lean2/library/data/int/div.lean

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/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Jeremy Avigad
Definitions and properties of div and mod, following the SSReflect library.
Following SSReflect and the SMTlib standard, we define a mod b so that 0 ≤ a mod b < |b| when b ≠ 0.
-/
import data.int.order data.nat.div
open [coercions] [reduce_hints] nat
open [declarations] [classes] nat (succ)
open algebra
open eq.ops
namespace int
/- definitions -/
protected definition divide (a b : ) : :=
sign b *
(match a with
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| of_nat m := of_nat (m div (nat_abs b))
| -[1+m] := -[1+ ((m:nat) div (nat_abs b))]
end)
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definition int_has_divide [reducible] [instance] [priority int.prio] : has_divide int :=
has_divide.mk int.divide
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lemma of_nat_div_eq (m : nat) (b : ) : (of_nat m) div b = sign b * of_nat (m div (nat_abs b)) :=
rfl
lemma neg_succ_div_eq (m: nat) (b : ) : -[1+m] div b = sign b * -[1+ (m div (nat_abs b))] :=
rfl
lemma divide.def (a b : ) : a div b =
sign b *
(match a with
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| of_nat m := of_nat (m div (nat_abs b))
| -[1+m] := -[1+ ((m:nat) div (nat_abs b))]
end) :=
rfl
protected definition modulo (a b : ) : := a - a div b * b
definition int_has_modulo [reducible] [instance] [priority int.prio] : has_modulo int :=
has_modulo.mk int.modulo
lemma modulo.def (a b : ) : a mod b = a - a div b * b :=
rfl
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notation [priority int.prio] a ≡ b `[mod `:0 c:0 `]` := a mod c = b mod c
/- div -/
theorem of_nat_div (m n : nat) : of_nat (m div n) = (of_nat m) div (of_nat n) :=
nat.cases_on n
(begin rewrite [of_nat_div_eq, of_nat_zero, sign_zero, zero_mul, nat.div_zero] end)
(take (n : nat), by rewrite [of_nat_div_eq, sign_of_succ, one_mul])
theorem neg_succ_of_nat_div (m : nat) {b : } (H : b > 0) :
-[1+m] div b = -(m div b + 1) :=
calc
-[1+m] div b = sign b * _ : rfl
... = -[1+(m div (nat_abs b))] : by rewrite [sign_of_pos H, one_mul]
... = -(m div b + 1) : by rewrite [of_nat_div_eq, sign_of_pos H, one_mul]
theorem div_neg (a b : ) : a div -b = -(a div b) :=
begin
induction a,
rewrite [*of_nat_div_eq, sign_neg, neg_mul_eq_neg_mul, nat_abs_neg],
rewrite [*neg_succ_div_eq, sign_neg, neg_mul_eq_neg_mul, nat_abs_neg],
end
theorem div_of_neg_of_pos {a b : } (Ha : a < 0) (Hb : b > 0) : a div b = -((-a - 1) div b + 1) :=
obtain (m : nat) (H1 : a = -[1+m]), from exists_eq_neg_succ_of_nat Ha,
calc
a div b = -(m div b + 1) : by rewrite [H1, neg_succ_of_nat_div _ Hb]
... = -((-a -1) div b + 1) : by rewrite [H1, neg_succ_of_nat_eq', neg_sub, sub_neg_eq_add,
add.comm 1, add_sub_cancel]
theorem div_nonneg {a b : } (Ha : a ≥ 0) (Hb : b ≥ 0) : a div b ≥ 0 :=
obtain (m : ) (Hm : a = m), from exists_eq_of_nat Ha,
obtain (n : ) (Hn : b = n), from exists_eq_of_nat Hb,
calc
a div b = m div n : by rewrite [Hm, Hn]
... ≥ 0 : by rewrite -of_nat_div; apply trivial
theorem div_nonpos {a b : } (Ha : a ≥ 0) (Hb : b ≤ 0) : a div b ≤ 0 :=
calc
a div b = -(a div -b) : by rewrite [div_neg, neg_neg]
... ≤ 0 : neg_nonpos_of_nonneg (div_nonneg Ha (neg_nonneg_of_nonpos Hb))
