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feat(library/data/int/div.lean): add theorems about div

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This commit is contained in:
Jeremy Avigad 2015-03-12 23:43:19 -04:00 • committed by Leonardo de Moura
parent a24f8dce67
commit ec05e83a2a
4 changed files with 248 additions and 38 deletions
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3
library/data/int/basic.lean
View file

@ -789,4 +789,7 @@ have H1 : m = (#nat m - n + n), from (nat.sub_add_cancel H)⁻¹,
have H2 : m = (#nat m - n) + n, from congr_arg of_nat H1,
sub_eq_of_eq_add H2
theorem neg_succ_of_nat_eq' (m : ℕ) : -[m +1] = -m - 1 :=
by rewrite [neg_succ_of_nat_eq, -of_nat_add_of_nat, neg_add]
end int

224
library/data/int/div.lean
View file

@ -1,12 +1,13 @@
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: data.int.div
Author: Jeremy Avigad
Definitions and properties of div, mod, gcd, lcm, coprime. Following the SSReflect library
(and the SMT lib standard), we define a mod b so that 0 ≤ a mod b < |b| when b ≠ 0.
Definitions and properties of div, mod, gcd, lcm, coprime, 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
@ -34,6 +35,8 @@ definition modulo (a b : ℤ) : ℤ := a - a div b * b
notation a mod b := modulo a b
notation a = b [mod c] := a mod c = b mod c
/- div -/
theorem of_nat_div_of_nat (m n : nat) : m div n = of_nat (#nat m div n) :=
@ -49,12 +52,33 @@ calc
... = -(m div b + 1) : by rewrite [↑divide, sign_of_pos H, one_mul]
theorem div_neg (a b : ℤ) : a div -b = -(a div b) :=
by rewrite [↑divide, sign_neg, neg_mul_eq_neg_mul, nat_abs_neg]
theorem div_of_neg_of_pos {a b : ℤ} (Ha : a < 0) (Hb : b > 0) : a div b = -((-a - 1) div b + 1) :=
obtain m (H1 : a = -[m +1]), from exists_eq_neg_succ_of_nat Ha,
calc
a div -b = sign (-b) * _ : rfl
... = -(sign b) * _ : sign_neg
... = -(sign b * _) : neg_mul_eq_neg_mul
... = -(sign b * _) : nat_abs_neg
... = -(a div b) : rfl
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 : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) : a div b ≥ 0 :=
obtain (m : ℕ) (Hm : a = m), from exists_eq_of_nat Ha,
obtain (n : ℕ) (Hn : b = n), from exists_eq_of_nat Hb,
calc
a div b = (#nat m div n) : by rewrite [Hm, Hn, of_nat_div_of_nat]
... ≥ 0 : trivial
theorem div_nonpos {a b : ℤ} (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 : ℤ} (Ha : a < 0) (Hb : b > 0) : a div b < 0 :=
have H1 : -a - 1 ≥ 0, from le_sub_one_of_lt (neg_pos_of_neg Ha),
have H2 : (-a - 1) div b + 1 > 0, from lt_add_one_of_le (div_nonneg H1 (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 H2
theorem zero_div (b : ℤ) : 0 div b = 0 :=
calc
@ -65,9 +89,45 @@ calc
theorem div_zero (a : ℤ) : a div 0 = 0 :=
by rewrite [↑divide, sign_zero, zero_mul]
theorem div_one (a : ℤ) :a div 1 = a :=
assert H : 1 > 0, from dec_trivial,
int.cases_on a
(take m, by rewrite [of_nat_div_of_nat, nat.div_one])
(take m, by rewrite [!neg_succ_of_nat_div H, of_nat_div_of_nat, nat.div_one])
theorem eq_div_mul_add_mod {a b : ℤ} : a = a div b * b + a mod b :=
!add.comm ▸ eq_add_of_sub_eq rfl
