-
Notifications
You must be signed in to change notification settings - Fork 298
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
feat(data/int/gcd): extended gcd to integers (#218)
Resurrected by @johoelzl. The original commit was not available anymore.
- Loading branch information
Showing
2 changed files
with
173 additions
and
1 deletion.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,155 @@ | ||
/- | ||
Copyright (c) 2018 Guy Leroy. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Sangwoo Jo (aka Jason), Guy Leroy | ||
Lemmas and extended definitions and properties of gcd and lcm for integers. | ||
-/ | ||
|
||
import data.int.basic data.nat.basic data.nat.gcd | ||
|
||
namespace int | ||
|
||
/- gcd -/ | ||
|
||
@[simp] theorem gcd_self (i : ℤ) : gcd i i = nat_abs i := | ||
by cases i; simp [gcd, mod_self] | ||
|
||
@[simp] theorem gcd_zero_left (i : ℤ) : gcd 0 i = nat_abs i := by simp [gcd] | ||
|
||
@[simp] theorem gcd_zero_right (i : ℤ) : gcd i 0 = nat_abs i := | ||
by cases i; simp [gcd] | ||
|
||
theorem gcd_dvd_left (i j : ℤ) : (gcd i j : ℤ) ∣ i := | ||
dvd_nat_abs.mp (coe_nat_dvd.mpr (nat.gcd_dvd_left (nat_abs i) (nat_abs j))) | ||
|
||
theorem gcd_dvd_right (i j : ℤ) : (gcd i j : ℤ) ∣ j := | ||
dvd_nat_abs.mp (coe_nat_dvd.mpr (nat.gcd_dvd_right (nat_abs i) (nat_abs j))) | ||
|
||
theorem gcd_dvd (i j : ℤ) : ((gcd i j : ℤ) ∣ i) ∧ ((gcd i j : ℤ) ∣ j) := | ||
⟨gcd_dvd_left i j, gcd_dvd_right i j⟩ | ||
|
||
theorem dvd_gcd {i j k : ℤ} : k ∣ i → k ∣ j → k ∣ gcd i j := | ||
by intros H1 H2; | ||
exact nat_abs_dvd.mp (coe_nat_dvd.mpr (nat.dvd_gcd (nat_abs_dvd_abs_iff.mpr H1) | ||
(nat_abs_dvd_abs_iff.mpr H2))) | ||
|
||
theorem gcd_comm (i j : ℤ) : gcd i j = gcd j i := | ||
nat.gcd_comm (nat_abs i) (nat_abs j) | ||
|
||
theorem gcd_assoc (i j k : ℤ) : gcd (gcd i j) k = gcd i (gcd j k) := | ||
nat.gcd_assoc (nat_abs i) (nat_abs j) (nat_abs k) | ||
|
||
@[simp] theorem gcd_one_left (i : ℤ) : gcd 1 i = 1 := nat.gcd_one_left _ | ||
|
||
@[simp] theorem gcd_one_right (i : ℤ) : gcd i 1 = 1 := | ||
eq.trans (gcd_comm i 1) $ gcd_one_left i | ||
|
||
theorem gcd_mul_left (i j k : ℤ) : gcd (i * j) (i * k) = nat_abs i * gcd j k := | ||
by rw [gcd, nat_abs_mul, nat_abs_mul]; | ||
exact nat.gcd_mul_left (nat_abs i) (nat_abs j) (nat_abs k) | ||
|
||
theorem gcd_mul_right (i j k : ℤ) : gcd (i * j) (k * j) = gcd i k * nat_abs j := | ||
by rw [gcd, nat_abs_mul, nat_abs_mul]; | ||
exact nat.gcd_mul_right (nat_abs i) (nat_abs j) (nat_abs k) | ||
|
||
theorem gcd_pos_of_non_zero_left {i : ℤ} (j : ℤ) (i_non_zero : i ≠ 0) : gcd i j > 0 := | ||
nat.gcd_pos_of_pos_left (nat_abs j) (nat_abs_pos_of_ne_zero i_non_zero) | ||
|
||
theorem gcd_pos_of_non_zero_right (i : ℤ) {j : ℤ} (j_non_zero : j ≠ 0) : gcd i j > 0 := | ||
nat.gcd_pos_of_pos_right (nat_abs i) (nat_abs_pos_of_ne_zero j_non_zero) | ||
|
||
theorem eq_zero_of_gcd_eq_zero_left {i j : ℤ} (H : gcd i j = 0) : i = 0 := | ||
eq_zero_of_nat_abs_eq_zero (nat.eq_zero_of_gcd_eq_zero_left H) | ||
|
||
theorem eq_zero_of_gcd_eq_zero_right {i j : ℤ} (H : gcd i j = 0) : j = 0 := | ||
eq_zero_of_nat_abs_eq_zero (nat.eq_zero_of_gcd_eq_zero_right H) | ||
|
||
theorem gcd_div {i j k : ℤ} (H1 : k ∣ i) (H2 : k ∣ j) : | ||
gcd (i / k) (j / k) = gcd i j / nat_abs k := | ||
by rw [gcd, nat_abs_div i k H1, nat_abs_div j k H2]; | ||
exact nat.gcd_div (nat_abs_dvd_abs_iff.mpr H1) (nat_abs_dvd_abs_iff.mpr H2) | ||
|
||
theorem gcd_dvd_gcd_of_dvd_left {i k : ℤ} (j : ℤ) (H : i ∣ k) : gcd i j ∣ gcd k j := | ||