theorem div_neg' {a b : } (Ha : a < 0) (Hb : b > 0) : a div b < 0 :=
have -a - 1 ≥ 0, from le_sub_one_of_lt (neg_pos_of_neg Ha),
have (-a - 1) div b + 1 > 0, from lt_add_one_of_le (div_nonneg this (le_of_lt Hb)),
calc
a div b = -((-a - 1) div b + 1) : div_of_neg_of_pos Ha Hb
... < 0 : neg_neg_of_pos this
theorem zero_div (b : ) : 0 div b = 0 :=
by rewrite [of_nat_div_eq, nat.zero_div, of_nat_zero, mul_zero]
theorem div_zero (a : ) : a div 0 = 0 :=
by rewrite [divide.def, sign_zero, zero_mul]
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theorem div_one (a : ) : a div 1 = a :=
assert (1 : int) > 0, from dec_trivial,
int.cases_on a
(take m : nat, by rewrite [-of_nat_one, -of_nat_div, nat.div_one])
(take m : nat, by rewrite [!neg_succ_of_nat_div this, -of_nat_one, -of_nat_div, nat.div_one])
theorem eq_div_mul_add_mod (a b : ) : a = a div b * b + a mod b :=
!add.comm ▸ eq_add_of_sub_eq rfl
theorem div_eq_zero_of_lt {a b : } : 0 ≤ a → a < b → a div b = 0 :=
int.cases_on a
(take (m : nat), assume H,
int.cases_on b
(take (n : nat),
assume H : m < n,
show m div n = 0,
by rewrite [-of_nat_div, nat.div_eq_zero_of_lt (lt_of_of_nat_lt_of_nat H)])
(take (n : nat),
assume H : m < -[1+n],
have H1 : ¬(m < -[1+n]), from dec_trivial,
absurd H H1))
(take (m : nat),
assume H : 0 ≤ -[1+m],
have ¬ (0 ≤ -[1+m]), from dec_trivial,
absurd H this)
theorem div_eq_zero_of_lt_abs {a b : } (H1 : 0 ≤ a) (H2 : a < abs b) : a div b = 0 :=
lt.by_cases
(suppose b < 0,
assert a < -b, from abs_of_neg this ▸ H2,
calc
a div b = - (a div -b) : by rewrite [div_neg, neg_neg]
... = 0 : by rewrite [div_eq_zero_of_lt H1 this, neg_zero])
(suppose b = 0, this⁻¹ ▸ !div_zero)
(suppose b > 0,
have a < b, from abs_of_pos this ▸ H2,
div_eq_zero_of_lt H1 this)
private theorem add_mul_div_self_aux1 {a : } {k : } (n : ) (H1 : a ≥ 0) (H2 : k > 0) :
(a + n * k) div k = a div k + n :=
obtain (m : nat) (Hm : a = of_nat m), from exists_eq_of_nat H1,
begin
subst Hm,
rewrite [-of_nat_mul, -of_nat_add, -*of_nat_div, -of_nat_add, !nat.add_mul_div_self H2]
end
private theorem add_mul_div_self_aux2 {a : } {k : } (n : ) (H1 : a < 0) (H2 : k > 0) :
(a + n * k) div k = a div k + n :=
obtain m (Hm : a = -[1+m]), from exists_eq_neg_succ_of_nat H1,
or.elim (nat.lt_or_ge m (n * k))
(assume m_lt_nk : m < n * k,
assert H3 : m + 1 ≤ n * k, from nat.succ_le_of_lt m_lt_nk,
assert H4 : m div k + 1 ≤ n,
from nat.succ_le_of_lt (nat.div_lt_of_lt_mul m_lt_nk),
have (-[1+m] + n * k) div k = -[1+m] div k + n, from calc
(-[1+m] + n * k) div k
= of_nat ((k * n - (m + 1)) div k) :
by rewrite [add.comm, neg_succ_of_nat_eq, of_nat_div, algebra.mul.comm k n,
of_nat_sub H3]
... = of_nat (n - m div k - 1) :
nat.mul_sub_div_of_lt (!nat.mul_comm ▸ m_lt_nk)
... = -[1+m] div k + n :
by rewrite [nat.sub_sub, of_nat_sub H4, int.add_comm, sub_eq_add_neg,
!neg_succ_of_nat_div (of_nat_lt_of_nat_of_lt H2),
of_nat_add, of_nat_div],
Hm⁻¹ ▸ this)
(assume nk_le_m : n * k ≤ m,
have -[1+m] div k + n = (-[1+m] + n * k) div k, from calc
-[1+m] div k + n
= -(of_nat ((m - n * k + n * k) div k) + 1) + n :
by rewrite [neg_succ_of_nat_div m (of_nat_lt_of_nat_of_lt H2),
nat.sub_add_cancel nk_le_m, of_nat_div]
... = -(of_nat ((m - n * k) div k + n) + 1) + n : nat.add_mul_div_self H2
