <= a
H0:0 < b
H1:a < b * q
b * (a / b) < b * q
apply le_lt_trans with a; auto.a, b, q:t
H:0 <= a
H0:0 < b
H1:a < b * q
b * (a / b) <= a
apply mul_div_le; auto.
Qed.

Theorem div_le_upper_bound:
  forall a b q, 0<=a -> 0<b -> a <= b*q -> a/b <= q.forall a b q : t,
0 <= a -> 0 < b -> a <= b * q -> a / b <= q
Proof.forall a b q : t,
0 <= a -> 0 < b -> a <= b * q -> a / b <= q
intros.a, b, q:t
H:0 <= a
H0:0 < b
H1:a <= b * q
a / b <= q
rewrite (mul_le_mono_pos_l _ _ b) by order.a, b, q:t
H:0 <= a
H0:0 < b
H1:a <= b * q
b * (a / b) <= b * q
apply le_trans with a; auto.a, b, q:t
H:0 <= a
H0:0 < b
H1:a <= b * q
b * (a / b) <= a
apply mul_div_le; auto.
Qed.

Theorem div_le_lower_bound:
  forall a b q, 0<=a -> 0<b -> b*q <= a -> q <= a/b.forall a b q : t,
0 <= a -> 0 < b -> b * q <= a -> q <= a / b
Proof.forall a b q : t,
0 <= a -> 0 < b -> b * q <= a -> q <= a / b
intros a b q Ha Hb H.a, b, q:t
Ha:0 <= a
Hb:0 < b
H:b * q <= a
q <= a / b
destruct (lt_ge_cases 0 q).a, b, q:t
Ha:0 <= a
Hb:0 < b
H:b * q <= a
H0:0 < q
q <= a / b
a, b, q:t
Ha:0 <= a
Hb:0 < b
H:b * q <= a
H0:q <= 0
q <= a / b
-a, b, q:t
Ha:0 <= a
Hb:0 < b
H:b * q <= a
H0:0 < q
q <= a / b rewrite <- (div_mul q b); try order.a, b, q:t
Ha:0 <= a
Hb:0 < b
H:b * q <= a
H0:0 < q
q * b / b <= a / b
  apply div_le_mono; auto.a, b, q:t
Ha:0 <= a
Hb:0 < b
H:b * q <= a
H0:0 < q
0 <= q * b <= a
  rewrite mul_comm; split; auto.a, b, q:t
Ha:0 <= a
Hb:0 < b
H:b * q <= a
H0:0 < q
0 <= b * q
  apply lt_le_incl, mul_pos_pos; auto.
-a, b, q:t
Ha:0 <= a
Hb:0 < b
H:b * q <= a
H0:q <= 0
q <= a / b apply le_trans with 0; auto; apply div_pos; auto.
Qed.

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.forall p q r : t,
0 <= p -> 0 < q <= r -> p / r <= p / q
Proof.forall p q r : t,
0 <= p -> 0 < q <= r -> p / r <= p / q
 intros p q r Hp (Hq,Hqr).p, q, r:t
Hp:0 <= p
Hq:0 < q
Hqr:q <= r
p / r <= p / q
 apply div_le_lower_bound; auto.p, q, r:t
Hp:0 <= p
Hq:0 < q
Hqr:q <= r
q * (p / r) <= p
 rewrite (div_mod p r) at 2 by order.p, q, r:t
Hp:0 <= p
Hq:0 < q
Hqr:q <= r
q * (p / r) <= r * (p / r) + p mod r
 apply le_trans with (r*(p/r)).p, q, r:t
Hp:0 <= p
Hq:0 < q
Hqr:q <= r
q * (p / r) <= r * (p / r)
p, q, r:t
Hp:0 <= p
Hq:0 < q
Hqr:q <= r
r * (p / r) <= r * (p / r) + p mod r
 -p, q, r:t
Hp:0 <= p
Hq:0 < q
Hqr:q <= r
q * (p / r) <= r * (p / r) apply mul_le_mono_nonneg_r; try order.p, q, r:t
Hp:0 <= p
Hq:0 < q
Hqr:q <= r
0 <= p / r
   apply div_pos; order.
 -p, q, r:t
Hp:0 <= p
Hq:0 < q
Hqr:q <= r
r * (p / r) <= r * (p / r) + p mod r rewrite <- (add_0_r (r*(p/r))) at 1.p, q, r:t
Hp:0 <= p
Hq:0 < q
Hqr:q <= r
r * (p / r) + 0 <= r * (p / r) + p mod r
   rewrite <- add_le_mono_l.p, q, r:t
Hp:0 <= p
Hq:0 < q
Hqr:q <= r
0 <= p mod r destruct (mod_bound_pos p r); order.
Qed.