theorem div_eq_zero_of_lt {a b : ℤ} : 0 ≤ a → a < b → a div b = 0 :=
int.cases_on a
(take m, assume H,
int.cases_on b
(take n,
assume H : m < n,
calc
m div n = #nat m div n : of_nat_div_of_nat
... = 0 : nat.div_eq_zero_of_lt (lt_of_of_nat_lt_of_nat H))
(take n,
assume H : m < -[ n +1],
have H1 : ¬(m < -[ n +1]), from dec_trivial,
absurd H H1))
(take m,
assume H : 0 ≤ -[ m +1],
have H1 : ¬ (0 ≤ -[ m +1]), from dec_trivial,
absurd H H1)
theorem div_eq_zero_of_lt_abs {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < abs b) : a div b = 0 :=
lt.by_cases
(assume H : b < 0,
assert H3 : a < -b, from abs_of_neg H ▸ H2,
calc
a div b = - (a div -b) : by rewrite [div_neg, neg_neg]
... = 0 : by rewrite [div_eq_zero_of_lt H1 H3, neg_zero])
(assume H : b = 0, H⁻¹ ▸ !div_zero)
(assume H : b > 0,
have H3 : a < b, from abs_of_pos H ▸ H2,
div_eq_zero_of_lt H1 H3)
/- mod -/
theorem of_nat_mod_of_nat (m n : nat) : m mod n = (#nat m mod n) :=
@ -109,6 +169,11 @@ by rewrite [↑modulo, zero_div, zero_mul, sub_zero]
theorem mod_zero (a : ℤ) : a mod 0 = a :=
by rewrite [↑modulo, mul_zero, sub_zero]
theorem mod_one (a : ℤ) : 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 : ℕ) (b : ℤ) : m mod (abs b) = (#nat m mod (nat_abs b)) :=
calc
m mod (abs b) = m mod (nat_abs b) : of_nat_nat_abs
@ -128,7 +193,8 @@ have H2 : a mod (abs b) ≥ 0, from
int.cases_on a
(take m, (of_nat_mod_abs m b)⁻¹ ▸ !of_nat_nonneg)
(take m,
have H3 : 1 + m mod (abs b) ≤ (abs b), from (!add.comm ▸ add_one_le_of_lt (of_nat_mod_abs_lt m H)),
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
-[ m +1] 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]
@ -151,17 +217,19 @@ have H2 : a mod (abs b) < abs b, from
/- both div and mod -/
private theorem add_mul_div_self_right_aux1 {a : ℤ} {k : ℕ} (n : ℕ) (H1 : a ≥ 0) (H2 : #nat k > 0) :
private theorem add_mul_div_self_aux1 {a : ℤ} {k : ℕ} (n : ℕ)
(H1 : a ≥ 0) (H2 : #nat k > 0) :
(a + n * k) div k = a div k + n :=
obtain m (Hm : a = of_nat m), from exists_eq_of_nat H1,
Hm⁻¹ ▸ (calc
(m + n * k) div k = (#nat (m + n * k)) div k : rfl
... = (#nat (m + n * k) div k) : of_nat_div_of_nat
... = (#nat m div k + n) : !nat.add_mul_div_self_right H2
... = (#nat m div k + n) : !nat.add_mul_div_self H2
... = (#nat m div k) + n : rfl
... = m div k + n : of_nat_div_of_nat)
private theorem add_mul_div_self_right_aux2 {a : ℤ} {k : ℕ} (n : ℕ) (H1 : a < 0) (H2 : #nat k > 0) :
private theorem add_mul_div_self_aux2 {a : ℤ} {k : ℕ} (n : ℕ)
(H1 : a < 0) (H2 : #nat k > 0) :
(a + n * k) div k = a div k + n :=
obtain m (Hm : a = -[m +1]), from exists_eq_neg_succ_of_nat H1,
or.elim (nat.lt_or_ge m (#nat n * k))
@ -189,7 +257,7 @@ or.elim (nat.lt_or_ge m (#nat n * k))
neg_succ_of_nat_div m (of_nat_lt_of_nat H2)
... = -((#nat m div k) + 1) + n : of_nat_div_of_nat
... = -((#nat (m - n * k + n * k) div k) + 1) + n : nat.sub_add_cancel nk_le_m
... = -((#nat (m - n * k) div k + n) + 1) + n : nat.add_mul_div_self_right H2
... = -((#nat (m - n * k) div k + n) + 1) + n : nat.add_mul_div_self H2
... = -((#nat m - n * k) div k + 1) :
