Documentation

Mathlib.Algebra.GCDMonoid.Nat

ℕ and ℤ are normalized GCD monoids. #

Main statements #

Tags #

natural numbers, integers, normalization monoid, gcd monoid, greatest common divisor

@[instance_reducible]

instance instGCDMonoidNat :

ℕ is a GCDMonoid.

theorem gcd_eq_nat_gcd (m n : ℕ) :

gcd m n = m.gcd n

theorem lcm_eq_nat_lcm (m n : ℕ) :

lcm m n = m.lcm n

@[instance_reducible]

instance instStrongNormalizedGCDMonoidNat :

StrongNormalizedGCDMonoid ℕ

@[instance_reducible]

instance Int.strongNormalizationMonoid :

StrongNormalizationMonoid ℤ

@[deprecated Int.strongNormalizationMonoid (since := "2026-07-08")]

def Int.normalizationMonoid :

StrongNormalizationMonoid ℤ

Alias of Int.strongNormalizationMonoid.

Instances For

theorem Int.normUnit_eq (z : ℤ) :

normUnit z = if 0 ≤ z then 1 else -1

theorem Int.normalize_of_nonneg {z : ℤ} (h : 0 ≤ z) :

normalize z = z

theorem Int.normalize_of_nonpos {z : ℤ} (h : z ≤ 0) :

normalize z = -z

theorem Int.normalize_coe_nat (n : ℕ) :

normalize ↑n = ↑n

theorem Int.abs_eq_normalize (z : ℤ) :

|z| = normalize z

theorem Int.nonneg_of_normalize_eq_self {z : ℤ} (hz : normalize z = z) :

0 ≤ z

theorem Int.nonneg_iff_normalize_eq_self (z : ℤ) :

normalize z = z ↔ 0 ≤ z

theorem Int.eq_of_associated_of_nonneg {a b : ℤ} (h : Associated a b) (ha : 0 ≤ a) (hb : 0 ≤ b) :

a = b

@[instance_reducible]

instance Int.instGCDMonoid :

@[instance_reducible]

instance Int.instStrongNormalizedGCDMonoid :

StrongNormalizedGCDMonoid ℤ

theorem Int.coe_gcd (i j : ℤ) :

↑(i.gcd j) = GCDMonoid.gcd i j

theorem Int.coe_lcm (i j : ℤ) :

↑(i.lcm j) = GCDMonoid.lcm i j

theorem Int.natAbs_gcd (i j : ℤ) :

(GCDMonoid.gcd i j).natAbs = i.gcd j

theorem Int.natAbs_lcm (i j : ℤ) :

(GCDMonoid.lcm i j).natAbs = i.lcm j

theorem Int.gcd_nonneg (i j : ℤ) :

0 ≤ GCDMonoid.gcd i j

theorem Int.lcm_nonneg (i j : ℤ) :

0 ≤ GCDMonoid.lcm i j

theorem Int.exists_unit_of_abs (a : ℤ) :

∃ (u : ℤ ), ∃ (x : IsUnit u), ↑a.natAbs = u * a

theorem Int.gcd_eq_natAbs {a b : ℤ} :

a.gcd b = a.natAbs.gcd b.natAbs

def associatesIntEquivNat :

Associates ℤ ≃ ℕ

Maps an associate class of integers consisting of -n, n to n : ℕ

Instances For

theorem Int.associated_natAbs (k : ℤ) :

Associated k ↑k.natAbs

theorem Int.associated_iff_natAbs {a b : ℤ} :

Associated a b ↔ a.natAbs = b.natAbs

theorem Int.associated_iff {a b : ℤ} :

Associated a b ↔ a = b ∨ a = -b