... = -(of_nat (m - n * k) div k + 1) :
by rewrite [of_nat_add, *neg_add, add.right_comm, neg_add_cancel_right,
of_nat_div]
... = -[1+(m - n * k)] div k :
neg_succ_of_nat_div _ (of_nat_lt_of_nat_of_lt H2)
... = -(of_nat(m - n * k) + 1) div k : rfl
... = -(of_nat m - of_nat(n * k) + 1) div k : of_nat_sub nk_le_m
... = (-(of_nat m + 1) + n * k) div k :
by rewrite [sub_eq_add_neg, -*add.assoc, *neg_add, neg_neg, add.right_comm]
... = (-[1+m] + n * k) div k : rfl,
Hm⁻¹ ▸ this⁻¹)
private theorem add_mul_div_self_aux3 (a : ) {b c : } (H1 : b ≥ 0) (H2 : c > 0) :
(a + b * c) div c = a div c + b :=
obtain (n : nat) (Hn : b = of_nat n), from exists_eq_of_nat H1,
obtain (k : nat) (Hk : c = of_nat k), from exists_eq_of_nat (le_of_lt H2),
have knz : k ≠ 0, from assume kz, !lt.irrefl (kz ▸ Hk ▸ H2),
have kgt0 : (#nat k > 0), from nat.pos_of_ne_zero knz,
have H3 : (a + n * k) div k = a div k + n, from
or.elim (lt_or_ge a 0)
(assume Ha : a < 0, add_mul_div_self_aux2 _ Ha kgt0)
(assume Ha : a ≥ 0, add_mul_div_self_aux1 _ Ha kgt0),
Hn⁻¹ ▸ Hk⁻¹ ▸ H3
private theorem add_mul_div_self_aux4 (a b : ) {c : } (H : c > 0) :
(a + b * c) div c = a div c + b :=
or.elim (le.total 0 b)
(assume H1 : 0 ≤ b, add_mul_div_self_aux3 _ H1 H)
(assume H1 : 0 ≥ b,
eq.symm (calc
a div c + b = (a + b * c + -b * c) div c + b :
by rewrite [-neg_mul_eq_neg_mul, add_neg_cancel_right]
... = (a + b * c) div c + - b + b :
add_mul_div_self_aux3 _ (neg_nonneg_of_nonpos H1) H
... = (a + b * c) div c : neg_add_cancel_right))
theorem add_mul_div_self (a b : ) {c : } (H : c ≠ 0) : (a + b * c) div c = a div c + b :=
lt.by_cases
(assume H1 : 0 < c, !add_mul_div_self_aux4 H1)
(assume H1 : 0 = c, absurd H1⁻¹ H)
(assume H1 : 0 > c,
have H2 : -c > 0, from neg_pos_of_neg H1,
calc
(a + b * c) div c = - ((a + -b * -c) div -c) : by rewrite [div_neg, neg_mul_neg, neg_neg]
... = -(a div -c + -b) : !add_mul_div_self_aux4 H2
... = a div c + b : by rewrite [div_neg, neg_add, *neg_neg])
theorem add_mul_div_self_left (a : ) {b : } (c : ) (H : b ≠ 0) :
(a + b * c) div b = a div b + c :=
!mul.comm ▸ !add_mul_div_self H
theorem mul_div_cancel (a : ) {b : } (H : b ≠ 0) : a * b div b = a :=
calc
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a * b div b = (0 + a * b) div b : algebra.zero_add
... = 0 div b + a : !add_mul_div_self H
... = a : by rewrite [zero_div, zero_add]
theorem mul_div_cancel_left {a : } (b : ) (H : a ≠ 0) : a * b div a = b :=
!mul.comm ▸ mul_div_cancel b H
theorem div_self {a : } (H : a ≠ 0) : a div a = 1 :=
!mul_one ▸ !mul_div_cancel_left H
/- mod -/
theorem of_nat_mod (m n : nat) : m mod n = of_nat (m mod n) :=
have H : m = of_nat (m mod n) + m div n * n, from calc
m = of_nat (m div n * n + m mod n) : nat.eq_div_mul_add_mod
... = of_nat (m div n) * n + of_nat (m mod n) : rfl
... = m div n * n + of_nat (m mod n) : of_nat_div
... = of_nat (m mod n) + m div n * n : add.comm,
calc
m mod n = m - m div n * n : rfl
... = of_nat (m mod n) : sub_eq_of_eq_add H
theorem neg_succ_of_nat_mod (m : ) {b : } (bpos : b > 0) :
-[1+m] mod b = b - 1 - m mod b :=
calc
-[1+m] mod b = -(m + 1) - -[1+m] div b * b : rfl
... = -(m + 1) - -(m div b + 1) * b : neg_succ_of_nat_div _ bpos
... = -m + -1 + (b + m div b * b) :
by rewrite [neg_add, -neg_mul_eq_neg_mul, sub_neg_eq_add, right_distrib,