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.forall a b c : t,
0 <= a ->
0 <= a + b * c ->
0 < c -> (a + b * c) mod c == a mod c
Proof.forall a b c : t,
0 <= a ->
0 <= a + b * c ->
0 < c -> (a + b * c) mod c == a mod c
 intros.a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
(a + b * c) mod c == a mod c
 symmetry.a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
a mod c == (a + b * c) mod c
 apply mod_unique with (a/c+b); auto.a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
0 <= a mod c < c
a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
a + b * c == c * (a / c + b) + a mod c
 -a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
0 <= a mod c < c apply mod_bound_pos; auto.
 -a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
a + b * c == c * (a / c + b) + a mod c rewrite mul_add_distr_l, add_shuffle0, <- div_mod by order.a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
a + b * c == a + c * b
   now rewrite mul_comm.
Qed.

Lemma div_add : forall a b c, 0<=a -> 0<=a+b*c -> 0<c ->
 (a + b * c) / c == a / c + b.forall a b c : t,
0 <= a ->
0 <= a + b * c ->
0 < c -> (a + b * c) / c == a / c + b
Proof.forall a b c : t,
0 <= a ->
0 <= a + b * c ->
0 < c -> (a + b * c) / c == a / c + b
 intros.a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
(a + b * c) / c == a / c + b
 apply (mul_cancel_l _ _ c); try order.a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
c * ((a + b * c) / c) == c * (a / c + b)
 apply (add_cancel_r _ _ ((a+b*c) mod c)).a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
c * ((a + b * c) / c) + (a + b * c) mod c ==
c * (a / c + b) + (a + b * c) mod c
 rewrite <- div_mod, mod_add by order.a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
a + b * c == c * (a / c + b) + a mod c
 rewrite mul_add_distr_l, add_shuffle0, <- div_mod by order.a, b, c:t
H:0 <= a
H0:0 <= a + b * c
H1:0 < c
a + b * c == a + c * b
 now rewrite mul_comm.
Qed.

Lemma div_add_l: forall a b c, 0<=c -> 0<=a*b+c -> 0<b ->
 (a * b + c) / b == a + c / b.forall a b c : t,
0 <= c ->
0 <= a * b + c ->
0 < b -> (a * b + c) / b == a + c / b
Proof.forall a b c : t,
0 <= c ->
0 <= a * b + c ->
0 < b -> (a * b + c) / b == a + c / b
 intros a b c.a, b, c:t
0 <= c ->
0 <= a * b + c ->
0 < b -> (a * b + c) / b == a + c / b rewrite (add_comm _ c), (add_comm a).a, b, c:t
0 <= c ->
0 <= c + a * b ->
0 < b -> (c + a * b) / b == c / b + a
 intros.a, b, c:t
H:0 <= c
H0:0 <= c + a * b
H1:0 < b
(c + a * b) / b == c / b + a apply div_add; auto.
Qed.

Cancellations.

Lemma div_mul_cancel_r : forall a b c, 0<=a -> 0<b -> 0<c ->
 (a*c)/(b*c) == a/b.forall a b c : t,
0 <= a -> 0 < b -> 0 < c -> a * c / (b * c) == a / b
Proof.forall a b c : t,
0 <= a -> 0 < b -> 0 < c -> a * c / (b * c) == a / b
 intros.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
a * c / (b * c) == a / b
 symmetry.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
a / b == a * c / (b * c)
 apply div_unique with ((a mod b)*c).a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
0 <= a * c
a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
0 <= a mod b * c < b * c
a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
a * c == b * c * (a / b) + a mod b * c
 -a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
0 <= a * c apply mul_nonneg_nonneg; order.
 -a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
0 <= a mod b * c < b * c split.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
0 <= a mod b * c
a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
a mod b * c < b * c
   +a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
0 <= a mod b * c apply mul_nonneg_nonneg; destruct (mod_bound_pos a b); order.
   +a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
a mod b * c < b * c rewrite <- mul_lt_mono_pos_r; auto.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
a mod b < b destruct (mod_bound_pos a b); auto.
 -a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
a * c == b * c * (a / b) + a mod b * c rewrite (div_mod a b) at 1 by order.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
(b * (a / b) + a mod b) * c ==
b * c * (a / b) + a mod b * c
   rewrite mul_add_distr_r.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
b * (a / b) * c + a mod b * c ==
b * c * (a / b) + a mod b * c
   rewrite add_cancel_r.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
b * (a / b) * c == b * c * (a / b)
   rewrite <- 2 mul_assoc.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
b * (a / b * c) == b * (c * (a / b)) now rewrite (mul_comm c).
Qed.