by rewrite [-of_nat_add_of_nat, *neg_add, add.right_comm, neg_add_cancel_right,
of_nat_div_of_nat]
@ -201,7 +269,7 @@ or.elim (nat.lt_or_ge m (#nat n * k))
by rewrite [sub_eq_add_neg, -*add.assoc, *neg_add, neg_neg, add.right_comm]
... = (-[m +1] + n * k) div k : rfl)))
private theorem add_mul_div_self_right_aux3 (a : ℤ) {b c : ℤ} (H1 : b ≥ 0) (H2 : c > 0) :
private theorem add_mul_div_self_aux3 (a : ℤ) {b c : ℤ} (H1 : b ≥ 0) (H2 : c > 0) :
(a + b * c) div c = a div c + b :=
obtain n (Hn : b = of_nat n), from exists_eq_of_nat H1,
obtain k (Hk : c = of_nat k), from exists_eq_of_nat (le_of_lt H2),
@ -209,31 +277,143 @@ 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_right_aux2 _ Ha kgt0)
(assume Ha : a ≥ 0, add_mul_div_self_right_aux1 _ Ha kgt0),
(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_right_aux4 (a b : ℤ) {c : ℤ} (H : c > 0) :
private theorem add_mul_div_self_aux4 (a b : ℤ) {c : ℤ} (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_right_aux3 _ H1 H)
(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_right_aux3 _ (neg_nonneg_of_nonpos H1) H
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_right (a b : ℤ) {c : ℤ} (H : c ≠ 0) : (a + b * c) div c = a div c + b :=
theorem add_mul_div_self (a b : ℤ) {c : ℤ} (H : c ≠ 0) : (a + b * c) div c = a div c + b :=
lt.by_cases
(assume H1 : 0 < c, !add_mul_div_self_right_aux4 H1)
(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_right_aux4 H2
... = -(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 : ℤ) {b : ℤ} (c : ℤ) (H : b ≠ 0) :
(a + b * c) div b = a div b + c :=
!mul.comm ▸ !add_mul_div_self H
theorem mul_div_cancel (a : ℤ) {b : ℤ} (H : b ≠ 0) : a * b div b = a :=
calc
a * b div b = (0 + a * b) div b : zero_add
... = 0 div b + a : !add_mul_div_self H
... = a : by rewrite [zero_div, zero_add]
theorem mul_div_cancel_left {a : ℤ} (b : ℤ) (H : a ≠ 0) : a * b div a = b :=
!mul.comm ▸ mul_div_cancel b H
theorem div_self {a : ℤ} (H : a ≠ 0) : a div a = 1 :=
!mul_one ▸ !mul_div_cancel_left H
theorem mod_self {a : ℤ} : 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 : ℤ) {b : ℤ} (H : b > 0) : a mod b < b :=
!abs_of_pos H ▸ !mod_lt (ne.symm (ne_of_lt H))
theorem mul_div_mul_of_pos_aux {a : ℤ} (b : ℤ) {c : ℤ}
(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, 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 : ℤ} (b c : ℤ) (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⁻¹, 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 : ℤ) {b : ℤ} (c : ℤ) (H : b > 0) : a * b div (c * b) = a div c :=
!mul.comm ▸ !mul.comm ▸ !mul_div_mul_of_pos H
theorem div_mul_le (a : ℤ) {b : ℤ} (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 lt_div_add_one_mul_self (a : ℤ) {b : ℤ} (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.mp' !lt_add_iff_sub_lt_left H
... = (a div b + 1) * b : by rewrite [mul.right_distrib, one_mul]
theorem div_le_of_nonneg_of_nonneg {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) : a div b ≤ a :=
obtain (m : ℕ) (Hm : a = m), from exists_eq_of_nat Ha,
obtain (n : ℕ) (Hn : b = n), from exists_eq_of_nat Hb,
calc