one_mul, (add.comm b)]
... = b + -1 + (-m + m div b * b) :
by rewrite [-*add.assoc, add.comm (-m), add.right_comm (-1), (add.comm b)]
... = b - 1 - m mod b :
by rewrite [(modulo.def), *sub_eq_add_neg, neg_add, neg_neg]
theorem mod_neg (a b : ) : a mod -b = a mod b :=
calc
a mod -b = a - (a div -b) * -b : rfl
... = a - -(a div b) * -b : div_neg
... = a - a div b * b : neg_mul_neg
... = a mod b : rfl
theorem mod_abs (a b : ) : a mod (abs b) = a mod b :=
abs.by_cases rfl !mod_neg
theorem zero_mod (b : ) : 0 mod b = 0 :=
by rewrite [(modulo.def), zero_div, zero_mul, sub_zero]
theorem mod_zero (a : ) : a mod 0 = a :=
by rewrite [(modulo.def), mul_zero, sub_zero]
theorem mod_one (a : ) : a mod 1 = 0 :=
calc
a mod 1 = a - a div 1 * 1 : rfl
... = 0 : by rewrite [mul_one, div_one, sub_self]
private lemma of_nat_mod_abs (m : ) (b : ) : m mod (abs b) = of_nat (m mod (nat_abs b)) :=
calc
m mod (abs b) = m mod (nat_abs b) : of_nat_nat_abs
... = of_nat (m mod (nat_abs b)) : of_nat_mod
private lemma of_nat_mod_abs_lt (m : ) {b : } (H : b ≠ 0) : m mod (abs b) < (abs b) :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : (#nat nat_abs b > 0), from lt_of_of_nat_lt_of_nat (!of_nat_nat_abs⁻¹ ▸ H1),
calc
m mod (abs b) = of_nat (m mod (nat_abs b)) : of_nat_mod_abs m b
... < nat_abs b : of_nat_lt_of_nat_of_lt (!nat.mod_lt H2)
... = abs b : of_nat_nat_abs _
theorem mod_eq_of_lt {a b : } (H1 : 0 ≤ a) (H2 : a < b) : a mod b = a :=
obtain (m : nat) (Hm : a = of_nat m), from exists_eq_of_nat H1,
obtain (n : nat) (Hn : b = of_nat n), from exists_eq_of_nat (le_of_lt (lt_of_le_of_lt H1 H2)),
begin
revert H2,
rewrite [Hm, Hn, of_nat_mod, of_nat_lt_of_nat_iff, of_nat_eq_of_nat_iff],
apply nat.mod_eq_of_lt
end
theorem mod_nonneg (a : ) {b : } (H : b ≠ 0) : a mod b ≥ 0 :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : a mod (abs b) ≥ 0, from
int.cases_on a
(take m : nat, (of_nat_mod_abs m b)⁻¹ ▸ of_nat_nonneg (nat.modulo m (nat_abs b)))
(take m : nat,
have H3 : 1 + m mod (abs b) ≤ (abs b),
from (!add.comm ▸ add_one_le_of_lt (of_nat_mod_abs_lt m H)),
calc
-[1+m] mod (abs b) = abs b - 1 - m mod (abs b) : neg_succ_of_nat_mod _ H1
... = abs b - (1 + m mod (abs b)) : by rewrite [*sub_eq_add_neg, neg_add, add.assoc]
... ≥ 0 : iff.mpr !sub_nonneg_iff_le H3),
!mod_abs ▸ H2
theorem mod_lt (a : ) {b : } (H : b ≠ 0) : a mod b < (abs b) :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : a mod (abs b) < abs b, from
int.cases_on a
(take m, of_nat_mod_abs_lt m H)
(take m : nat,
have H3 : abs b ≠ 0, from assume H', H (eq_zero_of_abs_eq_zero H'),
have H4 : 1 + m mod (abs b) > 0,
from add_pos_of_pos_of_nonneg dec_trivial (mod_nonneg _ H3),
calc
-[1+m] mod (abs b) = abs b - 1 - m mod (abs b) : neg_succ_of_nat_mod _ H1
... = abs b - (1 + m mod (abs b)) : by rewrite [*sub_eq_add_neg, neg_add, add.assoc]
... < abs b : sub_lt_self _ H4),
!mod_abs ▸ H2
theorem add_mul_mod_self {a b c : } : (a + b * c) mod c = a mod c :=
decidable.by_cases
(assume cz : c = 0, by rewrite [cz, mul_zero, add_zero])
(assume cnz, by rewrite [(modulo.def), !add_mul_div_self cnz, right_distrib,
sub_add_eq_sub_sub_swap, add_sub_cancel])
theorem add_mul_mod_self_left (a b c : ) : (a + b * c) mod b = a mod b :=
!mul.comm ▸ !add_mul_mod_self
theorem add_mod_self {a b : } : (a + b) mod b = a mod b :=