Lemma div_mul_cancel_l : forall a b c, 0<=a -> 0<b -> 0<c ->
 (c*a)/(c*b) == a/b.forall a b c : t,
0 <= a -> 0 < b -> 0 < c -> c * a / (c * b) == a / b
Proof.forall a b c : t,
0 <= a -> 0 < b -> 0 < c -> c * a / (c * b) == a / b
 intros.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
c * a / (c * b) == a / b rewrite !(mul_comm c); apply div_mul_cancel_r; auto.
Qed.

Lemma mul_mod_distr_l: forall a b c, 0<=a -> 0<b -> 0<c ->
  (c*a) mod (c*b) == c * (a mod b).forall a b c : t,
0 <= a ->
0 < b -> 0 < c -> (c * a) mod (c * b) == c * (a mod b)
Proof.forall a b c : t,
0 <= a ->
0 < b -> 0 < c -> (c * a) mod (c * b) == c * (a mod b)
 intros.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
(c * a) mod (c * b) == c * (a mod b)
 rewrite <- (add_cancel_l _ _ ((c*b)* ((c*a)/(c*b)))).a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
c * b * (c * a / (c * b)) + (c * a) mod (c * b) ==
c * b * (c * a / (c * b)) + c * (a mod b)
 rewrite <- div_mod.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
c * a == c * b * (c * a / (c * b)) + c * (a mod b)
a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
c * b ~= 0
 -a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
c * a == c * b * (c * a / (c * b)) + c * (a mod b) rewrite div_mul_cancel_l; auto.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
c * a == c * b * (a / b) + c * (a mod b)
   rewrite <- mul_assoc, <- mul_add_distr_l, mul_cancel_l by order.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
a == b * (a / b) + a mod b
   apply div_mod; order.
 -a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
c * b ~= 0 rewrite <- neq_mul_0; intuition; order.
Qed.

Lemma mul_mod_distr_r: forall a b c, 0<=a -> 0<b -> 0<c ->
  (a*c) mod (b*c) == (a mod b) * c.forall a b c : t,
0 <= a ->
0 < b -> 0 < c -> (a * c) mod (b * c) == a mod b * c
Proof.forall a b c : t,
0 <= a ->
0 < b -> 0 < c -> (a * c) mod (b * c) == a mod b * c
 intros.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 < c
(a * c) mod (b * c) == a mod b * c rewrite !(mul_comm _ c); now rewrite mul_mod_distr_l.
Qed.

Operations modulo.

Theorem mod_mod: forall a n, 0<=a -> 0<n ->
 (a mod n) mod n == a mod n.forall a n : t,
0 <= a -> 0 < n -> (a mod n) mod n == a mod n
Proof.forall a n : t,
0 <= a -> 0 < n -> (a mod n) mod n == a mod n
 intros.a, n:t
H:0 <= a
H0:0 < n
(a mod n) mod n == a mod n destruct (mod_bound_pos a n); auto.a, n:t
H:0 <= a
H0:0 < n
H1:0 <= a mod n
H2:a mod n < n
(a mod n) mod n == a mod n now rewrite mod_small_iff.
Qed.

