Further lemmas about the integers

The distinction between this file and Data.Int.Order.Basic is not particularly clear. They are separated by now to minimize the porting requirements for tactics during the transition to mathlib4. Now that Data.Rat.Order has been ported, please feel free to reorganize these two files.

nat abs

theorem Int.natAbs_eq_iff_mul_self_eq {a : ℤ} {b : ℤ} : Int.natAbs a = Int.natAbs b ↔ a * a = b * b

theorem Int.natAbs_lt_iff_mul_self_lt {a : ℤ} {b : ℤ} : Int.natAbs a < Int.natAbs b ↔ a * a < b * b

theorem Int.natAbs_le_iff_mul_self_le {a : ℤ} {b : ℤ} : Int.natAbs a ≤ Int.natAbs b ↔ a * a ≤ b * b

theorem Int.dvd_div_of_mul_dvd {a : ℤ} {b : ℤ} {c : ℤ} (h : a * b ∣ c) : b ∣ c / a

theorem Int.pow_right_injective {a : ℤ} (h : 1 < Int.natAbs a) : Function.Injective fun (x : ℕ) => a ^ x

units

theorem Int.eq_zero_of_abs_lt_dvd {m : ℤ} {x : ℤ} (h1 : m ∣ x) (h2 : |x| < m) : x = 0