by rewrite -(int.mul_one b) at {1}; apply add_mul_mod_self_left
theorem add_mod_self_left {a b : } : (a + b) mod a = b mod a :=
!add.comm ▸ !add_mod_self
theorem mod_add_mod (m n k : ) : (m mod n + k) mod n = (m + k) mod n :=
by rewrite [eq_div_mul_add_mod m n at {2}, add.assoc, add.comm (m div n * n), add_mul_mod_self]
theorem add_mod_mod (m n k : ) : (m + n mod k) mod k = (m + n) mod k :=
by rewrite [add.comm, mod_add_mod, add.comm]
theorem add_mod_eq_add_mod_right {m n k : } (i : ) (H : m mod n = k mod n) :
(m + i) mod n = (k + i) mod n :=
by rewrite [-mod_add_mod, -mod_add_mod k, H]
theorem add_mod_eq_add_mod_left {m n k : } (i : ) (H : m mod n = k mod n) :
(i + m) mod n = (i + k) mod n :=
by rewrite [add.comm, add_mod_eq_add_mod_right _ H, add.comm]
theorem mod_eq_mod_of_add_mod_eq_add_mod_right {m n k i : }
(H : (m + i) mod n = (k + i) mod n) :
m mod n = k mod n :=
assert H1 : (m + i + (-i)) mod n = (k + i + (-i)) mod n, from add_mod_eq_add_mod_right _ H,
by rewrite [*add_neg_cancel_right at H1]; apply H1
theorem mod_eq_mod_of_add_mod_eq_add_mod_left {m n k i : } :
(i + m) mod n = (i + k) mod n → m mod n = k mod n :=
by rewrite [add.comm i m, add.comm i k]; apply mod_eq_mod_of_add_mod_eq_add_mod_right
theorem mul_mod_left (a b : ) : (a * b) mod b = 0 :=
by rewrite [-zero_add (a * b), add_mul_mod_self, zero_mod]
theorem mul_mod_right (a b : ) : (a * b) mod a = 0 :=
!mul.comm ▸ !mul_mod_left
theorem mod_self {a : } : a mod a = 0 :=
decidable.by_cases
(assume H : a = 0, H⁻¹ ▸ !mod_zero)
(assume H : a ≠ 0,
calc
a mod a = a - a div a * a : rfl
... = 0 : by rewrite [!div_self H, one_mul, sub_self])
theorem mod_lt_of_pos (a : ) {b : } (H : b > 0) : a mod b < b :=
!abs_of_pos H ▸ !mod_lt (ne.symm (ne_of_lt H))
/- properties of div and mod -/
theorem mul_div_mul_of_pos_aux {a : } (b : ) {c : }
(H1 : a > 0) (H2 : c > 0) : a * b div (a * c) = b div c :=
have H3 : a * c ≠ 0, from ne.symm (ne_of_lt (mul_pos H1 H2)),
have H4 : a * (b mod c) < a * c, from mul_lt_mul_of_pos_left (!mod_lt_of_pos H2) H1,
have H5 : a * (b mod c) ≥ 0, from mul_nonneg (le_of_lt H1) (!mod_nonneg (ne.symm (ne_of_lt H2))),
calc
a * b div (a * c) = a * (b div c * c + b mod c) div (a * c) : eq_div_mul_add_mod
... = (a * (b mod c) + a * c * (b div c)) div (a * c) :
by rewrite [!add.comm, int.mul_left_distrib, mul.comm _ c, -!mul.assoc]
... = a * (b mod c) div (a * c) + b div c : !add_mul_div_self_left H3
... = 0 + b div c : {!div_eq_zero_of_lt H5 H4}
... = b div c : zero_add
theorem mul_div_mul_of_pos {a : } (b c : ) (H : a > 0) : a * b div (a * c) = b div c :=
lt.by_cases
(assume H1 : c < 0,
have H2 : -c > 0, from neg_pos_of_neg H1,
calc
a * b div (a * c) = - (a * b div (a * -c)) :
by rewrite [-neg_mul_eq_mul_neg, div_neg, neg_neg]
... = - (b div -c) : mul_div_mul_of_pos_aux _ H H2
... = b div c : by rewrite [div_neg, neg_neg])
(assume H1 : c = 0,
calc
a * b div (a * c) = 0 : by rewrite [H1, mul_zero, div_zero]
... = b div c : by rewrite [H1, div_zero])
(assume H1 : c > 0,
mul_div_mul_of_pos_aux _ H H1)
theorem mul_div_mul_of_pos_left (a : ) {b : } (c : ) (H : b > 0) :
a * b div (c * b) = a div c :=
!mul.comm ▸ !mul.comm ▸ !mul_div_mul_of_pos H
-- TODO: something strange here: why doesn't !modulo.def or !(modulo.def) work?