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.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a mod n * b) mod n == (a * b) mod n
Proof.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a mod n * b) mod n == (a * b) mod n
 intros a b n Ha Hb Hn.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
(a mod n * b) mod n == (a * b) mod n symmetry.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
(a * b) mod n == (a mod n * b) mod n
 generalize (mul_nonneg_nonneg _ _ Ha Hb).a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
0 <= a * b -> (a * b) mod n == (a mod n * b) mod n
 rewrite (div_mod a n) at 1 2 by order.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
0 <= (n * (a / n) + a mod n) * b ->
((n * (a / n) + a mod n) * b) mod n ==
(a mod n * b) mod n
 rewrite add_comm, (mul_comm n), (mul_comm _ b).a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
0 <= b * (a mod n + a / n * n) ->
(b * (a mod n + a / n * n)) mod n ==
(a mod n * b) mod n
 rewrite mul_add_distr_l, mul_assoc.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
0 <= b * (a mod n) + b * (a / n) * n ->
(b * (a mod n) + b * (a / n) * n) mod n ==
(a mod n * b) mod n
 intros.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= b * (a mod n) + b * (a / n) * n
(b * (a mod n) + b * (a / n) * n) mod n ==
(a mod n * b) mod n rewrite mod_add; auto.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= b * (a mod n) + b * (a / n) * n
(b * (a mod n)) mod n == (a mod n * b) mod n
a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= b * (a mod n) + b * (a / n) * n
0 <= b * (a mod n)
 -a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= b * (a mod n) + b * (a / n) * n
(b * (a mod n)) mod n == (a mod n * b) mod n now rewrite mul_comm.
 -a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= b * (a mod n) + b * (a / n) * n
0 <= b * (a mod n) apply mul_nonneg_nonneg; destruct (mod_bound_pos a n); auto.
Qed.

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.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a * (b mod n)) mod n == (a * b) mod n
Proof.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a * (b mod n)) mod n == (a * b) mod n
 intros.a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
(a * (b mod n)) mod n == (a * b) mod n rewrite !(mul_comm a).a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
(b mod n * a) mod n == (b * a) mod n apply mul_mod_idemp_l; auto.
Qed.

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.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a * b) mod n == (a mod n * (b mod n)) mod n
Proof.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a * b) mod n == (a mod n * (b mod n)) mod n
  intros.a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
(a * b) mod n == (a mod n * (b mod n)) mod n rewrite mul_mod_idemp_l, mul_mod_idemp_r; trivial.a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
(a * b) mod n == (a * b) mod n
a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
0 <= b mod n -a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
(a * b) mod n == (a * b) mod n reflexivity.
  -a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
0 <= b mod n now destruct (mod_bound_pos b n).
Qed.

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.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a mod n + b) mod n == (a + b) mod n
Proof.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a mod n + b) mod n == (a + b) mod n
 intros a b n Ha Hb Hn.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
(a mod n + b) mod n == (a + b) mod n symmetry.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
(a + b) mod n == (a mod n + b) mod n
 generalize (add_nonneg_nonneg _ _ Ha Hb).a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
0 <= a + b -> (a + b) mod n == (a mod n + b) mod n
 rewrite (div_mod a n) at 1 2 by order.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
0 <= n * (a / n) + a mod n + b ->
(n * (a / n) + a mod n + b) mod n ==
(a mod n + b) mod n
 rewrite <- add_assoc, add_comm, mul_comm.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
0 <= a mod n + b + a / n * n ->
(a mod n + b + a / n * n) mod n == (a mod n + b) mod n
 intros.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= a mod n + b + a / n * n
(a mod n + b + a / n * n) mod n == (a mod n + b) mod n rewrite mod_add; trivial.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= a mod n + b + a / n * n
(a mod n + b) mod n == (a mod n + b) mod n
a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= a mod n + b + a / n * n
0 <= a mod n + b -a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= a mod n + b + a / n * n
(a mod n + b) mod n == (a mod n + b) mod n reflexivity.
 -a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= a mod n + b + a / n * n
0 <= a mod n + b apply add_nonneg_nonneg; auto.a, b, n:t
Ha:0 <= a
Hb:0 <= b
Hn:0 < n
H:0 <= a mod n + b + a / n * n
0 <= a mod n destruct (mod_bound_pos a n); auto.
Qed.

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.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a + b mod n) mod n == (a + b) mod n
Proof.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a + b mod n) mod n == (a + b) mod n
 intros.a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
(a + b mod n) mod n == (a + b) mod n rewrite !(add_comm a).a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
(b mod n + a) mod n == (b + a) mod n apply add_mod_idemp_l; auto.
Qed.

