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

source
instance instGCDMonoidNat :
GCDMonoid ℕ

ℕ is a gcd_monoid.

Equations
  • One or more equations did not get rendered due to their size.
source
theorem gcd_eq_nat_gcd (m n : ℕ) :
gcd m n = m.gcd n
source
theorem lcm_eq_nat_lcm (m n : ℕ) :
lcm m n = m.lcm n
source
instance instNormalizedGCDMonoidNat :
NormalizedGCDMonoid ℕ
Equations
source
instance Int.normalizationMonoid :
NormalizationMonoid ℤ
Equations
  • One or more equations did not get rendered due to their size.
source
theorem Int.normUnit_eq (z : ℤ) :
normUnit z = if 0 ≤ z then 1 else -1
source
theorem Int.normalize_of_nonneg {z : ℤ} (h : 0 ≤ z) :
normalize z = z
source
theorem Int.normalize_of_nonpos {z : ℤ} (h : z ≤ 0) :
normalize z = -z
source
theorem Int.normalize_coe_nat (n : ℕ) :
normalize ↑n = ↑n
source
theorem Int.abs_eq_normalize (z : ℤ) :
|z| = normalize z
source
theorem Int.nonneg_of_normalize_eq_self {z : ℤ} (hz : normalize z = z) :
0 ≤ z
source
theorem Int.nonneg_iff_normalize_eq_self (z : ℤ) :
normalize z = z ↔ 0 ≤ z
source
theorem Int.eq_of_associated_of_nonneg {a b : ℤ} (h : Associated a b) (ha : 0 ≤ a) (hb : 0 ≤ b) :
a = b
source
instance Int.instGCDMonoid :
GCDMonoid ℤ
Equations
  • One or more equations did not get rendered due to their size.
source
instance Int.instNormalizedGCDMonoid :
NormalizedGCDMonoid ℤ
Equations
source
theorem Int.coe_gcd (i j : ℤ) :
↑(i.gcd j) = gcd i j
source
theorem Int.coe_lcm (i j : ℤ) :
↑(i.lcm j) = lcm i j
source
theorem Int.natAbs_gcd (i j : ℤ) :
(gcd i j).natAbs = i.gcd j
source
theorem Int.natAbs_lcm (i j : ℤ) :
(lcm i j).natAbs = i.lcm j
source
theorem Int.exists_unit_of_abs (a : ℤ) :
∃ (u : ℤ), ∃ (x : IsUnit u), ↑a.natAbs = u * a
source
theorem Int.gcd_eq_natAbs {a b : ℤ} :
a.gcd b = a.natAbs.gcd b.natAbs
source
def associatesIntEquivNat :
Associates ℤ ≃ ℕ

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

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    source
    theorem Int.associated_natAbs (k : ℤ) :
    Associated k ↑k.natAbs
    source
    theorem Int.associated_iff_natAbs {a b : ℤ} :
    Associated a b ↔ a.natAbs = b.natAbs
    source
    theorem Int.associated_iff {a b : ℤ} :
    Associated a b ↔ a = b ∨ a = -b