theorem mul_mod_mul_of_pos {a : } (b c : ) (H : a > 0) : a * b mod (a * c) = a * (b mod c) :=
by rewrite [(modulo.def), modulo.def, !mul_div_mul_of_pos H, mul_sub_left_distrib, mul.left_comm]
theorem lt_div_add_one_mul_self (a : ) {b : } (H : b > 0) : a < (a div b + 1) * b :=
have H : a - a div b * b < b, from !mod_lt_of_pos H,
calc
a < a div b * b + b : iff.mpr !lt_add_iff_sub_lt_left H
... = (a div b + 1) * b : by rewrite [right_distrib, one_mul]
theorem div_le_of_nonneg_of_nonneg {a b : } (Ha : a ≥ 0) (Hb : b ≥ 0) : a div b ≤ a :=
obtain (m : ) (Hm : a = m), from exists_eq_of_nat Ha,
obtain (n : ) (Hn : b = n), from exists_eq_of_nat Hb,
calc
a div b = of_nat (m div n) : by rewrite [Hm, Hn, of_nat_div]
... ≤ m : of_nat_le_of_nat_of_le !nat.div_le_self
... = a : Hm
theorem abs_div_le_abs (a b : ) : abs (a div b) ≤ abs a :=
have H : ∀a b, b > 0 → abs (a div b) ≤ abs a, from
take a b,
assume H1 : b > 0,
or.elim (le_or_gt 0 a)
(assume H2 : 0 ≤ a,
have H3 : 0 ≤ b, from le_of_lt H1,
calc
abs (a div b) = a div b : abs_of_nonneg (div_nonneg H2 H3)
... ≤ a : div_le_of_nonneg_of_nonneg H2 H3
... = abs a : abs_of_nonneg H2)
(assume H2 : a < 0,
have H3 : -a - 1 ≥ 0, from le_sub_one_of_lt (neg_pos_of_neg H2),
have H4 : (-a - 1) div b + 1 ≥ 0,
from add_nonneg (div_nonneg H3 (le_of_lt H1)) (of_nat_le_of_nat_of_le !nat.zero_le),
have H5 : (-a - 1) div b ≤ -a - 1, from div_le_of_nonneg_of_nonneg H3 (le_of_lt H1),
calc
abs (a div b) = abs ((-a - 1) div b + 1) : by rewrite [div_of_neg_of_pos H2 H1, abs_neg]
... = (-a - 1) div b + 1 : abs_of_nonneg H4
... ≤ -a - 1 + 1 : add_le_add_right H5 _
... = abs a : by rewrite [sub_add_cancel, abs_of_neg H2]),
lt.by_cases
(assume H1 : b < 0,
calc
abs (a div b) = abs (a div -b) : by rewrite [div_neg, abs_neg]
... ≤ abs a : H _ _ (neg_pos_of_neg H1))
(assume H1 : b = 0,
calc
abs (a div b) = 0 : by rewrite [H1, div_zero, abs_zero]
... ≤ abs a : abs_nonneg)
(assume H1 : b > 0, H _ _ H1)
theorem div_mul_cancel_of_mod_eq_zero {a b : } (H : a mod b = 0) : a div b * b = a :=
by rewrite [eq_div_mul_add_mod a b at {2}, H, add_zero]
theorem mul_div_cancel_of_mod_eq_zero {a b : } (H : a mod b = 0) : b * (a div b) = a :=
!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H
/- dvd -/
theorem dvd_of_of_nat_dvd_of_nat {m n : } : of_nat m of_nat n → (#nat m n) :=
nat.by_cases_zero_pos n
(assume H, dvd_zero m)
(take n' : ,
assume H1 : (#nat n' > 0),
have H2 : of_nat n' > 0, from of_nat_pos H1,
assume H3 : of_nat m of_nat n',
dvd.elim H3
(take c,
assume H4 : of_nat n' = of_nat m * c,
have H5 : c > 0, from pos_of_mul_pos_left (H4 ▸ H2) !of_nat_nonneg,
obtain k (H6 : c = of_nat k), from exists_eq_of_nat (le_of_lt H5),
have H7 : n' = (#nat m * k), from (of_nat.inj (H6 ▸ H4)),
dvd.intro H7⁻¹))
theorem of_nat_dvd_of_nat_of_dvd {m n : } (H : #nat m n) : of_nat m of_nat n :=
dvd.elim H
(take k, assume H1 : #nat n = m * k,
dvd.intro (H1⁻¹ ▸ rfl))
theorem of_nat_dvd_of_nat_iff (m n : ) : of_nat m of_nat n ↔ m n :=
iff.intro dvd_of_of_nat_dvd_of_nat of_nat_dvd_of_nat_of_dvd
theorem dvd.antisymm {a b : } (H1 : a ≥ 0) (H2 : b ≥ 0) : a b → b a → a = b :=
begin
rewrite [-abs_of_nonneg H1, -abs_of_nonneg H2, -*of_nat_nat_abs],
rewrite [*of_nat_dvd_of_nat_iff, *of_nat_eq_of_nat_iff],
apply nat.dvd.antisymm
end
theorem dvd_of_mod_eq_zero {a b : } (H : b mod a = 0) : a b :=