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.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a + b) mod n == (a mod n + b mod n) mod n
Proof.forall a b n : t,
0 <= a ->
0 <= b ->
0 < n -> (a + b) mod n == (a mod n + b mod n) mod n
  intros.a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
(a + b) mod n == (a mod n + b mod n) mod n rewrite add_mod_idemp_l, add_mod_idemp_r; trivial.a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
(a + b) mod n == (a + b) mod n
a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
0 <= b mod n -a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
(a + b) mod n == (a + b) mod n reflexivity.
  -a, b, n:t
H:0 <= a
H0:0 <= b
H1:0 < n
0 <= b mod n now destruct (mod_bound_pos b n).
Qed.

Lemma div_div : forall a b c, 0<=a -> 0<b -> 0<c ->
 (a/b)/c == a/(b*c).forall a b c : t,
0 <= a -> 0 < b -> 0 < c -> a / b / c == a / (b * c)
Proof.forall a b c : t,
0 <= a -> 0 < b -> 0 < c -> a / b / c == a / (b * c)
 intros a b c Ha Hb Hc.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
a / b / c == a / (b * c)
 apply div_unique with (b*((a/b) mod c) + a mod b); trivial.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
0 <= b * ((a / b) mod c) + a mod b < b * c
a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
a ==
b * c * (a / b / c) + (b * ((a / b) mod c) + a mod b)
 (* begin 0<= ... <b*c *)
 -a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
0 <= b * ((a / b) mod c) + a mod b < b * c destruct (mod_bound_pos (a/b) c), (mod_bound_pos a b); auto using div_pos.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
H:0 <= (a / b) mod c
H0:(a / b) mod c < c
H1:0 <= a mod b
H2:a mod b < b
0 <= b * ((a / b) mod c) + a mod b < b * c
   split.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
H:0 <= (a / b) mod c
H0:(a / b) mod c < c
H1:0 <= a mod b
H2:a mod b < b
0 <= b * ((a / b) mod c) + a mod b
a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
H:0 <= (a / b) mod c
H0:(a / b) mod c < c
H1:0 <= a mod b
H2:a mod b < b
b * ((a / b) mod c) + a mod b < b * c
   +a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
H:0 <= (a / b) mod c
H0:(a / b) mod c < c
H1:0 <= a mod b
H2:a mod b < b
0 <= b * ((a / b) mod c) + a mod b apply add_nonneg_nonneg; auto.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
H:0 <= (a / b) mod c
H0:(a / b) mod c < c
H1:0 <= a mod b
H2:a mod b < b
0 <= b * ((a / b) mod c)
     apply mul_nonneg_nonneg; order.
   +a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
H:0 <= (a / b) mod c
H0:(a / b) mod c < c
H1:0 <= a mod b
H2:a mod b < b
b * ((a / b) mod c) + a mod b < b * c apply lt_le_trans with (b*((a/b) mod c) + b).a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
H:0 <= (a / b) mod c
H0:(a / b) mod c < c
H1:0 <= a mod b
H2:a mod b < b
b * ((a / b) mod c) + a mod b <
b * ((a / b) mod c) + b
a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
H:0 <= (a / b) mod c
H0:(a / b) mod c < c
H1:0 <= a mod b
H2:a mod b < b
b * ((a / b) mod c) + b <= b * c
     *a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
H:0 <= (a / b) mod c
H0:(a / b) mod c < c
H1:0 <= a mod b
H2:a mod b < b
b * ((a / b) mod c) + a mod b <
b * ((a / b) mod c) + b rewrite <- add_lt_mono_l; auto.
     *a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
H:0 <= (a / b) mod c
H0:(a / b) mod c < c
H1:0 <= a mod b
H2:a mod b < b
b * ((a / b) mod c) + b <= b * c rewrite <- mul_succ_r, <- mul_le_mono_pos_l, le_succ_l; auto.
       (* end 0<= ... < b*c *)
 -a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
a ==
b * c * (a / b / c) + (b * ((a / b) mod c) + a mod b) rewrite (div_mod a b) at 1 by order.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
b * (a / b) + a mod b ==
b * c * (a / b / c) + (b * ((a / b) mod c) + a mod b)
   rewrite add_assoc, add_cancel_r.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
b * (a / b) ==
b * c * (a / b / c) + b * ((a / b) mod c)
   rewrite <- mul_assoc, <- mul_add_distr_l, mul_cancel_l by order.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
a / b == c * (a / b / c) + (a / b) mod c
   apply div_mod; order.
Qed.