a div b = #nat m div n : by rewrite [Hm, Hn, of_nat_div_of_nat]
... ≤ m : of_nat_le_of_nat !nat.div_le
... = a : Hm
theorem abs_div_le_abs (a b : ℤ) : 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)) trivial,
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)
/- ltz_divLR -/
end int

19
library/data/int/order.lean
View file

@ -595,6 +595,25 @@ obtain n (H1 : a + succ n = b), from lt.elim H,
have H2 : a + 1 + n = b, by rewrite [-H1, add.assoc, add.comm 1],
le.intro H2
theorem lt_add_one_of_le {a b : ℤ} (H : a ≤ b) : a < b + 1 :=
lt_add_of_le_of_pos H trivial
theorem le_of_lt_add_one {a b : ℤ} (H : a < b + 1) : a ≤ b :=
have H1 : a + 1 ≤ b + 1, from add_one_le_of_lt H,
le_of_add_le_add_right H1
theorem sub_one_le_of_lt {a b : ℤ} (H : a ≤ b) : a - 1 < b :=
lt_of_add_one_le (!sub_add_cancel⁻¹ ▸ H)
theorem lt_of_sub_one_le {a b : ℤ} (H : a - 1 < b) : a ≤ b :=
!sub_add_cancel ▸ add_one_le_of_lt H
theorem le_sub_one_of_lt {a b : ℤ} (H : a < b) : a ≤ b - 1 :=
le_of_lt_add_one (!sub_add_cancel⁻¹ ▸ H)
theorem lt_of_le_sub_one {a b : ℤ} (H : a ≤ b - 1) : a < b :=
!sub_add_cancel ▸ (lt_add_one_of_le H)
theorem of_nat_nonneg (n : ℕ) : of_nat n ≥ 0 := trivial
theorem of_nat_pos {n : ℕ} (Hpos : #nat n > 0) : of_nat n > 0 :=

40
library/data/nat/div.lean
View file

@ -42,16 +42,16 @@ divide_def 0 b ⬝ if_neg (λ h, and.rec_on h (λ l r, absurd (lt_of_lt_of_le l
theorem div_eq_succ_sub_div {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a div b = succ ((a - b) div b) :=
divide_def a b ⬝ if_pos (and.intro h₁ h₂)
theorem add_div_self_right (x : ℕ) {z : ℕ} (H : z > 0) : (x + z) div z = succ (x div z) :=
theorem add_div_self (x : ℕ) {z : ℕ} (H : z > 0) : (x + z) div z = succ (x div z) :=
calc
(x + z) div z = if 0 < z ∧ z ≤ x + z then (x + z - z) div z + 1 else 0 : !divide_def
... = (x + z - z) div z + 1 : if_pos (and.intro H (le_add_left z x))
... = succ (x div z) : {!add_sub_cancel}
theorem add_div_self_left {x : ℕ} (z : ℕ) (H : x > 0) : (x + z) div x = succ (z div x) :=
!add.comm ▸ !add_div_self_right H
!add.comm ▸ !add_div_self H
theorem add_mul_div_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) div z = x div z + y :=
theorem add_mul_div_self {x y z : ℕ} (H : z > 0) : (x + y * z) div z = x div z + y :=
nat.induction_on y
(calc (x + zero * z) div z = (x + zero) div z : zero_mul
... = x div z : add_zero
@ -59,16 +59,16 @@ nat.induction_on y
(take y,
assume IH : (x + y * z) div z = x div z + y, calc
(x + succ y * z) div z = (x + y * z + z) div z : by simp
... = succ ((x + y * z) div z) : !add_div_self_right H
... = succ ((x + y * z) div z) : !add_div_self H
... = x div z + succ y : by simp)
theorem add_mul_div_self_left (x z : ℕ) {y : ℕ} (H : y > 0) : (x + y * z) div y = x div y + z :=
!mul.comm ▸ add_mul_div_self_right H
!mul.comm ▸ add_mul_div_self H
theorem mul_div_cancel (m : ℕ) {n : ℕ} (H : n > 0) : m * n div n = m :=
calc
m * n div n = (0 + m * n) div n : zero_add
... = 0 div n + m : add_mul_div_self_right H
... = 0 div n + m : add_mul_div_self H
... = 0 + m : zero_div
... = m : zero_add
@ -97,16 +97,16 @@ modulo_def 0 b ⬝ if_neg (λ h, and.rec_on h (λ l r, absurd (lt_of_lt_of_le l