dvd.intro (!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H)
theorem mod_eq_zero_of_dvd {a b : } (H : a b) : b mod a = 0 :=
dvd.elim H (take z, assume H1 : b = a * z, H1⁻¹ ▸ !mul_mod_right)
theorem dvd_iff_mod_eq_zero (a b : ) : a b ↔ b mod a = 0 :=
iff.intro mod_eq_zero_of_dvd dvd_of_mod_eq_zero
definition dvd.decidable_rel [instance] : decidable_rel dvd :=
take a n, decidable_of_decidable_of_iff _ (iff.symm !dvd_iff_mod_eq_zero)
theorem div_mul_cancel {a b : } (H : b a) : a div b * b = a :=
div_mul_cancel_of_mod_eq_zero (mod_eq_zero_of_dvd H)
theorem mul_div_cancel' {a b : } (H : a b) : a * (b div a) = b :=
!mul.comm ▸ !div_mul_cancel H
theorem mul_div_assoc (a : ) {b c : } (H : c b) : (a * b) div c = a * (b div c) :=
decidable.by_cases
(assume cz : c = 0, by rewrite [cz, *div_zero, mul_zero])
(assume cnz : c ≠ 0,
obtain d (H' : b = d * c), from exists_eq_mul_left_of_dvd H,
by rewrite [H', -mul.assoc, *(!mul_div_cancel cnz)])
theorem div_dvd_div {a b c : } (H1 : a b) (H2 : b c) : b div a c div a :=
have H3 : b = b div a * a, from (div_mul_cancel H1)⁻¹,
have H4 : c = c div a * a, from (div_mul_cancel (dvd.trans H1 H2))⁻¹,
decidable.by_cases
(assume H5 : a = 0,
have H6: c div a = 0, from (congr_arg _ H5 ⬝ !div_zero),
H6⁻¹ ▸ !dvd_zero)
(assume H5 : a ≠ 0,
dvd_of_mul_dvd_mul_right H5 (H3 ▸ H4 ▸ H2))
theorem div_eq_iff_eq_mul_right {a b : } (c : ) (H : b ≠ 0) (H' : b a) :
a div b = c ↔ a = b * c :=
iff.intro
(assume H1, by rewrite [-H1, mul_div_cancel' H'])
(assume H1, by rewrite [H1, !mul_div_cancel_left H])
theorem div_eq_iff_eq_mul_left {a b : } (c : ) (H : b ≠ 0) (H' : b a) :
a div b = c ↔ a = c * b :=
!mul.comm ▸ !div_eq_iff_eq_mul_right H H'
theorem eq_mul_of_div_eq_right {a b c : } (H1 : b a) (H2 : a div b = c) :
a = b * c :=
calc
a = b * (a div b) : mul_div_cancel' H1
... = b * c : H2
theorem div_eq_of_eq_mul_right {a b c : } (H1 : b ≠ 0) (H2 : a = b * c) :
a div b = c :=
calc
a div b = b * c div b : H2
... = c : !mul_div_cancel_left H1
theorem eq_mul_of_div_eq_left {a b c : } (H1 : b a) (H2 : a div b = c) :
a = c * b :=
!mul.comm ▸ !eq_mul_of_div_eq_right H1 H2
theorem div_eq_of_eq_mul_left {a b c : } (H1 : b ≠ 0) (H2 : a = c * b) :
a div b = c :=
div_eq_of_eq_mul_right H1 (!mul.comm ▸ H2)
theorem neg_div_of_dvd {a b : } (H : b a) : -a div b = -(a div b) :=
decidable.by_cases
(assume H1 : b = 0, by rewrite [H1, *div_zero, neg_zero])
(assume H1 : b ≠ 0,
dvd.elim H
(take c, assume H' : a = b * c,
by rewrite [H', neg_mul_eq_mul_neg, *!mul_div_cancel_left H1]))
theorem sign_eq_div_abs (a : ) : sign a = a div (abs a) :=
decidable.by_cases
(suppose a = 0, by subst a)
(suppose a ≠ 0,
have abs a ≠ 0, from assume H, this (eq_zero_of_abs_eq_zero H),
have abs a a, from abs_dvd_of_dvd !dvd.refl,
eq.symm (iff.mpr (!div_eq_iff_eq_mul_left `abs a ≠ 0` this) !eq_sign_mul_abs))
theorem le_of_dvd {a b : } (bpos : b > 0) (H : a b) : a ≤ b :=
or.elim !le_or_gt
(suppose a ≤ 0, le.trans this (le_of_lt bpos))
(suppose a > 0,
obtain c (Hc : b = a * c), from exists_eq_mul_right_of_dvd H,
have a * c > 0, by rewrite -Hc; exact bpos,
have c > 0, from pos_of_mul_pos_left this (le_of_lt `a > 0`),
show a ≤ b, from calc
a = a * 1 : mul_one
... ≤ a * c : mul_le_mul_of_nonneg_left (add_one_le_of_lt `c > 0`) (le_of_lt `a > 0`)
... = b : Hc)
/- div and ordering -/
theorem div_mul_le (a : ) {b : } (H : b ≠ 0) : a div b * b ≤ a :=