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).forall a b c : t,
0 <= a ->
0 < b ->
0 < c ->
a mod (b * c) == a mod b + b * ((a / b) mod c)
Proof.forall a b c : t,
0 <= a ->
0 < b ->
0 < c ->
a mod (b * c) == a mod b + b * ((a / b) mod c)
 intros a b c Ha Hb Hc.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
a mod (b * c) == a mod b + b * ((a / b) mod c)
 apply add_cancel_l with (b*c*(a/(b*c))).a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
b * c * (a / (b * c)) + a mod (b * c) ==
b * c * (a / (b * c)) +
(a mod b + b * ((a / b) mod c))
 rewrite <- div_mod by (apply neq_mul_0; split; order).a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
a ==
b * c * (a / (b * c)) +
(a mod b + b * ((a / b) mod c))
 rewrite <- div_div by trivial.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
a ==
b * c * (a / b / c) + (a mod b + b * ((a / b) mod c))
 rewrite add_assoc, add_shuffle0, <- mul_assoc, <- mul_add_distr_l.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
a == b * (c * (a / b / c) + (a / b) mod c) + a mod b
 rewrite <- div_mod by order.a, b, c:t
Ha:0 <= a
Hb:0 < b
Hc:0 < c
a == b * (a / b) + a mod b
 apply div_mod; order.
Qed.

A last inequality:

Theorem div_mul_le:
 forall a b c, 0<=a -> 0<b -> 0<=c -> c*(a/b) <= (c*a)/b.forall a b c : t,
0 <= a -> 0 < b -> 0 <= c -> c * (a / b) <= c * a / b
Proof.forall a b c : t,
0 <= a -> 0 < b -> 0 <= c -> c * (a / b) <= c * a / b
 intros.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 <= c
c * (a / b) <= c * a / b
 apply div_le_lower_bound; auto.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 <= c
0 <= c * a
a, b, c:t
H:0 <= a
H0:0 < b
H1:0 <= c
b * (c * (a / b)) <= c * a
 -a, b, c:t
H:0 <= a
H0:0 < b
H1:0 <= c
0 <= c * a apply mul_nonneg_nonneg; auto.
 -a, b, c:t
H:0 <= a
H0:0 < b
H1:0 <= c
b * (c * (a / b)) <= c * a rewrite mul_assoc, (mul_comm b c), <- mul_assoc.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 <= c
c * (b * (a / b)) <= c * a
   apply mul_le_mono_nonneg_l; auto.a, b, c:t
H:0 <= a
H0:0 < b
H1:0 <= c
b * (a / b) <= a
   apply mul_div_le; auto.
Qed.

mod is related to divisibility

Lemma mod_divides : forall a b, 0<=a -> 0<b ->
 (a mod b == 0 <-> exists c, a == b*c).forall a b : t,
0 <= a ->
0 < b -> a mod b == 0 <-> (exists c : t, a == b * c)
Proof.forall a b : t,
0 <= a ->
0 < b -> a mod b == 0 <-> (exists c : t, a == b * c)
 split.a, b:t
H:0 <= a
H0:0 < b
a mod b == 0 -> exists c : t, a == b * c
a, b:t
H:0 <= a
H0:0 < b
(exists c : t, a == b * c) -> a mod b == 0
 -a, b:t
H:0 <= a
H0:0 < b
a mod b == 0 -> exists c : t, a == b * c intros.a, b:t
H:0 <= a
H0:0 < b
H1:a mod b == 0
exists c : t, a == b * c exists (a/b).a, b:t
H:0 <= a
H0:0 < b
H1:a mod b == 0
a == b * (a / b) rewrite div_exact; auto.
 -a, b:t
H:0 <= a
H0:0 < b
(exists c : t, a == b * c) -> a mod b == 0 intros (c,Hc).a, b:t
H:0 <= a
H0:0 < b
c:t
Hc:a == b * c
a mod b == 0 rewrite Hc, mul_comm.a, b:t
H:0 <= a
H0:0 < b
c:t
Hc:a == b * c
(c * b) mod b == 0 apply mod_mul; auto.a, b:t
H:0 <= a
H0:0 < b
c:t
Hc:a == b * c
0 <= c
   rewrite (mul_le_mono_pos_l _ _ b); auto.a, b:t
H:0 <= a
H0:0 < b
c:t
Hc:a == b * c
b * 0 <= b * c nzsimpl.a, b:t
H:0 <= a
H0:0 < b
c:t
Hc:a == b * c
0 <= b * c order.
Qed.

End NZDivProp.