int.coe_nat_dvd.1 $ dvd_gcd (dvd.trans (gcd_dvd_left i j) H) (gcd_dvd_right i j) | ||
|
||
theorem gcd_dvd_gcd_of_dvd_right {i k : ℤ} (j : ℤ) (H : i ∣ k) : gcd j i ∣ gcd j k := | ||
int.coe_nat_dvd.1 $ dvd_gcd (gcd_dvd_left j i) (dvd.trans (gcd_dvd_right j i) H) | ||
|
||
theorem gcd_dvd_gcd_mul_left (i j k : ℤ) : gcd i j ∣ gcd (k * i) j := | ||
gcd_dvd_gcd_of_dvd_left _ (dvd_mul_left _ _) | ||
|
||
theorem gcd_dvd_gcd_mul_right (i j k : ℤ) : gcd i j ∣ gcd (i * k) j := | ||
gcd_dvd_gcd_of_dvd_left _ (dvd_mul_right _ _) | ||
|
||
theorem gcd_dvd_gcd_mul_left_right (i j k : ℤ) : gcd i j ∣ gcd i (k * j) := | ||
gcd_dvd_gcd_of_dvd_right _ (dvd_mul_left _ _) | ||
|
||
theorem gcd_dvd_gcd_mul_right_right (i j k : ℤ) : gcd i j ∣ gcd i (j * k) := | ||
gcd_dvd_gcd_of_dvd_right _ (dvd_mul_right _ _) | ||
|
||
theorem gcd_eq_left {i j : ℤ} (H : i ∣ j) : gcd i j = nat_abs i := | ||
nat.dvd_antisymm (by unfold gcd; exact nat.gcd_dvd_left _ _) | ||
(by unfold gcd; exact nat.dvd_gcd (dvd_refl _) (nat_abs_dvd_abs_iff.mpr H)) | ||
|
||
theorem gcd_eq_right {i j : ℤ} (H : j ∣ i) : gcd i j = nat_abs j := | ||
by rw [gcd_comm, gcd_eq_left H] | ||
|
||
/- lcm -/ | ||
|
||
def lcm (i j : ℤ) : ℕ := nat_abs(i * j) / (gcd i j) | ||
|
||
theorem lcm_def (i j : ℤ) : lcm i j = nat.lcm (nat_abs i) (nat_abs j) := | ||
by rw [lcm, nat.lcm, gcd, nat_abs_mul] | ||
|
||
theorem lcm_comm (i j : ℤ) : lcm i j = lcm j i := | ||
by delta lcm; rw [mul_comm, gcd_comm] | ||
|
||
theorem lcm_zero_left (i : ℤ) : lcm 0 i = 0 := | ||
by rw [lcm, zero_mul, gcd_zero_left]; by simp | ||
|
||
theorem lcm_zero_right (i : ℤ) : lcm i 0 = 0 := lcm_comm 0 i ▸ lcm_zero_left i | ||
|
||
theorem lcm_one_left (i : ℤ) : lcm 1 i = nat_abs i := | ||
by rw [lcm, one_mul, gcd_one_left, nat.div_one] | ||
|
||
theorem lcm_one_right (i : ℤ) : lcm i 1 = nat_abs i := | ||
by unfold lcm; simp | ||
|
||
theorem lcm_self (i : ℤ) : lcm i i = nat_abs i := | ||
by rw [lcm, gcd_self, nat_abs_mul, nat.mul_div_assoc, mul_comm, nat.div_mul_cancel]; | ||
simp; simp | ||
|
||
theorem dvd_lcm_left (i j : ℤ) : i ∣ lcm i j := | ||
nat_abs_dvd.mp (coe_nat_dvd.mpr (eq.subst (eq.symm (lcm_def i j)) | ||
(nat.dvd_lcm_left (nat_abs i) (nat_abs j)))) | ||
|
||
theorem dvd_lcm_right (i j : ℤ) : j ∣ lcm i j := | ||
lcm_comm j i ▸ dvd_lcm_left j i | ||
|
||
theorem gcd_mul_lcm (i j : ℤ) : gcd i j * lcm i j = nat_abs (i * j) := | ||
begin | ||
rw [lcm, mul_comm, nat.div_mul_cancel], | ||
exact eq.subst (eq.symm (nat_abs_mul i j)) | ||
(dvd_mul_of_dvd_left (coe_nat_dvd.mp (dvd_nat_abs.mpr (gcd_dvd_left i j))) _), | ||
end | ||
|
||
theorem lcm_dvd {i j k : ℤ} (H1 : i ∣ k) (H2 : j ∣ k) : (lcm i j : ℤ) ∣ k := | ||
dvd_nat_abs.mp (coe_nat_dvd.mpr (eq.subst (eq.symm (lcm_def i j)) | ||
(nat.lcm_dvd (nat_abs_dvd_abs_iff.mpr H1) | ||
(nat_abs_dvd_abs_iff.mpr H2)))) | ||
|
||
theorem lcm_assoc (i j k : ℤ) : lcm (lcm i j) k = lcm i (lcm j k) := | ||
by rw [lcm_def, lcm_def, lcm_def, lcm_def]; | ||
exact nat.lcm_assoc (nat_abs i) (nat_abs j) (nat_abs k) | ||
|
||
/- lemmas -/ | ||
|
||
theorem dvd_of_mul_dvd_mul_left {i j k : ℤ} (k_non_zero : k ≠ 0) (H : k * i ∣ k * j) : i ∣ j := | ||
dvd.elim H (λl H1, by rw mul_assoc at H1; exact ⟨_, eq_of_mul_eq_mul_left k_non_zero H1⟩) | ||
|
||
theorem dvd_of_mul_dvd_mul_right {i j k : ℤ} (k_non_zero : k ≠ 0) (H : i * k ∣ j * k) : i ∣ j := | ||
by rw [mul_comm i k, mul_comm j k] at H; exact dvd_of_mul_dvd_mul_left k_non_zero H | ||
|
||
|
||
end int |