calc
a = a div b * b + a mod b : eq_div_mul_add_mod
... ≥ a div b * b : le_add_of_nonneg_right (!mod_nonneg H)
theorem div_le_of_le_mul {a b c : } (H : c > 0) (H' : a ≤ b * c) : a div c ≤ b :=
le_of_mul_le_mul_right (calc
a div c * c = a div c * c + 0 : add_zero
... ≤ a div c * c + a mod c : add_le_add_left (!mod_nonneg (ne_of_gt H))
... = a : eq_div_mul_add_mod
... ≤ b * c : H') H
theorem div_le_self (a : ) {b : } (H1 : a ≥ 0) (H2 : b ≥ 0) : a div b ≤ a :=
or.elim (lt_or_eq_of_le H2)
(assume H3 : b > 0,
have H4 : b ≥ 1, from add_one_le_of_lt H3,
have H5 : a ≤ a * b, from calc
a = a * 1 : mul_one
... ≤ a * b : !mul_le_mul_of_nonneg_left H4 H1,
div_le_of_le_mul H3 H5)
(assume H3 : 0 = b,
by rewrite [-H3, div_zero]; apply H1)
theorem mul_le_of_le_div {a b c : } (H1 : c > 0) (H2 : a ≤ b div c) : a * c ≤ b :=
calc
a * c ≤ b div c * c : !mul_le_mul_of_nonneg_right H2 (le_of_lt H1)
... ≤ b : !div_mul_le (ne_of_gt H1)
theorem le_div_of_mul_le {a b c : } (H1 : c > 0) (H2 : a * c ≤ b) : a ≤ b div c :=
have H3 : a * c < (b div c + 1) * c, from
calc
a * c ≤ b : H2
... = b div c * c + b mod c : eq_div_mul_add_mod
... < b div c * c + c : add_lt_add_left (!mod_lt_of_pos H1)
... = (b div c + 1) * c : by rewrite [right_distrib, one_mul],
le_of_lt_add_one (lt_of_mul_lt_mul_right H3 (le_of_lt H1))
theorem le_div_iff_mul_le {a b c : } (H : c > 0) : a ≤ b div c ↔ a * c ≤ b :=
iff.intro (!mul_le_of_le_div H) (!le_div_of_mul_le H)
theorem div_le_div {a b c : } (H : c > 0) (H' : a ≤ b) : a div c ≤ b div c :=
le_div_of_mul_le H (le.trans (!div_mul_le (ne_of_gt H)) H')
theorem div_lt_of_lt_mul {a b c : } (H : c > 0) (H' : a < b * c) : a div c < b :=
lt_of_mul_lt_mul_right
(calc
a div c * c = a div c * c + 0 : add_zero
... ≤ a div c * c + a mod c : add_le_add_left (!mod_nonneg (ne_of_gt H))
... = a : eq_div_mul_add_mod
... < b * c : H')
(le_of_lt H)
theorem lt_mul_of_div_lt {a b c : } (H1 : c > 0) (H2 : a div c < b) : a < b * c :=
assert H3 : (a div c + 1) * c ≤ b * c,
from !mul_le_mul_of_nonneg_right (add_one_le_of_lt H2) (le_of_lt H1),
have H4 : a div c * c + c ≤ b * c, by rewrite [right_distrib at H3, one_mul at H3]; apply H3,
calc
a = a div c * c + a mod c : eq_div_mul_add_mod
... < a div c * c + c : add_lt_add_left (!mod_lt_of_pos H1)
... ≤ b * c : H4
theorem div_lt_iff_lt_mul {a b c : } (H : c > 0) : a div c < b ↔ a < b * c :=
iff.intro (!lt_mul_of_div_lt H) (!div_lt_of_lt_mul H)
theorem div_le_iff_le_mul_of_div {a b : } (c : ) (H : b > 0) (H' : b a) :
a div b ≤ c ↔ a ≤ c * b :=
by rewrite [propext (!le_iff_mul_le_mul_right H), !div_mul_cancel H']
theorem le_mul_of_div_le_of_div {a b c : } (H1 : b > 0) (H2 : b a) (H3 : a div b ≤ c) :
a ≤ c * b :=
iff.mp (!div_le_iff_le_mul_of_div H1 H2) H3
theorem div_pos_of_pos_of_dvd {a b : } (H1 : a > 0) (H2 : b ≥ 0) (H3 : b a) : a div b > 0 :=
have H4 : b ≠ 0, from
(assume H5 : b = 0,
have H6 : a = 0, from eq_zero_of_zero_dvd (H5 ▸ H3),
ne_of_gt H1 H6),
have H6 : (a div b) * b > 0, by rewrite (div_mul_cancel H3); apply H1,
pos_of_mul_pos_right H6 H2
theorem div_eq_div_of_dvd_of_dvd {a b c d : } (H1 : b a) (H2 : d c) (H3 : b ≠ 0)
(H4 : d ≠ 0) (H5 : a * d = b * c) :
a div b = c div d :=
begin
apply div_eq_of_eq_mul_right H3,
rewrite [-!mul_div_assoc H2],
apply eq.symm,
apply div_eq_of_eq_mul_left H4,
apply eq.symm H5
end
end int