theorem mod_eq_sub_mod {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a mod b = (a - b) mod b :=
modulo_def a b ⬝ if_pos (and.intro h₁ h₂)
theorem add_mod_left {x z : ℕ} (H : z > 0) : (x + z) mod z = x mod z :=
theorem add_mod_self {x z : ℕ} (H : z > 0) : (x + z) mod z = x mod z :=
calc
(x + z) mod z = if 0 < z ∧ z ≤ x + z then (x + z - z) mod z else _ : modulo_def
... = (x + z - z) mod z : if_pos (and.intro H (le_add_left z x))
... = x mod z : add_sub_cancel
theorem add_mod_right {x z : ℕ} (H : x > 0) : (x + z) mod x = z mod x :=
!add.comm ▸ add_mod_left H
theorem add_mod_self_left {x z : ℕ} (H : x > 0) : (x + z) mod x = z mod x :=
!add.comm ▸ add_mod_self H
theorem add_mul_mod_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) mod z = x mod z :=
theorem add_mul_mod_self {x y z : ℕ} (H : z > 0) : (x + y * z) mod z = x mod z :=
nat.induction_on y
(calc (x + zero * z) mod z = (x + zero) mod z : zero_mul
... = x mod z : add_zero)
@ -115,17 +115,17 @@ nat.induction_on y
calc
(x + succ y * z) mod z = (x + (y * z + z)) mod z : succ_mul
... = (x + y * z + z) mod z : add.assoc
... = (x + y * z) mod z : add_mod_left H
... = (x + y * z) mod z : add_mod_self H
... = x mod z : IH)
theorem add_mul_mod_self_left {x y z : ℕ} (H : y > 0) : (x + y * z) mod y = x mod y :=
!mul.comm ▸ add_mul_mod_self_right H
!mul.comm ▸ add_mul_mod_self H
theorem mul_mod_left {m n : ℕ} : (m * n) mod n = 0 :=
by_cases_zero_pos n (by simp)
(take n,
assume npos : n > 0,
(by simp) ▸ (@add_mul_mod_self_right 0 m _ npos))
(by simp) ▸ (@add_mul_mod_self 0 m _ npos))
theorem mul_mod_right {m n : ℕ} : (m * n) mod m = 0 :=
!mul.comm ▸ !mul_mod_left
@ -192,10 +192,10 @@ theorem eq_remainder {y : ℕ} (H : y > 0) {q1 r1 q2 r2 : ℕ} (H1 : r1 < y) (H2
(H3 : q1 * y + r1 = q2 * y + r2) : r1 = r2 :=
calc
r1 = r1 mod y : by simp
... = (r1 + q1 * y) mod y : (add_mul_mod_self_right H)⁻¹
... = (r1 + q1 * y) mod y : (add_mul_mod_self H)⁻¹
... = (q1 * y + r1) mod y : add.comm
... = (r2 + q2 * y) mod y : by simp
... = r2 mod y : add_mul_mod_self_right H
... = r2 mod y : add_mul_mod_self H
... = r2 : by simp
theorem eq_quotient {y : ℕ} (H : y > 0) {q1 r1 q2 r2 : ℕ} (H1 : r1 < y) (H2 : r2 < y)
@ -298,6 +298,14 @@ or.elim (eq_zero_or_pos k)
... ≤ k * n : H
... = n * k : nat.mul.comm) H1)
theorem div_le (m n : ℕ) : m div n ≤ m :=
nat.cases_on n (!div_zero⁻¹ ▸ !zero_le)
take n,
have H : m ≤ succ n * m, from calc
m = 1 * m : one_mul
... ≤ succ n * m : mul_le_mul_right (succ_le_succ !zero_le),
div_le_of_le_mul H
theorem mul_sub_div_of_lt {m n k : ℕ} (H : k < m * n) :
(m * n - (k + 1)) div m = n - k div m - 1 :=
have H1 : k div m < n, from div_lt_of_lt_mul H,
@ -318,7 +326,7 @@ calc
by rewrite [H3 at {1}, mul.right_distrib, nat.one_mul]
... = ((n - k div m - 1) * m + (m - (k mod m + 1))) div m : {add_sub_assoc H5 _}
... = (m - (k mod m + 1)) div m + (n - k div m - 1) :
by rewrite [add.comm, (add_mul_div_self_right H4)]
by rewrite [add.comm, (add_mul_div_self H4)]
... = n - k div m - 1 :
by rewrite [div_eq_zero_of_lt H6, zero_add]